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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

1 answer

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:

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