Question

Question #81645

Answer 1

graph{y=x^2-x-6 [-9.96, 10.04, -7.64, 2.36]}

This is the graph of `y=x^2-x-6`

Explanation:

`y=(x+2)(x-3)`

`y=x^2-x-6`

We can plot a parabola by substituting x=-5 to +5, which will give
24, 14, 6, 0, -4, -6, -6, -4, 0, 14, 24.

By graphing the points, we can link the points to get a parabola.

Note:
`y=x^2-x-6`
`y=(x-0.5)^2-0.25-6`
`y=(x-0.5)^2-6.25`

Hence, the vertex of the parabola is at `(0.5,-6.25)`

Answer 2

Refer to the explanation.

Explanation:

Graph:

`y=(x+2)(x-3)`

In order to graph a parabola, useful points are the vertex and x-intercepts (if it crosses the x-axis), and other points determined by plugging in different values for `x` and solving for `y`.

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

The given equation is in the factored form of a quadratic equation. We can substitute `0` for `y` and solve for `x`, which will give us the two x-intercepts.

`(x+2)=0`

`x=-2`

`(x-3)=0`

`x=3`

`color(red)("x-intercepts":` `(-2,0), (3,0)`

Now we need to convert the equation to standard form by FOILing the binomials.

http://www.mathcaptain.com/algebra/foil-method.html

`y=x^2-x-6` is a quadratic equation in standard form:

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

where:

`a=1`, `b=-1`, `c=-6`

For a quadratic equation in standard form, the `x`-value of the vertex is the same as the axis of symmetry.

Axis of Symmetry: vertical line that divides the parabola into equal halves

Formula for axis of symmetry for a quadratic equation in standard form:

`x=(-b)/(2a)`

`x=(-(-1))/(2*1)`

`x=1/2=0.5`

Vertex: minimum or maximum point on the parabola

Substitute `0.5` for `x` in the equation and solve for `y`.

`y=(0.5)^2-0.5-6`

`y=0.25-0.5-6=-6.25`

`color(blue)("vertex":` `(0.5,-6.25)`

Other points

First point:

Substitute `-3` for `x` and solve for `y`.

`y=(-3)^2-(-3)-6`

`y=9+3-6=6`

`color(green)("first point:"` `(-3,6)`

Second point:

Substitute `4` for `x` and solve for `y`.

`y=4^2-4-6`

`y=16-4-6=6`

`color(purple)"second point:"` `(4,6)`

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

graph{y=x^2-x-6 [-10, 10, -5, 5]}