Transformations of Polynomial Functions

4.7 Transformations of Polynomial Functions A. VOCABULARY & CONCEPTS 1. Transformations Transformation f(x) Notation Examples Horizontal Translation g(x) = (x - 5)^2 5 units right Graph shifts left or right. g(x) = (x + 2)^2 2 units left Vertical Translation g(x) = x^2 + 1 1 unit up Graph shifts up or down. g(x) = x^2 - 4 4 units down Reflection g(x) = (-x)^2 = x^2 over y-axis Graph flips over x- or y-axis. g(x) = -x^2 over x-axis Horizontal Stretch or Shrink g(x) = (2x)^2 shrink by 1/2 Graph stretches away from or g(x) = (1/2x)^2 stretch by 2 shrinks toward y-axis. Vertical Stretch or Shrink g(x) = 8x^2 stretch by 8 Graph stretches away from or g(x) = (1/4)x^2 shrink by 1/4 shrinks toward x-axis. B. EXAMPLES 1. Describe the transformation of f(x) = x^3 represented by g(x) = (x + 5)^3 + 2. Then graph each function. g(x) f(x) X -2 -8 -1 -1 0 0 1 1 2 8 Horizontal translation left 5 units and vertical translation up 2 units 2. Describe the transformation of f(x) = x^4 represented by g(x) = (x - 3)^4 - 1. Then graph each function. 3 1 \[f(x)\] XY Horizontal translation right 3 units and vertical translation down I unit \[g(x)\] 00 216 3. Graph the function g(x) = -f(x - 3) on the same coordinate plane as f(x). reflection over x-axis Horizontal translation right 3 units   AYA 4 f(x) 2 -2 2 4 6 8 x -2 -4 \[g(x)\] 4. Write a rule for g then graph each function. Describe the graph of g as a transformation of the graph of f. a. f(x) = x^3 + 8; g(x) = f(-x) - 9 g(x) = f(-x) - 9 = (-x)^3 + 8 - 9 rule: g(x) = -x^3 - 1 b. f(x) = x^3 + x^2 + 1; g(x) = f(-x) g(x) = f(-x) = (-x)^3 + (-x)^2 + 1 = -x^3 + x^2 + 1 rule: g(x) = -x^3 + x^2 + 1 c. f(x) = 2x^5 - 2x^3 + x^2 + 1; g(x) = 5f(x) g(x) = 5f(x) = 5(2x^5 - 2x^3 + x^2 + 1) = 10x^5 - 5x^3 + 5 rule: g(x) = 10x^5 - 5x^3 + 5 f(x) y-intercept is 1 (0,1) g(x) y-intercept is 5 (0,5) 5. Write a rule for g that represents the indicated transformations of the graph of f. a. f(x) = x^3 - 6x^2 + 5; translation 1 unit left, followed by a reflection in the x-axis and a vertical stretch by a factor of 2 Step 1 f(x + 1) = (x + 1)^3 - 6(x + 1)^2 + 5 Step 2 -f(x + 1) = -(x + 1)^3 + 6(x + 1)^2 - 5 Step 3 2[-f(x + 1)] = -2(x + 1)^3 + 12(x + 1) - 10 g(x) = -2(x + 1)^3 + 12(x + 1) - 10 b. f(x) = 3x^4 + x^3 + 3x^2 + 12; horizontal shrink by a factor of 1/3 and a translation 8 units down, followed by a reflection in the y-axis Step 1 f(3x) = 3(3x)^4 + (3x)^3 + 3(3x)^2 + 12 Step 2 f(3x) - 8 = 3(3x)^4 + (3x)^3 + 3(3x)^2 + 12 - 8 Step 3 f(-3x) = 3(-3x)^4 + (-3x)^3 + 3(-3x)^2 + 4 g(x) = 3(-3x)^4 + (-3x)^3 + 3(-3x)^2 + 4