Question

part of the graph of the function $f(x) = (x - 1)(x + 7)$ is shown. which statements about the function are true? choose three correct answers. 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:

Step1: Find x-intercepts

The function is given in factored form $f(x)=(x-1)(x+7)$. Set $f(x)=0$, so $x-1=0$ or $x+7=0$. The x-intercepts are $x=1$ and $x=-7$.

Step2: Calculate vertex x-coordinate

The x-coordinate of the vertex of a quadratic is the midpoint of the x-intervals: $x=\frac{1+(-7)}{2}=\frac{-6}{2}=-3$.

Step3: Calculate vertex y-coordinate

Substitute $x=-3$ into $f(x)$:

$$\begin{align*} f(-3)&=(-3-1)(-3+7)\\ &=(-4)(4)\\ &=-16 \end{align*}$$

The vertex is $(-3,-16)$.

Step4: Analyze increasing interval

The quadratic opens upwards (leading coefficient positive, since expanding $f(x)=x^2+6x-7$ has $a=1>0$). It increases to the right of the vertex, so for $x>-3$.

Step5: Analyze positive intervals

For upward opening quadratic, the graph is positive when $x <$ left intercept or $x >$ right intercept, so $x<-7$ and $x>1$.

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 $x > 1$.