Question

Part of the graph of the function f(x) = (x – 1)(x + 7) is shown below.

Which statements about the function are true? Select three options.

The vertex of the function is at (–4,–15).
The vertex of the function is at (–3,–16).
The graph is increasing on the interval x > –3.
The graph is positive only on the intervals where x < –7 and where
x > 1.
The graph is negative on the interval x < –4.

Answer 1

The vertex of the function is at (–3,–16)

The graph is increasing on the interval x > –3

The graph is positive only on the intervals where x < –7 and where

x > 1.

Explanation:

The graph of
`f(x)=(x-1)(x+7)` has clear zeroes at
`x=1` and
`x=-7`, showing that
`f(x) > 0` when
`x < -7` and
`x > 1`. To determine where the vertex is, we can complete the square:

`f(x)=(x-1)(x+7)\\y=x^2+6x-7\\y+16=x^2+6x-7+16\\y+16=x^2+6x+9\\y+16=(x+3)^2\\y=(x+3)^2-16`

So, we can see the vertex is (-3,-16), meaning that where
`x > -3`, the function will be increasing on that interval