10.5 Graphing quadratic equations

Recognize the graph of a quadratic equation in two variables

We have graphed equations of the form $Ax+By=C$ . We called equations like this linear equations because their graphs are straight lines.

Now, we will graph equations of the form $y=a{x}^2+bx+c$ . We call this kind of equation a quadratic equation in two variables    .

Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations.

Let’s look first at graphing the quadratic equation $y=x^2$ . We will choose integer values of $x$ between $\mathrm{-2}$ and 2 and find their $y$ values. See [link] .

Notice when we let $x=1$ and $x=\mathrm{-1}$ , we got the same value for $y$ .

The same thing happened when we let $x=2$ and $x=\mathrm{-2}$ .

Now, we will plot the points to show the graph of $y=x^2$ . See [link] .

The graph is not a line. This figure is called a parabola    . Every quadratic equation has a graph that looks like this.

In [link] you will practice graphing a parabola by plotting a few points.

Graph $y=x^2-1$ .

Solution

We will graph the equation by plotting points.

Choose integers values for x , substitute them into the equation and solve for y .
Record the values of the ordered pairs in the chart.
Plot the points, and then connect them with a smooth curve. The result will be the graph of the equation $y=x^2-1$ .

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How do the equations $y=x^2$ and $y=x^2-1$ differ? What is the difference between their graphs? How are their graphs the same?

All parabolas of the form $y=a{x}^2+bx+c$ open upwards or downwards. See [link] .

Notice that the only difference in the two equations is the negative sign before the $x^2$ in the equation of the second graph in [link] . When the $x^2$ term is positive, the parabola    opens upward, and when the $x^2$ term is negative, the parabola opens downward.

Parabola orientation

For the quadratic equation $y=a{x}^2+bx+c$ , if:

Determine whether each parabola opens upward or downward:

ⓐ $y=\mathrm{-3}{x}^2+2x-4$ ⓑ $y=6x^2+7x-9$

Solution

ⓐ Find the value of " a ".	Since the “a” is negative, the parabola will open downward.
ⓑ Find the value of " a ".	Since the “a” is positive, the parabola will open upward.

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Find the axis of symmetry and vertex of a parabola

Look again at [link] . Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry    of the parabola.