3.8 Transforming Polynomial Functions
Objectives:
F.IF.7c: Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x +k) for
specific values of k…
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or nonviable options in a
modeling contest.
A.CED.2:
For the board: You will be able to transform polynomial functions.
Bell Work 3.8:
Let g be the indicated transformation of f(x) = x2. Write the rule for g.
1. horizontal translation 1 unit right.
2. vertical stretch by a factor of 3.
3. vertical compression by a factor of ½.
4. vertical translation 4 units down.
Anticipatory set:
Transformation
Vertical translation
Horizontal translation
Vertical stretch/compression
Horizontal stretch/compression
Reflection
Transformations of f(x)
f(x) Notation
f(x) + k
g(x) = x3 + 3
g(x) = x3 – 4
f(x + h)
g(x) = (x – 2)3
g(x) = (x + 1)3
a ∙ f(x)
g(x) = 6x3
g(x) = ½ x3
f(ax)
g(x) = (1/5 x)3
g(x) = (3x)3
-f(x) or f(-x)
g(x) = -x3
g(x) = (-x)3
Examples
3 units up
4 units down
2 units right
1 unit left
stretch by a factor of 6
compression by a factor of ½
stretch by a factor of 5
compression by a factor of 1/3
across the x-axis
across the y-axis
Instruction:
Open the book to page 204 and read example 1.
Example: For f(x) = x3 – 6, write the rule for each function and describe the transformation.
a. g(x) = f(x) – 2: translation down 2: g(x) = x3 – 6 – 2
g(x) = x3 – 8
3
b. h(x) = f(x + 3): translation left 3: h(x) = (x + 3) – 6
White Board Activity:
Practice: For f(x) = x3 + 4, write the rule for each function and Describe the transformation.
a. g(x) = f(x) – 5: translation down 5: g(x) = x3 + 4 – 5
g(x) = x3 – 5
b. g(x) = f(x + 2): translation left 2: g(x) = (x + 2)3 + 4
Read example 2 on page 205.
Example: Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation.
a. reflect f across the x axis.
g(x) = -f(x) = -(x3 + 5x2 – 8x + 1) = -x3 – 5x2 + 8x – 1
b. reflect f across the y axis.
g(x) = f(-x) = (-x)3 + 5(-x)2 – (-x) + 1 = -x3 + 5x2 + x + 1
White Board Activity:
Practice: Let f(x) = x3 – 2x2 – x + 2. Write a function g(x) that performs each transformation.
a. Reflect f(x) across the x-axis.
g(x) = -f(x) = -(x3 – 2x2 – x + 2) = -x3 + 2x2 + x – 2
b. Reflect f(x) across the y-axis.
g(x) = f(-x) = (-x)3 – 2(-x)2 – (-x) + 2 = -x3 – 2x2 + x + 2
Read example 3 on page 205.
Example: Let f(x) = 2x4 – 6x2 + 1. Describe g as a transformation of f.
a. g(x) = ½ f(x): vertical compression by a factor of ½ .
b. h(x) = f(1/3 x): horizontal stretch by a factor of 3.
White Board Activity:
Practice: Let f(x) = 16x4 – 24x2 + 4. Describe g as a transformation of f.
a. g(x) = ¼ f(x): vertical compression by a factor of ¼ .
b. g(x) = f(½ x): horizontal stretch by a factor of 2.
Read example 4 on page 206.
Example: Write a function that transforms f(x) = 6x3 – 3 in each of the following ways.
Support your solution by using a graphing calculator.
a. Compress vertically by 1/3 and shift 2 units right.
g(x) = 1/3 f(x – 2) = 1/3[6(x – 2)3 – 3] = 2(x – 2)3 – 1
= 2[x3 + 3x2(-2) +3x(-2)2 + (-2)3] – 3 = 2(x3 – 6x2 + 12x – 8) – 1
= 2x3 – 12x2 + 24x – 16 – 1 = 2x3 – 12x2 + 24x – 17.
b. Reflect across the y-axis and shift 2 units down.
h(x) = f(-x) – 2 = 6(-x)3 – 3 – 2 = -6x3 – 5
White Board Activity:
Practice: Write a function that transforms f(x) 8x3 – 2 in each of the following ways.
Support your solution by using a graphing calculator.
a. Compress vertically by a factor of ½, and move the x-intercept 3 units right.
g(x) = ½f(x – 3) = ½[8(x – 3)3 – 2] = 4(x – 3)3 – 1
= 4[x3 + 3x2(-3) + 3x(-3)2 + (-3)3] – 1 = 4(x3 – 9x2 + 27x – 27) – 1
= 4x3 – 36x2 + 108x – 108 – 1 = 4x3 – 36x2 + 108 – 109
b. Reflect over the x-axis, and move the x-intercept 4 units left.
h(x) = f[-(x + 4)] = 8[-(x + 4)]3 – 2 = -8(x + 4)3 – 2
= -8[x3 + 3x2(4) + 3x(4)2 + 43] – 2 = -8[x3 + 12x2 + 48x + 64] – 2
= -8x3 – 96x3 + 384x + 512 – 2 = -8x3 – 96x3 + 384x + 510
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 207 – 208 prob. 1 – 6, 10 – 12, 14 – 18, 22 – 24. Do not graph.
For a Grade:
Text: pgs. 207 – 208 prob. 14, 18, 22. Do not graph.