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 , where the original function has x-intercepts at and . Let's go through each part:
(a) What are the x-intercepts of the graph of ?
- A horizontal shift to the left by 6 units occurs when is replaced by .
- The original x-intercepts are and . Shifting them left by 6 gives: So, the x-intercepts are and .
(b) What are the x-intercepts of the graph of ?
- A horizontal shift to the right by 7 units occurs when is replaced by .
- Shifting the original x-intercepts and to the right by 7 gives: So, the x-intercepts are and .
(c) What are the x-intercepts of the graph of ?
- Multiplying the function by 3 (vertical stretch) does not affect the x-intercepts, because x-intercepts occur where , and multiplying by 3 still keeps .
- Therefore, the x-intercepts remain and .
(d) What are the x-intercepts of the graph of ?
- Reflecting across the y-axis occurs when is replaced by .
- The x-intercepts change signs because the function is reflected across the y-axis. The original intercepts and become: So, the x-intercepts are and .
Let me know if you need further clarification or have more questions. Here are five related questions for further understanding:
- What happens to the x-intercepts when the graph is shifted vertically?
- How does a reflection across the x-axis affect the x-intercepts?
- What effect does multiplying a function by a constant have on its x-intercepts?
- How do you determine the y-intercepts after a horizontal shift?
- Can an absolute value transformation affect the x-intercepts?
Tip: Horizontal shifts are applied by adjusting the -values, while vertical transformations do not change the x-intercepts unless the function crosses the x-axis differently.
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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