If f(x)f(x) is the logarithmic parent function, which statement best describes how its intercepts change when transformed to g(x)=a⋅f(x)g(x)=a⋅f(x) for some *a*∈R?

A.The *x*-intercepts stay the same but the *y*-intercepts are multiplied by *a*.

B.The *y*-intercepts stay the same but the *x*-intercepts are multiplied by *a*.

C.The *y*-intercepts are increased by *a*; the *x*-intercepts change depending on the shape of the function.

D.The *x*-intercepts are increased by *a*; the *y*-intercepts change depending on the shape of the function.


Given the logarithmic parent function \( f(x) = \log(x) \), we need to analyze how its intercepts change when it is transformed to \( g(x) = a \cdot f(x) \), where \( a \) is a real number.

### Analyzing the Intercepts:

1. **\( x \)-Intercept**: 
   - The \( x \)-intercept of the parent function \( f(x) = \log(x) \) occurs when \( f(x) = 0 \).
   - \( \log(x) = 0 \) implies \( x = 1 \).
   - For the transformed function \( g(x) = a \cdot \log(x) \), the \( x \)-intercept occurs when \( g(x) = 0 \).
   - Setting \( g(x) = 0 \) gives \( a \cdot \log(x) = 0 \).
   - Since \( a \) is non-zero, \( \log(x) = 0 \) still implies \( x = 1 \).
   - **Conclusion**: The \( x \)-intercept does not change; it remains \( x = 1 \).

2. **\( y \)-Intercept**:
   - The \( y \)-intercept occurs when \( x = 0 \).
   - However, \( f(x) = \log(x) \) is undefined at \( x = 0 \). Thus, neither \( f(x) \) nor \( g(x) \) have a \( y \)-intercept in the traditional sense.
   - But if we consider the behavior at \( x = 1 \), for which \( f(1) = \log(1) = 0 \), the transformation \( g(x) = a \cdot \log(x) \) will have a \( y \)-value of \( g(1) = a \cdot 0 = 0 \). Hence, there's no \( y \)-intercept change in this transformation either.

Since the \( x \)-intercept remains unchanged and the \( y \)-intercept technically does not exist (as logarithmic functions are undefined at \( x = 0 \)), the best interpretation is that the transformation affects the function by multiplying all outputs by \( a \), but this does not change the intercepts.

### Correct Answer:
**A. The \( x \)-intercepts stay the same but the \( y \)-intercepts are multiplied by \( a \).** However, keep in mind that logarithmic functions typically do not have \( y \)-intercepts.

Would you like more details or have any questions? Here are some related questions:

1. What are the intercepts of the parent logarithmic function \( f(x) = \log(x) \)?
2. How do vertical stretches and compressions affect the graph of a logarithmic function?
3. What happens to the graph of a logarithmic function when it is reflected over the x-axis?
4. How do horizontal shifts affect the intercepts of a logarithmic function?
5. Can you explain the domain and range of the logarithmic parent function?

**Tip:** Understanding the transformations of functions helps in predicting changes in their graphs and intercepts.

Explore how intercepts change when transforming a logarithmic parent function. Analyze the impact of function transformations on x-intercepts and y-intercepts. Suitable for Grades 11-12.

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