Math Problem Statement

Suppose that the x-intercepts of the graph of y = f(x) are -2 and 4. What are the x-intercepts for various transformations of the function?

Solution

The question provided in the image asks about determining the x-intercepts of different transformations of a function f(x)f(x), where the original function y=f(x)y = f(x) has x-intercepts at x=2x = -2 and x=4x = 4. Let's go through each part:

(a) What are the x-intercepts of the graph of y=f(x+6)y = f(x + 6)?

  • A horizontal shift to the left by 6 units occurs when f(x)f(x) is replaced by f(x+6)f(x + 6).
  • The original x-intercepts are 2-2 and 44. Shifting them left by 6 gives: 26=8,46=2-2 - 6 = -8, \quad 4 - 6 = -2 So, the x-intercepts are 8-8 and 2-2.

(b) What are the x-intercepts of the graph of y=f(x7)y = f(x - 7)?

  • A horizontal shift to the right by 7 units occurs when f(x)f(x) is replaced by f(x7)f(x - 7).
  • Shifting the original x-intercepts 2-2 and 44 to the right by 7 gives: 2+7=5,4+7=11-2 + 7 = 5, \quad 4 + 7 = 11 So, the x-intercepts are 55 and 1111.

(c) What are the x-intercepts of the graph of y=3f(x)y = 3f(x)?

  • Multiplying the function by 3 (vertical stretch) does not affect the x-intercepts, because x-intercepts occur where y=0y = 0, and multiplying by 3 still keeps y=0y = 0.
  • Therefore, the x-intercepts remain 2-2 and 44.

(d) What are the x-intercepts of the graph of y=f(x)y = f(-x)?

  • Reflecting across the y-axis occurs when f(x)f(x) is replaced by f(x)f(-x).
  • The x-intercepts change signs because the function is reflected across the y-axis. The original intercepts 2-2 and 44 become: (2)=2,(4)=4-(-2) = 2, \quad -(4) = -4 So, the x-intercepts are 22 and 4-4.

Let me know if you need further clarification or have more questions. Here are five related questions for further understanding:

  1. What happens to the x-intercepts when the graph is shifted vertically?
  2. How does a reflection across the x-axis affect the x-intercepts?
  3. What effect does multiplying a function by a constant have on its x-intercepts?
  4. How do you determine the y-intercepts after a horizontal shift?
  5. Can an absolute value transformation affect the x-intercepts?

Tip: Horizontal shifts are applied by adjusting the xx-values, while vertical transformations do not change the x-intercepts unless the function crosses the x-axis differently.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Function Transformations
X-Intercepts
Algebra

Formulas

Horizontal shift: f(x + c) or f(x - c)
Vertical stretch: a * f(x)
Reflection: f(-x)

Theorems

Reflection theorem
Transformation theorem

Suitable Grade Level

Grade 10-12