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

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

C. The graph is increasing on the interval x > –3.

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

x > 1.

Answer 2

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 roots of this function are the values that make f(x) = 0. As the equation is already factored, we can see that the roots are x1 = 1 and x2 = -7

The x-coordinate of the vertix can be calculated as:

x_v = (x1 + x2)/2 = -6/2 = -3

So the first statement: "The vertex of the function is at (–4,–15)" is wrong.

The y_coordinate of the vertex is:

f(x_v) = (x_v - 1)*(x_v + 7) = (-3 -1) * (-3 +7) = (-4) * (4) = -16

So the second statemente: "The vertex of the function is at (–3,–16)" is correct.

for values of x greater than x_v = -3, the graph increases, as the equation represents a parabola with concavity upwards. So the third statement is correct.

As the roots are 1 and -7 and the concavity is upwards, the parabola have positive values for x<-7 and x>1. So the fourth statement is correct.

For values of x< -4, the parabola can have both positive or negative values (it has positive values for x < -7) so the fifth statement is wrong.