Question

A quadratic function is represented using the equation f (x) = -2 (x - 4)^2 + 6. Complete the inequality to show the range of the function.

Answer 1

To determine the range of the quadratic function \( f(x) = -2(x - 4)^2 + 6 \), we first identify its vertex and the direction it opens.

The function is in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case:

- \(a = -2\) (which is negative, indicating the parabola opens downwards)
- \(h = 4\)
- \(k = 6\)

Thus, the vertex of the parabola is at the point \((4, 6)\).

Since the parabola opens downwards, the maximum value of the function is at the vertex, which is 6. As \(x\) moves away from 4 in either direction, the value of \(f(x)\) will decrease without bound.

Therefore, the range of the function can be expressed using the inequality:

\[
f(x) \leq 6
\]

In interval notation, the range is:

\[
(-\infty, 6]
\]

So, the completed inequality showing the range of the function is:

\[
f(x) \leq 6
\]
And in interval notation: \((- \infty, 6]\).