Question

1. $f(x)=(x + 5)(x + 1)$

Explanation:

Step1: Find x-intercepts

Set $f(x)=0$, solve $(x+5)(x+1)=0$.
Solutions: $x=-5$ and $x=-1$, so intercepts at $(-5,0)$ and $(-1,0)$.

Step2: Find vertex x-coordinate

Average the x-intercepts:
$$x = \frac{-5 + (-1)}{2} = -3$$

Step3: Find vertex y-coordinate

Substitute $x=-3$ into $f(x)$:
$$f(-3)=(-3+5)(-3+1)=(2)(-2)=-4$$
Vertex at $(-3,-4)$.

Step4: Find y-intercept

Set $x=0$:
$$f(0)=(0+5)(0+1)=5$$
Y-intercept at $(0,5)$.

Step5: Plot points and draw parabola

Plot $(-5,0)$, $(-1,0)$, $(-3,-4)$, $(0,5)$, then sketch the upward-opening parabola through these points.

Answer:

The graph is an upward-opening parabola with x-intercepts at $(-5, 0)$ and $(-1, 0)$, vertex at $(-3, -4)$, and y-intercept at $(0, 5)$, passing smoothly through all plotted points.