part of the graph of the function $f(x)=(x + 4)(x - 6)$ is shown below. which statements about the function…

part of the graph of the function $f(x)=(x + 4)(x - 6)$ is shown below. which statements about the function are true? select two options. the vertex of the function is at $(1,-25)$. the vertex of the function is at $(1,-24)$. the graph is increasing only on the interval $-4<x<6$. the graph is positive only on one interval, where $x < - 4$. the graph is negative on the entire interval $-4<x<6$.

part of the graph of the function $f(x)=(x + 4)(x - 6)$ is shown below. which statements about the function are true? select two options. the vertex of the function is at $(1,-25)$. the vertex of the function is at $(1,-24)$. the graph is increasing only on the interval $-4<x<6$. the graph is positive only on one interval, where $x < - 4$. the graph is negative on the entire interval $-4<x<6$.

Answer

Explanation:

Step1: Expand the function

$f(x)=(x + 4)(x - 6)=x^{2}-6x+4x - 24=x^{2}-2x - 24$.

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 $a = 1$, $b=-2$, so $x=-\frac{-2}{2\times1}=1$.

Step3: Find the y - coordinate of the vertex

Substitute $x = 1$ into $f(x)=x^{2}-2x - 24$, we get $f(1)=1^{2}-2\times1 - 24=1 - 2-24=-25$. So the vertex is $(1,-25)$.

Step4: Analyze the sign of the function

Set $f(x)=(x + 4)(x - 6)=0$, the roots are $x=-4$ and $x = 6$. The parabola opens upward (since $a = 1>0$). The function is negative when $-4<x<6$ and positive when $x<-4$ or $x>6$. The function is increasing for $x>1$ and decreasing for $x<1$.

Answer:

The vertex of the function is at $(1,-25)$; The graph is negative on the entire interval $-4 < x < 6$.