part of the graph of the function f(x)=(x - 1)(x + 7) is shown below. which statements about the function…

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.

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

Explanation:

Step1: Expand the function

$f(x)=(x - 1)(x + 7)=x^{2}+7x-x - 7=x^{2}+6x - 7$.

Step2: Find the x - coordinate of the vertex

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex is $x=-\frac{b}{2a}$. Here $a = 1$, $b = 6$, so $x=-\frac{6}{2\times1}=-3$.

Step3: Find the y - coordinate of the vertex

Substitute $x=-3$ into $f(x)=x^{2}+6x - 7$, we get $f(-3)=(-3)^{2}+6\times(-3)-7=9 - 18 - 7=-16$. So the vertex is at $(-3,-16)$.

Step4: Analyze the increasing - decreasing property

Since $a = 1>0$, the parabola opens upward. The function is increasing on the interval $x>-3$.

Step5: Analyze the positive - negative property

Set $f(x)=(x - 1)(x + 7)=0$, we get $x = 1$ or $x=-7$. The function is positive when $x<-7$ or $x>1$, and negative when $-7<x<1$.

Answer:

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 $x>1$.