Connecting Intercepts with Factoring

MELT: Math1

Way back in the day, you learned to find the x- and y-intercepts of linear functions. These were useful points to find because sometimes they are the answers to important questions. More about this in a minute. Remember that  to find a y-intercept, substitute x = 0 and solve, and  to find an x-intercept, substitute y = 0 and solve. Review this skill by finding both intercepts for the equations below. Be sure to show your work.

Here is an oldie, but a goodie. Suppose that the local elementary school is having a talent show. They are selling adult tickets for $5 and student tickets for $3. They are trying to raise $450 to purchase art supplies and some copy paper to make it through the end of the year. Write an equation that models this problem.

Using your model above, and showing your how many student tickets would have to be work, how many adult tickets would they sold in order to meet their goal if no adults have to sell to meet their goal if no came? students came?

Suppose that the value of a new car is $27,000, but that its value goes down each year by $2,250 each year (This loss of value over time is called depreciation.) Write a function, V(t), that models the value of this car in dollars after t years.

In the space below, find both intercepts of this function and explain in a sentence what each one means.

Stefanie Smith and Malen Braswell MELT: Math1

As you’ve reviewed above, intercepts often tell us useful information about a problem. Before we learn to find the intercepts of quadratic functions, let’s do a little thinking… Fill in the blanks below

Finding the y-intercepts of a quadratic function is easy: just substitute x = 0 and simplify. Practice this now, taking care to show your work.

After working these problems, do you notice a short-cut way to find the answer? Explain.

Finding the x-intercepts starts the same way, but so far, we haven’t learned how to solve an equation like , but don’t fret. We will in just a minute.

Go back to the three easy problems we answered at the bottom of the first page. Use your answers there to help you fill in the blanks in the statement below:

For two numbers x and y, if , then either ______, or ______.

This statement is called the Zero-Factor Property, and it is what we will use to solve quadratic equations. Here are two examples of how it works.

Practice this skill by solving the equations below. Be careful how you show your work.

This is all great and good, but doesn’t look a think like the four problems above. It can, though. All you have to do is______(hint: we just took a test on it.)

This process of solving quadratic equations we’re learning is called, “solving by factoring.” This method doesn’t work for every quadratic equation because they won’t all factor, but it works for many of them and it is the easiest “by hand” method to learn, so let’s get good at it. Solve each of these by factoring. Make sure your work looks like the example above.

Here comes the fun part. Solving problems. So far, we’ve learned how to find the vertex and the intercepts of a quadratic function. Suppose that a human cannon-baller is trying to set a new record for distance. He is going to launch himself off a building, up into the air, and down into a net down the street from the building. His safety engineer has calculated that his height h in feet at any time t seconds after he is launched can be modeled by

. Use this function and what you know about intercepts and vertices to answer the following questions. Be sure to show your work for each.

How high off the ground How high in the air did he How long did it take for was he the instant he was go before starting to come him to hit the net down the launched? down? street?

Stefanie Smith and Malen Braswell