# How do you draw the graph of y=3x^2-7?

Apr 7, 2015

Note that $y = 3 {x}^{2} - 7$ is a parabola (which opens upward) so we will need to calculate a few sample points and then draw a parabolic curve through those points.

For example you might calculate $f \left(x\right)$ for all integer values in the range $\left[- - 4 , + 4\right]$ (note symmetry about the Y-axis reduces the number of actual calculations).
$\left(- 4 , 41\right)$ and $\left(+ 4 , 41\right)$
$\left(- 3 , 20\right)$ and $\left(+ 3 , 20\right)$
$\left(- 2 , 5\right)$ and $\left(+ 2 , 5\right)$
$\left(- 1 , - 4\right)$ and $\left(+ 1 , - 4\right)$
$\left(0 , - 7\right)$

Mark these point on some graph paper and joint them in a smooth curve that should look something like:
graph{3x^2-7 [-9.71, 10.29, -7.16, 2.84]}

Apr 7, 2015

Refer to explanation.

#### Explanation:

Given:

$y = 3 {x}^{2} - 7$ is a quadratic equation in standard form:

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

where:

$a = 3$, $b = 0$, $c = - 7$

To graph a parabola you need to determine the following points:

axis of symmetry, vertex, y-intercept (if applicable), x-intercepts (if applicable), and additional points if needed.

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

Vertex: the maximum or minimum point of the parabola.

The formula for determining the axis of symmetry, which is also the $x$-coordinate of the vertex is:

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

$x = \frac{0}{2 \cdot 3}$

$x = 0$

To determine the $y$-coordinate by substituting $0$ for $x$ and solving for $y$.

$y = 3 {\left(0\right)}^{2} - 7$

$y = - 7$

The vertex is $\left(0 , - 7\right)$

Y-intercept: value of $y$ when $x = 0$.

In this particular case, the y-intercept is the same as the vertex, $\left(0 , - 7\right)$.

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

$0 = 3 x - 7$

Add $7$ to both sides.

$7 = 3 {x}^{2}$

Divide both sides by $3$.

$\frac{7}{3} = {x}^{2}$

$\pm \sqrt{\frac{7}{3}} = x$

$x = - \sqrt{\frac{7}{3}} ,$ $\sqrt{\frac{7}{3}}$

The x-intercepts are $\left(- \sqrt{\frac{7}{3}} , 0\right)$ and $\left(\sqrt{\frac{7}{3}} , 0\right)$.

The approximate x-intercepts are $\left(- 1.528 , 0\right)$ and $\left(1.528 , 0\right)$.

Additional points: Choose values for $x$ and solve for $y$.

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

graph{y=3x^2-7 [-9.88, 10.12, -7.88, 2.12]}