Question

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 $-4 < x < 6$.

Explanation:

Step1: Find vertex x-coordinate

For $f(x)=(x+4)(x-6)=x^2-2x-24$, vertex $x=-\frac{b}{2a}=-\frac{-2}{2(1)}=1$

Step2: Calculate vertex y-coordinate

Substitute $x=1$: $f(1)=(1+4)(1-6)=5(-5)=-25$

Step3: Analyze sign of the function

Roots at $x=-4,6$. Parabola opens up, so negative between roots: $-4

Step4: Analyze increasing interval

Parabola opens up, increasing for $x>1$, not just $-4

Step5: Analyze positive intervals

Positive when $x<-4$ and $x>6$, not just $x<-4$

Answer:

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