Question

Which exponential function, after being reflected over the x-axis, has a y-intercept at (0, −4) and the right side of the graph approaches y = 0?
Responses
A y = 4(2)^x
y = 4 ( 2 ) x
B y = 4







y = 4 ( 1 2 ) x
C y = −4
y = −4 ( 2 ) x
D y = −4

Answer 1

To find the exponential function that, when reflected over the x-axis, has a y-intercept at (0, −4) and its right side approaches \(y = 0\), we will need to analyze the given options.

1. **Understanding Reflection Over the X-Axis**: When a function \(f(x)\) is reflected over the x-axis, the new function will be \(-f(x)\). Thus, if we want the reflected function to have a y-intercept at (0, -4), then before reflection, it must have a y-intercept at (0, 4).

2. **Analyzing the Choices**:
- **A**: \(y = 4(2^x)\): This function has a y-intercept of 4 at \(x = 0\). When reflected, it becomes \(y = -4(2^x)\), which will have a y-intercept at (0, -4). This function approaches \(y = 0\) as \(x \to \infty\).
- **B**: \(y = 4(1/2)^x\): This function also has a y-intercept of 4. When reflected, it becomes \(y = -4(1/2)^x\), which has a y-intercept at (0, -4). However, this function approaches \(y = -\infty\) as \(x\) increases, which does not meet the criteria of approaching \(y = 0\).
- **C**: \(y = -4(2^x)\): This has a y-intercept of -4, so it does not fit our needs.
- **D**: \(y = -4\): This is a constant function with no exponential behavior and does not meet the criteria.

Based on this analysis, the correct function is:

**A**: \(y = 4(2^x)\)

Once reflected over the x-axis, it becomes \(y = -4(2^x)\), which has a y-intercept of (0, -4) and approaches \(y = 0\) as \(x \to \infty\).