Question

How do you graph `y=x^2-9`?

Answer 1

`" "`
Please read the explanation.

Explanation:

`" "`
A quadratic equation is of the form:

`color(red)(ax^2+bx+c=0`

We have : `color(red)(y=f(x)=x^2-9`

Set `color(blue)(y=0`

We have the quadratic equation:

`color(blue)(x^2-9=0`

Using the algebraic identity:

`color(green)(a^2-b^2 -= (a+b)(a-b)`

We can rewrite `color(blue)(x^2-9=0` as

`x^2-3^2=0`

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

`rArr (x+3)=0, (x-3)=0`

`rArr x= -3, x=3`

Hence, there are two solutions for `color(red)(x`

So, we have two x-intercepts:

`color(red)((-3,0) and (3,0)`

To find the y-intercept, set `color(blue)(x=0`

`rArr y=(0)^2-9=0`

`rArr y = -9`

Hence, the y-intercept: `color(red)((0,-9)`

The graph of `color(red)(y=f(x)=x^2-9` is given below:

Hope it helps.

Answer 2

Refer to the explanation.

Explanation:

Given:

`y=x^2-9` is a quadratic equation in standard form:

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

where:

`a=1`, `b=0`, and `c=-9`

To graph a quadratic equation in standard form, you need the vertex, y-intercept, x-intercepts (if real), and one or two additional points.

Vertex: maximum or minimum point `(x,y)` of the parabola

Since `a>0`, the vertex is the minimum point and the parabola opens upward.

The x-coordinate of the vertex is determined using the formula for the axis of symmetry:

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

`x=0/2`

`x=0`

To find the y-coordinate of the vertex, substitute `0` for `x` and solve for `y`.

`y=0^2-9`

`y=-9`

The vertex is `(0,-9)` Plot this point.

In this case, the vertex is also the y-intercept, which is the value of `y` when `x=0`.

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

Substitute `0` for `y` and solve for `x`.

`0=x^2-9`

Switch sides.

`x^2-9=0`

Factor `x^-9`

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

Set each binomial equal to `0` and solve.

`x+3=0`

`x=-3`

Point: `(-3,0)` Plot this point. `larr` first x-intercept

`x-3=0`

`x=3`

Point: `(3,0)` Plot this point. `larr` second x-intercept

For additional points, choose values for `x` and solve for `y`.

Plot all the points and sketch a parabola through the points. Do not connect the dots.

graph{y=x^2-9 [-11.13, 11.37, -9.885, 1.365]}