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

Jul 27, 2018

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#### Explanation:

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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

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

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

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

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

$\Rightarrow x = - 3 , x = 3$

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

$\Rightarrow y = {\left(0\right)}^{2} - 9 = 0$

$\Rightarrow y = - 9$

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.

Jul 27, 2018

Refer to the explanation.

#### Explanation:

Given:

$y = {x}^{2} - 9$ is a quadratic equation in standard form:

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

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 $\left(x , y\right)$ 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:

$x = \frac{- b}{2 a}$

$x = \frac{0}{2}$

$x = 0$

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 $\left(0 , - 9\right)$ 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 ${x}^{-} 9$

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

Set each binomial equal to $0$ and solve.

$x + 3 = 0$

$x = - 3$

Point: $\left(- 3 , 0\right)$ Plot this point. $\leftarrow$ first x-intercept

$x - 3 = 0$

$x = 3$

Point: $\left(3 , 0\right)$ Plot this point. $\leftarrow$ 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]}