# How do you graph and label the vertex and axis of symmetry of y=-3x(x+7)?

Nov 21, 2017

Summary:

Axis of symmetry: $x = - 3.5$

Vertex: $\left(- 3.5 , 36.75\right)$

X-intercepts: $\left(0 , 0\right)$ and $\left(- 7 , 0\right)$

Refer to the explanation for the process.

#### Explanation:

$y = - 3 x \left(x + 7\right)$

Expand.

$y = - 3 {x}^{2} - 21 x$ is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c$,

where:

$a = - 3$, $b = - 21$, and $c = 0$.

Axis of symmetry: vertical line that divides the parabola into two equal halves; $x = \frac{- b}{2 a}$

Plug in known values.

$x = \frac{- \left(- 21\right)}{2 \cdot - 3}$

$x = \frac{21}{- 6}$

$x = - \frac{21}{6}$

Reduce by dividing the numerator and denominator by $3$.

$x = - \frac{7}{2} = - 3.5$ $\leftarrow$ Axis of symmetry, also $x$-value of vertex

Vertex: maximum or minimum point $\left(x , y\right)$ of the parabola. Since $a < 0$, the vertex will be the maximum point and the parabola will open downward.

Substitute $- 3.5$ for $x$ and solve for $y$.

$y = - 3 {\left(- 3.5\right)}^{2} - 21 \left(- 3.5\right)$

$y = - 36.75 + 73.5 = 36.75$

Vertex: $\left(- 3.5 , 36.75\right)$

X-intercepts: values of $x$ when $y = 0$

$0 = - 3 {x}^{2} - 21$

Factor out the common term $3 x$.

$0 = - 3 x \left(x + 7\right)$

Divide both sides by $- 3$.

$0 = x \left(x + 7\right)$

Solutions for $x$.

$x = 0$

$x + 7 = 0$

$x = - 7$

x-intercepts: $\left(0 , 0\right)$ and $\left(- 7 , 0\right)$

Summary:

Axis of symmetry: $x = - 3.5$

Vertex: $\left(- 3.5 , 36.75\right)$

X-intercepts: $\left(0 , 0\right)$ and $\left(- 7 , 0\right)$

Plot the vertex and x-intercepts. Sketch a parabola through the points. Do not connect the dots.

graph{y=-3x^2-21x+0 [-19.92, 12.11, -2.67, 13.35]}