Question

Part of the graph of the function f(x) = (x - 1)(x + 7) isshown below.Complete the following infomationThe vertex of the function is at ___The graph is increasing on the interval ___The graph is positive on the interval ___The graph is negative on the interval ___

Answer 1

The function is given as

$f(x)=(x-1)(x+7)$

Let us put the quadratic formula in the form of

$ax^2+bx+c$

Hence, we will have

$\begin{gathered} f(x)=x^2-x+7x-7 \\ f(x)=x^2+6x-7 \end{gathered}$

Vertex of the function:

The vertex of a function is given as

$x=-\frac{b}{2a}$

From the equation, we have

a = 1

b = 6

c = -7

Hence, we calculate the vertex as

$\begin{gathered} x=-\frac{6}{2\times1} \\ x=-3 \end{gathered}$

Therefore, the vertex is at x = -3.

Interval where the graph is increasing:

Since the graph has a positive curve (because a > 0), the graph increases to the right of the vertex.

Therefore, the graph increases at the interval x > -3.

Interval where the graph is positive:

The graph is positive at all points where f(x) > 0.

$f(x)>0$

From the graph, we can see that f(x) is greater than 0 when x < -7 and x > 1.

Therefore, the graph is positive at the interval (x < 7 and x > 1)

Interval where the graph is negative:

The graph is negative at all points where f(x) < 0.

$f(x)<0$

From the graph, we can see that f(x) is less than 0 when x > -7 and x < 1.

Therefore, the graph is negative at the interval (-7 < x < 1)