Intercepts, Zeroes and Solutions

Now that you are able to graph a quadratic function with it's transformations, it is time to look at an important part of the graph. The intercepts of a quadratic function are usually points of interest.

The y-intercept

The y-intercept of any function is the point where the graph of the function crosses the y-axis. (Note that the graph CANNOT cross the y-axis in more than one place. If it did, it would not be a function!) On the graph shown here:

The y-intercept is y = 1. Finding intercepts on graphs can be tricky, however. If the graph does not happen to pass through an exact point on the y-axis, you may have trouble determining the EXACT location of the y-intercept. To always find the EXACT location, you can use algebra to help you.

Example

Find the y-intercept of the quadratic function y = x² + 3x + 7

Solution

You could graph this function, if you knew how to convert it to standard form (y = a(x-h)² + k), and then find the intercept from the graph. However, to find the intercept algebraically, you must remember the following fact:

The x-coordinate of ANY y-intercept is 0.

Look at the graph above. You can see that where the graph crosses the y-axis, the x has to be 0. By definition, ANY point on the y-axis has coordinates of (0,?), where ? is any point on the y-axis. We can use this fact to solve the equation for the y-intercept. Here is another rule:

To solve for the y-intercept in ANY equation, set x = 0 and solve for y.

Let's see this in action with the equation above. The algebraic steps are shown below:

This tells you that the y-intercept of this function is y = 7, without having to graph!

The x-intercept(s)

A function has an x-intercept (or intercepts) wherever it crosses the x-axis. Look at the graph below:

This graph has 2 x-intercepts at x = -1 and x = 3. A quadratic function will have at most two x-intercepts. It can have only one intercept (as shown on the graph below):

This function has one x-intercept at x = 1. A quadratic function can also have NO x-intercepts, as shown in the graph below:

Notice that this corresponds to the number of solutions a quadratic equation can have (2, 1 or 0). The x-intercepts of a quadratic function are also known as its solutions. They are also known as zeroes, for a reason we will see in a second.

As with y-intercepts, it may sometimes be difficult to read the x-intercepts off of a graph, especially if the graph does not nicely cross though an exact point on the grid. We will use algebra to solve for the EXACT value of the intercepts.

Example

Find the x-intercepts of y = x² - 3x - 7.

Solution

If we graph this function (which you are not expected to be able to do at this point - if you want you can skip ahead to the section on converting quadratic functions to standard form) you can see that it will be difficult to be sure of the EXACT values of the x-intercepts.

Are the intercepts at -1.5 and 4.5? They might be, but we don't know FOR SURE unless we solve for them algebraically.

To algebraically determine x-intercepts, set y = 0 and solve the equation for x.

You have already looked at solving quadratic equations in the equations and inequalities unit. Once you substitute 0 in for y in the equation above, you arrive at the following:

You should recognize that the last line above is a quadratic equation. You can solve this by factoring (which in this case is difficult) or by using the quadratic formula. We will pick the latter method. The work is shown below:

You can see that it would have been difficult (at best!) to come up with those exact values for the intercepts. Solving algebraically insures that you always know exactly where the intercepts lie.

You have seen that the intercepts of a quadratic function are also the solutions to the corresponding quadratic equation. That is why they are also called 'zeroes', as they are the values you get for x when you make the function equal to zero.

As you can see, this page does not look at this topic to the extent that your teacher does. There are questions in your booklet, but you may also try a couple of sample questions, or you can head back to the Quadratic Functions main page.