Given that the solutions to the equation \( f(x) = 0 \) are \( x = 8 \) and \( x = -4 \), we can express the quadratic function in its factored form as follows:
\[ f(x) = a(x - 8)(x + 4) \]
where \( a \) is a non-zero constant that determines the vertical stretch/compression and direction (upwards or downwards) of the parabola.
To find one specific function that could represent \( f \), we can choose \( a = 1 \):
\[ f(x) = 1(x - 8)(x + 4) = (x - 8)(x + 4) \]
Now multiply the factors:
\[ f(x) = x^2 + 4x - 8x - 32 = x^2 - 4x - 32 \]
Thus, one possible quadratic function that represents \( f \) is:
\[ f(x) = x^2 - 4x - 32 \]
You could also choose other values for \( a \), such as \( a = 2 \), \( a = -1 \), etc., leading to different quadratic functions, but the roots will remain the same. For example:
If \( a = 2 \): \[ f(x) = 2(x - 8)(x + 4) = 2x^2 - 8x - 64 \]
If \( a = -1 \): \[ f(x) = -1(x - 8)(x + 4) = -(x^2 - 4x - 32) = -x^2 + 4x + 32 \]
All of these functions would have the same roots \( x = 8 \) and \( x = -4 \).