Question

The graph of which function will have a maximum and a y-intercept of 4?

f(x) = 4x2 + 6x – 1
f(x) = –4x2 + 8x + 5
f(x) = –x2 + 2x + 4
f(x) = x2 + 4x – 4

Answer 1

To determine which function has a maximum and a y-intercept of 4, we can analyze each function one by one.

1. **Identifying the y-intercept**:
The y-intercept of a function \( f(x) \) is found by evaluating \( f(0) \).

2. **Identifying if the function has a maximum**:
A quadratic function \( f(x) = ax^2 + bx + c \) has a maximum if \( a < 0 \).

Now, let's analyze the provided functions:

1. **Function: \( f(x) = 4x^2 + 6x - 1 \)**
- Y-intercept: \( f(0) = 4(0)^2 + 6(0) - 1 = -1 \) (not 4)
- \( a = 4 \) (not a maximum)

2. **Function: \( f(x) = -4x^2 + 8x + 5 \)**
- Y-intercept: \( f(0) = -4(0)^2 + 8(0) + 5 = 5 \) (not 4)
- \( a = -4 \) (maximum)

3. **Function: \( f(x) = -x^2 + 2x + 4 \)**
- Y-intercept: \( f(0) = -0^2 + 2(0) + 4 = 4 \) (is 4)
- \( a = -1 \) (maximum)

4. **Function: \( f(x) = x^2 + 4x - 4 \)**
- Y-intercept: \( f(0) = 0^2 + 4(0) - 4 = -4 \) (not 4)
- \( a = 1 \) (not a maximum)

From our analysis, the function that has a maximum and a y-intercept of 4 is:

**\( f(x) = -x^2 + 2x + 4 \)**.