Quadratic Functions Examples

Graph the equation y = x² + 2.

Here are some points:

Here is a graph:

Connecting the dots in a "U'' shape gives us

This looks almost exactly like the graph of y = x², except we've moved the whole picture up by 2. We like the way it looks up there better. Brings the whole graph together, wouldn't you say?

Graph the quadratic equation y = -x² + 2x + 3.

We know what we have to find, so let's find it.

1. What are the intercepts?

First, the x-intercepts. We need to find the roots of the quadratic polynomial. If we find them, we can celebrate by drinking a root beer.

We need to find the solutions to the equation

0 = -x² + 2x + 3
= -(x² – 2x – 3).

This equation factors as

0 = -(x – 3)(x + 1),

so the solutions (and the x-intercepts) are

x = 3, x = -1.

We can graph these points:

The y-intercept is the constant term, 3, so we can graph that also:

2. What is the vertex?

The vertex occurs halfway between the x-intercepts -1 and 3, so at x = 1. When we plug x = 1 in to the quadratic equation, we find

-(1)² + 2(1) + 3 = 4,

so the vertex occurs at (1, 4).

3. Does the parabola open upwards or downwards?

Since the coefficient on the x² term is negative, the parabola opens downwards.

Putting together all the pieces, we find our graph:

We know this graphing stuff can be infectious, but be careful. We don't want you to get a graph infection.

Graph the quadratic equation y = 3x² + x – 3.

Let's find the useful things. Other than duct tape, a Swiss army knife, and a compass. We meant more within the realm of algebra. We should have been clearer.

1. What are the intercepts?

To find the x-intercepts, we need to use the quadratic equation because this polynomial doesn't factor nicely.

For the purposes of graphing, we can round these numbers to 0.8 and -1.2:

The y-intercept is the constant term of the quadratic equation, or -3:

2. What is the vertex?

The vertex is at , which in this case is .

When we plug  back into the equation, we find

The vertex is .

For the sake of graphing, as well as for the sake of our own sanity, we can round to (-0.2, -3.1).

3. Does the parabola open upwards or downwards?

Since the coefficient on the x² term is positive, the parabola opens upwards. Like a blossoming flower. That's so nice.

Putting everything together, we get our graph: