Graph Quadratic Functions Using Transformations – Intermediate Algebra (2024)

Quadratic Equations and Functions

Learning Objectives

By the end of this section, you will be able to:

  • Graph quadratic functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (1)
  • Graph quadratic functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (2)
  • Graph quadratic functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (3)
  • Graph quadratic functions using transformations
  • Find a quadratic function from its graph

Before you get started, take this readiness quiz.

  1. Graph the function Graph Quadratic Functions Using Transformations – Intermediate Algebra (4) by plotting points.

    If you missed this problem, review (Figure).

  2. Factor completely: Graph Quadratic Functions Using Transformations – Intermediate Algebra (5)

    If you missed this problem, review (Figure).

  3. Factor completely: Graph Quadratic Functions Using Transformations – Intermediate Algebra (6)

    If you missed this problem, review (Figure).

Graph Quadratic Functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (7)

In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (8) and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.

In the first example, we will graph the quadratic function Graph Quadratic Functions Using Transformations – Intermediate Algebra (9) by plotting points. Then we will see what effect adding a constant, k, to the equation will have on the graph of the new function Graph Quadratic Functions Using Transformations – Intermediate Algebra (10)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (11) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (12) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Plotting points will help us see the effect of the constants on the basic Graph Quadratic Functions Using Transformations – Intermediate Algebra (13) graph. We fill in the chart for all three functions.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (14)

The g(x) values are two more than the f(x) values. Also, the h(x) values are two less than the f(x) values. Now we will graph all three functions on the same rectangular coordinate system.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (15)

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (16) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (17) but shifted up 2 units.

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (18) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (19) but shifted down 2 units.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (20) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (21) on the same rectangular coordinate system.

Describe what effect adding a constant to the function has on the basic parabola.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (22)

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (23) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (24) but shifted up 1 unit. The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (25) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (26) but shifted down 1 unit.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (27) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (28) on the same rectangular coordinate system.

Describe what effect adding a constant to the function has on the basic parabola.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (29)

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (30) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (31) but shifted up 6 units. The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (32) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (33) but shifted down 6 units.

The last example shows us that to graph a quadratic function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (34) we take the basic parabola graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (35) and vertically shift it up Graph Quadratic Functions Using Transformations – Intermediate Algebra (36) or shift it down Graph Quadratic Functions Using Transformations – Intermediate Algebra (37).

This transformation is called a vertical shift.

Graph a Quadratic Function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (38) Using a Vertical Shift

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (39) shifts the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (40) vertically k units.

  • If k > 0, shift the parabola vertically up k units.
  • If k < 0, shift the parabola vertically down Graph Quadratic Functions Using Transformations – Intermediate Algebra (41) units.

Now that we have seen the effect of the constant, k, it is easy to graph functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (42) We just start with the basic parabola of Graph Quadratic Functions Using Transformations – Intermediate Algebra (43) and then shift it up or down.

It may be helpful to practice sketching Graph Quadratic Functions Using Transformations – Intermediate Algebra (44) quickly. We know the values and can sketch the graph from there.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (45)

Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (46) using a vertical shift.

We first draw the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (47) on

the grid.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (48)
Determine Graph Quadratic Functions Using Transformations – Intermediate Algebra (49).Graph Quadratic Functions Using Transformations – Intermediate Algebra (50)
Graph Quadratic Functions Using Transformations – Intermediate Algebra (51)
Shift the graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (52) down 3.Graph Quadratic Functions Using Transformations – Intermediate Algebra (53)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (54) using a vertical shift.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (55)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (56) using a vertical shift.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (57)

Graph Quadratic Functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (58)

In the first example, we graphed the quadratic function Graph Quadratic Functions Using Transformations – Intermediate Algebra (59) by plotting points and then saw the effect of adding a constant k to the function had on the resulting graph of the new function Graph Quadratic Functions Using Transformations – Intermediate Algebra (60)

We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function Graph Quadratic Functions Using Transformations – Intermediate Algebra (61)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (62) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (63) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Plotting points will help us see the effect of the constants on the basic Graph Quadratic Functions Using Transformations – Intermediate Algebra (64) graph. We fill in the chart for all three functions.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (65)

The g(x) values and the h(x) values share the common numbers 0, 1, 4, 9, and 16, but are shifted.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (66)Graph Quadratic Functions Using Transformations – Intermediate Algebra (67)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (68) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (69) on the same rectangular coordinate system.

Describe what effect adding a constant to the function has on the basic parabola.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (70)

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (71) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (72) but shifted left 2 units. The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (73) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (74) but shift right 2 units.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (75) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (76) on the same rectangular coordinate system.

Describe what effect adding a constant to the function has on the basic parabola.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (77)

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (78) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (79) but shifted left 5 units. The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (80) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (81) but shifted right 5 units.

The last example shows us that to graph a quadratic function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (82) we take the basic parabola graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (83) and shift it left (h > 0) or shift it right (h < 0).

This transformation is called a horizontal shift.

Graph a Quadratic Function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (84) Using a Horizontal Shift

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (85) shifts the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (86) horizontally Graph Quadratic Functions Using Transformations – Intermediate Algebra (87) units.

  • If h > 0, shift the parabola horizontally left h units.
  • If h < 0, shift the parabola horizontally right Graph Quadratic Functions Using Transformations – Intermediate Algebra (88) units.

Now that we have seen the effect of the constant, h, it is easy to graph functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (89) We just start with the basic parabola of Graph Quadratic Functions Using Transformations – Intermediate Algebra (90) and then shift it left or right.

The next example will require a horizontal shift.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (91) using a horizontal shift.

We first draw the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (92) on

the grid.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (93)
Determine h.Graph Quadratic Functions Using Transformations – Intermediate Algebra (94)
Graph Quadratic Functions Using Transformations – Intermediate Algebra (95)
Shift the graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (96) to the right 6 units.Graph Quadratic Functions Using Transformations – Intermediate Algebra (97)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (98) using a horizontal shift.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (99)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (100) using a horizontal shift.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (101)

Now that we know the effect of the constants h and k, we will graph a quadratic function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (102) by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (103) using transformations.

This function will involve two transformations and we need a plan.

Let’s first identify the constants h, k.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (104)

The h constant gives us a horizontal shift and the k gives us a vertical shift.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (105)

We first draw the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (106) on the grid.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (107)Graph Quadratic Functions Using Transformations – Intermediate Algebra (108)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (109) using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (110)

Graph Quadratic Functions of the Form Graph Quadratic Functions Using Transformations – Intermediate Algebra (113)

So far we graphed the quadratic function Graph Quadratic Functions Using Transformations – Intermediate Algebra (114) and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient a on the resulting graph of the new function Graph Quadratic Functions Using Transformations – Intermediate Algebra (115)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (116)

If we graph these functions, we can see the effect of the constant a, assuming a > 0.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (117)

To graph a function with constant a it is easiest to choose a few points on Graph Quadratic Functions Using Transformations – Intermediate Algebra (118) and multiply the y-values by a.

Graph of a Quadratic Function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (119)

The coefficient a in the function Graph Quadratic Functions Using Transformations – Intermediate Algebra (120) affects the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (121) by stretching or compressing it.

  • If Graph Quadratic Functions Using Transformations – Intermediate Algebra (122) the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (123) will be “wider” than the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (124)
  • If Graph Quadratic Functions Using Transformations – Intermediate Algebra (125), the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (126) will be “skinnier” than the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (127)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (128)

We will graph the functions Graph Quadratic Functions Using Transformations – Intermediate Algebra (129) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (130) on the same grid. We will choose a few points on Graph Quadratic Functions Using Transformations – Intermediate Algebra (131) and then multiply the y-values by 3 to get the points for Graph Quadratic Functions Using Transformations – Intermediate Algebra (132)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (133)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (134)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (135)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (136)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (137)

Graph Quadratic Functions Using Transformations

We have learned how the constants a, h, and k in the functions, Graph Quadratic Functions Using Transformations – Intermediate Algebra (138) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (139) affect their graphs. We can now put this together and graph quadratic functions Graph Quadratic Functions Using Transformations – Intermediate Algebra (140) by first putting them into the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (141) by completing the square. This form is sometimes known as the vertex form or standard form.

We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (142)

When we complete the square in a function with a coefficient of x2 that is not one, we have to factor that coefficient from just the x-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the x-terms.

Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.

Rewrite Graph Quadratic Functions Using Transformations – Intermediate Algebra (143) in the Graph Quadratic Functions Using Transformations – Intermediate Algebra (144) form by completing the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (145)
Separate the x terms from the constant.Graph Quadratic Functions Using Transformations – Intermediate Algebra (146)
Factor the coefficient of Graph Quadratic Functions Using Transformations – Intermediate Algebra (147), Graph Quadratic Functions Using Transformations – Intermediate Algebra (148).Graph Quadratic Functions Using Transformations – Intermediate Algebra (149)
Prepare to complete the square.Graph Quadratic Functions Using Transformations – Intermediate Algebra (150)
Take half of 2 and then square it to complete the

square. Graph Quadratic Functions Using Transformations – Intermediate Algebra (151)

The constant 1 completes the square in the

parentheses, but the parentheses is multiplied by

Graph Quadratic Functions Using Transformations – Intermediate Algebra (152). So we are really adding Graph Quadratic Functions Using Transformations – Intermediate Algebra (153) We must then

add 3 to not change the value of the function.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (154)
Rewrite the trinomial as a square and subtract the

constants.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (155)
The function is now in the Graph Quadratic Functions Using Transformations – Intermediate Algebra (156)

form.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (157)

Rewrite Graph Quadratic Functions Using Transformations – Intermediate Algebra (158) in the Graph Quadratic Functions Using Transformations – Intermediate Algebra (159) form by completing the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (160)

Rewrite Graph Quadratic Functions Using Transformations – Intermediate Algebra (161) in the Graph Quadratic Functions Using Transformations – Intermediate Algebra (162) form by completing the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (163)

Once we put the function into the Graph Quadratic Functions Using Transformations – Intermediate Algebra (164) form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (165) by using transformations.

Step 1. Rewrite the function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (166) vertex form by completing the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (167)
Separate the x terms from the constant.Graph Quadratic Functions Using Transformations – Intermediate Algebra (168)
Take half of 6 and then square it to complete the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (169)

We both add 9 and subtract 9 to not change the value of the function.Graph Quadratic Functions Using Transformations – Intermediate Algebra (170)
Rewrite the trinomial as a square and subtract the constants.Graph Quadratic Functions Using Transformations – Intermediate Algebra (171)
The function is now in the Graph Quadratic Functions Using Transformations – Intermediate Algebra (172) form.Graph Quadratic Functions Using Transformations – Intermediate Algebra (173)

Step 2: Graph the function using transformations.

Looking at the h, k values, we see the graph will take the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (174) and shift it to the left 3 units and down 4 units.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (175)

We first draw the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (176) on the grid.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (177)Graph Quadratic Functions Using Transformations – Intermediate Algebra (178)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (179) by using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (180)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (181) by using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (182)

We list the steps to take to graph a quadratic function using transformations here.

Graph a quadratic function using transformations.

  1. Rewrite the function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (183) form by completing the square.
  2. Graph the function using transformations.

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (184) by using transformations.

Step 1. Rewrite the function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (185) vertex form by completing the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (186)
Separate the x terms from the constant.Graph Quadratic Functions Using Transformations – Intermediate Algebra (187)
We need the coefficient of Graph Quadratic Functions Using Transformations – Intermediate Algebra (188) to be one.

We factor Graph Quadratic Functions Using Transformations – Intermediate Algebra (189) from the x-terms.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (190)
Take half of 2 and then square it to complete the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (191)

We add 1 to complete the square in the parentheses, but the parentheses is multiplied by Graph Quadratic Functions Using Transformations – Intermediate Algebra (192). Se we are really adding Graph Quadratic Functions Using Transformations – Intermediate Algebra (193). To not change the value of the function we add 2.Graph Quadratic Functions Using Transformations – Intermediate Algebra (194)
Rewrite the trinomial as a square and subtract the constants.Graph Quadratic Functions Using Transformations – Intermediate Algebra (195)
The function is now in the Graph Quadratic Functions Using Transformations – Intermediate Algebra (196) form.Graph Quadratic Functions Using Transformations – Intermediate Algebra (197)

Step 2. Graph the function using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (198)

We first draw the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (199) on the grid.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (200)Graph Quadratic Functions Using Transformations – Intermediate Algebra (201)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (202) by using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (203)

Graph Graph Quadratic Functions Using Transformations – Intermediate Algebra (204) by using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (205)

Now that we have completed the square to put a quadratic function into Graph Quadratic Functions Using Transformations – Intermediate Algebra (206) form, we can also use this technique to graph the function using its properties as in the previous section.

If we look back at the last few examples, we see that the vertex is related to the constants h and k.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (207)

In each case, the vertex is (h, k). Also the axis of symmetry is the line x = h.

We rewrite our steps for graphing a quadratic function using properties for when the function is in Graph Quadratic Functions Using Transformations – Intermediate Algebra (208) form.

Graph a quadratic function in the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (209) using properties.

  1. Rewrite the function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (210) form.
  2. Determine whether the parabola opens upward, a > 0, or downward, a < 0.
  3. Find the axis of symmetry, x = h.
  4. Find the vertex, (h, k).
  5. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
  6. Find the x-intercepts.
  7. Graph the parabola.

Rewrite Graph Quadratic Functions Using Transformations – Intermediate Algebra (211) in Graph Quadratic Functions Using Transformations – Intermediate Algebra (212) form and graph the function using properties.

Rewrite the function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (213)

form by completing the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (214)
Graph Quadratic Functions Using Transformations – Intermediate Algebra (215)
Graph Quadratic Functions Using Transformations – Intermediate Algebra (216)
Graph Quadratic Functions Using Transformations – Intermediate Algebra (217)
Identify the constants Graph Quadratic Functions Using Transformations – Intermediate Algebra (218)Graph Quadratic Functions Using Transformations – Intermediate Algebra (219)
Since Graph Quadratic Functions Using Transformations – Intermediate Algebra (220), the parabola opens upward.Graph Quadratic Functions Using Transformations – Intermediate Algebra (221)
The axis of symmetry is Graph Quadratic Functions Using Transformations – Intermediate Algebra (222).The axis of symmetry is Graph Quadratic Functions Using Transformations – Intermediate Algebra (223).
The vertex is Graph Quadratic Functions Using Transformations – Intermediate Algebra (224).The vertex is Graph Quadratic Functions Using Transformations – Intermediate Algebra (225).
Find the y-intercept by finding Graph Quadratic Functions Using Transformations – Intermediate Algebra (226).Graph Quadratic Functions Using Transformations – Intermediate Algebra (227)
Graph Quadratic Functions Using Transformations – Intermediate Algebra (228)
y-intercept Graph Quadratic Functions Using Transformations – Intermediate Algebra (229)
Find the point symmetric to Graph Quadratic Functions Using Transformations – Intermediate Algebra (230) across the

axis of symmetry.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (231)
Find the x-intercepts.The discriminant negative, so there are

no x-intercepts. Graph the parabola.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (232)

Rewrite Graph Quadratic Functions Using Transformations – Intermediate Algebra (233) in Graph Quadratic Functions Using Transformations – Intermediate Algebra (234) form and graph the function using properties.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (235)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (236)

Rewrite Graph Quadratic Functions Using Transformations – Intermediate Algebra (237) in Graph Quadratic Functions Using Transformations – Intermediate Algebra (238) form and graph the function using properties.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (239)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (240)

Find a Quadratic Function from its Graph

So far we have started with a function and then found its graph.

Now we are going to reverse the process. Starting with the graph, we will find the function.

Determine the quadratic function whose graph is shown.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (241)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (242)

Write the quadratic function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (243) form whose graph is shown.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (244)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (245)

Determine the quadratic function whose graph is shown.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (246)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (247)

Key Concepts

  • Graph a Quadratic Function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (248) Using a Vertical Shift
    • The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (249) shifts the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (250) vertically k units.
      • If k > 0, shift the parabola vertically up k units.
      • If k < 0, shift the parabola vertically down Graph Quadratic Functions Using Transformations – Intermediate Algebra (251) units.
  • Graph a Quadratic Function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (252) Using a Horizontal Shift
    • The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (253) shifts the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (254) horizontally h units.
      • If h > 0, shift the parabola horizontally left h units.
      • If h < 0, shift the parabola horizontally right Graph Quadratic Functions Using Transformations – Intermediate Algebra (255) units.
  • Graph of a Quadratic Function of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (256)
    • The coefficient a in the function Graph Quadratic Functions Using Transformations – Intermediate Algebra (257) affects the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (258) by stretching or compressing it.

      If Graph Quadratic Functions Using Transformations – Intermediate Algebra (259) then the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (260) will be “wider” than the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (261)

      If Graph Quadratic Functions Using Transformations – Intermediate Algebra (262) then the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (263) will be “skinnier” than the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (264)

  • How to graph a quadratic function using transformations
    1. Rewrite the function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (265) form by completing the square.
    2. Graph the function using transformations.
  • Graph a quadratic function in the vertex form Graph Quadratic Functions Using Transformations – Intermediate Algebra (266) using properties
    1. Rewrite the function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (267) form.
    2. Determine whether the parabola opens upward, a > 0, or downward, a < 0.
    3. Find the axis of symmetry, x = h.
    4. Find the vertex, (h, k).
    5. Find they-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
    6. Find the x-intercepts, if possible.
    7. Graph the parabola.

Practice Makes Perfect

Graph Quadratic Functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (268)

In the following exercises, graph the quadratic functions on the same rectangular coordinate system and describe what effect adding a constant, k, to the function has on the basic parabola.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (269) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (270)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (271)

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (272) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (273) but shifted up 4 units. The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (274) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (275) but shift down 4 units.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (276) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (277)

In the following exercises, graph each function using a vertical shift.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (278)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (279)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (280)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (281)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (282)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (283)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (284)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (285)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (286)

Graph Quadratic Functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (287)

In the following exercises, graph the quadratic functions on the same rectangular coordinate system and describe what effect adding a constant, Graph Quadratic Functions Using Transformations – Intermediate Algebra (288), inside the parentheses has

Graph Quadratic Functions Using Transformations – Intermediate Algebra (289) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (290)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (291)

The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (292) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (293) but shifted right 3 units. The graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (294) is the same as the graph of Graph Quadratic Functions Using Transformations – Intermediate Algebra (295) but shifted left 3 units.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (296) and Graph Quadratic Functions Using Transformations – Intermediate Algebra (297)

In the following exercises, graph each function using a horizontal shift.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (298)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (299)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (300)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (301)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (302)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (303)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (304)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (305)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (306)

In the following exercises, graph each function using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (307)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (308)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (309)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (310)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (311)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (312)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (313)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (314)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (315)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (316)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (317)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (318)

Graph Quadratic Functions of the form Graph Quadratic Functions Using Transformations – Intermediate Algebra (319)

In the following exercises, graph each function.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (320)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (321)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (322)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (323)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (324)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (325)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (326)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (327)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (328)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (329)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (330)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (331)

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the Graph Quadratic Functions Using Transformations – Intermediate Algebra (332) form by completing the square.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (333)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (334)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (335)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (336)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (337)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (338)

In the following exercises, rewrite each function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (339) form and graph it by using transformations.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (340)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (341)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (342)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (343)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (344)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (345)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (346)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (347)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (348)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (349)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (350)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (351)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (352)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (353)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (354)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (355)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (356)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (357)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (358)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (359)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (360)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (361)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (362)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (363)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (364)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (365)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (366)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (367)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (368)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (369)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (370)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (371)

In the following exercises, rewrite each function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (372) form and graph it using properties.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (373)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (374)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (375)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (376)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (377)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (378)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (379)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (380)

Matching

In the following exercises, match the graphs to one of the following functions: Graph Quadratic Functions Using Transformations – Intermediate Algebra (381) Graph Quadratic Functions Using Transformations – Intermediate Algebra (382) Graph Quadratic Functions Using Transformations – Intermediate Algebra (383) Graph Quadratic Functions Using Transformations – Intermediate Algebra (384) Graph Quadratic Functions Using Transformations – Intermediate Algebra (385) Graph Quadratic Functions Using Transformations – Intermediate Algebra (386) Graph Quadratic Functions Using Transformations – Intermediate Algebra (387) Graph Quadratic Functions Using Transformations – Intermediate Algebra (388)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (389)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (390)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (391)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (392)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (393)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (394)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (395)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (396)

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in Graph Quadratic Functions Using Transformations – Intermediate Algebra (397) form whose graph is shown.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (398)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (399)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (400)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (401)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (402)

Graph Quadratic Functions Using Transformations – Intermediate Algebra (403)

Writing Exercise

Graph the quadratic function Graph Quadratic Functions Using Transformations – Intermediate Algebra (404) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?

Answers will vary.

Graph the quadratic function Graph Quadratic Functions Using Transformations – Intermediate Algebra (405) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Graph Quadratic Functions Using Transformations – Intermediate Algebra (406)

After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

Graph Quadratic Functions Using Transformations – Intermediate Algebra (2024)
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