Question

example 3. using the table of values below, sketch the graph of y = -2(x + 1)(x + 5).
complete the table below:
vertex the zeros step pattern direction of opening optimal value axis of symmetry

Explanation:

Step1: Find the zeros

Set $y = 0$, then $-2(x + 1)(x + 5)=0$. By the zero - product property, $x+1 = 0$ or $x + 5=0$. So the zeros are $x=-1$ and $x=-5$.

Step2: Find the axis of symmetry

The axis of symmetry of a quadratic function in the form $y=a(x - r_1)(x - r_2)$ is given by $x=\frac{r_1 + r_2}{2}$, where $r_1$ and $r_2$ are the zeros. Here $r_1=-1$ and $r_2=-5$, so $x=\frac{-1+( - 5)}{2}=\frac{-6}{2}=-3$.

Step3: Find the vertex

Substitute $x=-3$ into $y=-2(x + 1)(x + 5)$. Then $y=-2(-3 + 1)(-3 + 5)=-2\times(-2)\times2 = 8$. So the vertex is $(-3,8)$.

Step4: Determine the direction of opening

Since $a=-2<0$, the parabola opens downwards.

Step5: Find the optimal value

Because the parabola opens downwards, the optimal value is the $y$-coordinate of the vertex, which is $y = 8$.

Step6: Find the step - pattern

For a quadratic function $y=a(x - h)^2+k$ (vertex form), starting from the vertex, the step - pattern is based on the value of $a$. For $y=-2(x + 3)^2+8$, when $x$ moves 1 unit away from the vertex $x=-3$ (i.e., $x=-2$ or $x=-4$), $y=-2( - 2+3)^2+8=-2\times1 + 8 = 6$ and $y=-2(-4 + 3)^2+8=-2\times1+8 = 6$. The step - pattern is over 1, down 2 (because $a=-2$).

Now complete the table:

vertex	the zeros	step pattern	direction of opening	optimal value	axis of symmetry

To complete the first table:
When $x=-6$, $y=-2(-6 + 1)(-6 + 5)=-2\times(-5)\times(-1)=-10$.
When $x=-5$, $y=-2(-5 + 1)(-5 + 5)=0$.
When $x=-4$, $y=-2(-4 + 1)(-4 + 5)=-2\times(-3)\times1 = 6$.
When $x=-3$, $y=-2(-3 + 1)(-3 + 5)=8$.
When $x=-2$, $y=-2(-2 + 1)(-2 + 5)=-2\times(-1)\times3 = 6$.
When $x=-1$, $y=-2(-1 + 1)(-1 + 5)=0$.
When $x = 0$, $y=-2(0 + 1)(0 + 5)=-10$.

$x$	$y=-2(x + 1)(x + 5)$
$-5$	$0$
$-4$	$6$
$-3$	$8$
$-2$	$6$
$-1$	$0$
$0$	$-10$

Answer:

The completed tables are shown above.