Question

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

Answer 1

Summary:

Axis of symmetry: `x=-3.5`

Vertex: `(-3.5,36.75)`

X-intercepts: `(0,0)` and `(-7,0)`

Refer to the explanation for the process.

Explanation:

`y=-3x(x+7)`

Expand.

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

`y=ax^2+bx+c`,

where:

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

Axis of symmetry: vertical line that divides the parabola into two equal halves; `x=(-b)/(2a)`

Plug in known values.

`x=(-(-21))/(2*-3)`

`x=21/(-6)`

`x=-21/6`

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

`x=-7/2=-3.5` `larr` Axis of symmetry, also `x`-value of vertex

Vertex: maximum or minimum point `(x,y)` 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(-3.5)^2-21(-3.5)`

`y=-36.75+73.5=36.75`

Vertex: `(-3.5,36.75)`

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

`0=-3x^2-21`

Factor out the common term `3x`.

`0=-3x(x+7)`

Divide both sides by `-3`.

`0=x(x+7)`

Solutions for `x`.

`x=0`

`x+7=0`

`x=-7`

x-intercepts: `(0,0)` and `(-7,0)`

Summary:

Axis of symmetry: `x=-3.5`

Vertex: `(-3.5,36.75)`

X-intercepts: `(0,0)` and `(-7,0)`

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]}