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# 9.3: Using Intercepts

Difficulty Level: At Grade Created by: CK-12

## Introduction

Comparing Field Trips

The students have completed all of their research and now it is time for them to figure out which field trip to attend. Mr. Thomas scheduled a meeting during homeroom when both teams would have the opportunity to present their findings. Both did this and then the students began to debate the merits of each trip.

“I think we should go to the Omni Theater because it is more educational,” Tasha stated strongly.

“This trip doesn’t have to be educational we can just have a fun field trip,” Casey said.

Arguments began between the students. Mr. Thomas whistled and all were quiet.

“Is there anything that the two have in common that we can think of?” Mr. Thomas asked.

“Yes. Is there a common fee between them both?”

The students began to think about this. Intercepts are places where two or more equations meet. In the case of the students, they wrote two equations one for each field trip. Is there a common cost in each?

Use this lesson to learn about intercepts and then return to this problem and Mr. Thomas’ question at the end of the lesson.

What You Will Learn

By the end of this lesson, you will understand how to complete the following skills.

• Find an \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercept of the graphs of a linear equation.
• Use intercepts to graph a linear equation.
• Interpret intercepts of the graph of a linear equation.
• Solve real – world problems using intercepts of lines.

Teaching Time

I. Find an \begin{align*}\underline{x}\end{align*}- and \begin{align*}\underline{y}\end{align*}- Intercept of the Graphs of a Linear Equation

In football, a player makes an interception when he catches a ball thrown by the other team that was not intended for him. Intercept means to catch or to interrupt. In graphs, we will find that most lines intercept the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-axes. We’ll call these points the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts and we will use them in a variety of ways.

Consider the graph below. As you know, the \begin{align*}x\end{align*}-axis is horizontal and the \begin{align*}y\end{align*}-axis is vertical. Do you see that the graph crosses, or intercepts, both the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-axes?

In looking at this graph, you can see that the line crosses the \begin{align*}x\end{align*}-axis and the \begin{align*}y\end{align*}-axis. These are the two intercepts of this graph.

The line crosses the \begin{align*}x\end{align*} – axis at -1.

The line crosses the \begin{align*}y\end{align*} – axis at 3.

These are the two intercepts.

We can figure out the two intercepts of any linear equation. All you have to do is to look for the place where the line crosses the two axis’.

That is a great question. Because a vertical line is \begin{align*}x =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}, it would not have a \begin{align*}y\end{align*} – intercept. A horizontal line has the equation \begin{align*}y =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}, so it does not have an \begin{align*}x\end{align*} – intercept. Therefore, we would only name the \begin{align*}x\end{align*} – intercept or the \begin{align*}y\end{align*} – intercept in these two examples. Here is a graph of both a horizontal line and a vertical line. You can see what we mean by looking at this example.

In these two graphs, \begin{align*}x\end{align*} is equal to 4 and \begin{align*}y\end{align*} is equal to -1. You can see that each of these special types of graphs only has one intercept.

II. Use Intercepts to Graph a Linear Equation

Now that you know how to identify an \begin{align*}x\end{align*} and a \begin{align*}y\end{align*}-intercept by looking at a graph, you can learn how to find the two intercepts in an equation and then use this information to graph the line. Think about it, if you know where the line crosses the \begin{align*}x\end{align*}-axis and where the line crosses the \begin{align*}y\end{align*} – axis, then you can easily graph the line. Let’s look at an example.

Example

Find the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts and then graph the equation \begin{align*}2x+3y=6\end{align*}.

First, notice that this is an equation in standard form. We will need to find the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercepts.

To find the \begin{align*}x\end{align*}-intercept, set \begin{align*}y\end{align*} equal to zero. Think about this and it makes perfect sense. If you have an intercept with the \begin{align*}x\end{align*} – axis, then it makes sense that the \begin{align*}y\end{align*} value is 0.

\begin{align*}2x+3y&=6\\ 2x+3 \cdot 0&=6\\ 2x&=6\\ x&=3\end{align*}

We now have the ordered pair (3, 0) or the \begin{align*}x\end{align*}-intercept 3.

To find the \begin{align*}y\end{align*}-intercept set \begin{align*}x\end{align*} equal to zero. Think about this and it makes perfect sense. If you have an intercept with the \begin{align*}y\end{align*} – axis, then it makes sense that the \begin{align*}x\end{align*} value is 0.

\begin{align*}2x+3y&=6\\ 2 \cdot 0+3y&=6\\ 3&=6\\ y&=2\end{align*}

We now have the ordered pair (0, 2) or the \begin{align*}y\end{align*}-intercept 2.

Now we can use the intercepts to graph this line.

Write down how you can find the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercepts in your notebook.

III. Interpret Intercepts of the Graph of a Linear Equation

Now we can find the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts of a graph by seeing where the graph crosses the axes. We can also graph an equation by calculating its \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts and plotting them on a coordinate plane. But let’s understand the significance of the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts. Recall that the \begin{align*}y\end{align*} value of the \begin{align*}x\end{align*}-intercept is always zero and the \begin{align*}x\end{align*} value of the \begin{align*}y\end{align*}-intercept is always zero. This is useful in understanding equations.

Consider a graph whose \begin{align*}x\end{align*}-intercept is 5. Not only does this indicate to us that the graph will cross the \begin{align*}x\end{align*}-axis at 5, but it necessitates that when \begin{align*}x\end{align*} is 5, the \begin{align*}y\end{align*} value is zero. Likewise, for whatever value the \begin{align*}y\end{align*}-intercept has, the \begin{align*}x\end{align*} value must be zero.

Example

Look at the following graph and interpret the intercepts of the graph.

Now let’s look at what information we can interpret from this graph. First, this is a graph of the equation \begin{align*}y=-2x-4\end{align*}.

Notice that the coordinates of the \begin{align*}y\end{align*} – intercept is (0, -4). We can see that the -4 can also be found in the equation itself. Notice how it is the value that is not connected to the \begin{align*}x\end{align*} variable. When looking at an equation and a graph, this is one way to determine the \begin{align*}y\end{align*} – intercept.

Now we can look at the value of the \begin{align*}x\end{align*}-intercept. In this case, it is (-2, 0). Don’t let this fool you, the \begin{align*}y\end{align*} – intercept can be found in the equation, but the \begin{align*}x\end{align*} – intercept is determined by the steepness of the line. Therefore, we will have to use the equation and a table of values to determine the \begin{align*}x\end{align*} – intercept. However, you can still determine it when looking at the graph of a line.

IV. Solve Real – World Problems Using Intercepts of Lines

Interpreting the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts is useful in real-world situations. We can figure out intercepts by looking at all kinds of different problems. Let’s look at an example.

Example

Martha likes to go to the park every day but it’s 6 o’clock and her parents are waiting for her at home. She has her bike but she sometimes walks it. She walks at 3mph and she rides her bike at 9mph. If she is 6 miles from home, how long might her parents have to wait?

\begin{align*}w=\end{align*} time (in hours) walking and \begin{align*}b=\end{align*} time(in hours) on her bike

\begin{align*}3w+9b &=6\\ 3 \cdot 0+9b &=6\\ 9b &=6\\ b &=\frac{2}{3}\end{align*}

If she only rides her bike, it will take her \begin{align*}\frac{2}{3}\end{align*} or 40 minutes.

\begin{align*}3 \cdot w + 9 \cdot 0 &=6\\ 3w &=6\\ w &=2\end{align*}

If she only walks, it will take her 2 hours! Hope she rides her bike.

Now let’s go back and look at the problem from the introduction.

## Real-Life Example Completed

Comparing Field Trips

Here is the problem from the introduction. Use what you have learned to determine the intercept.

The students have completed all of their research and now it is time for them to figure out which field trip to attend. Mr. Thomas scheduled a meeting during homeroom when both teams would have the opportunity to present their findings. Both did this and then the students began to debate the merits of each trip.

“I think we should go to the Omni Theater because it is more educational,” Tasha stated strongly.

“This trip doesn’t have to be educational we can just have a fun field trip,” Casey said.

Arguments began between the students. Mr. Thomas whistled and all were quiet.

“Is there anything that the two have in common that we can think of?” Mr. Thomas asked.

“Yes. Is there a common fee between them both?”

The students began to think about this. Intercepts are places where two or more equations meet. In the case of the students, they wrote two equations one for each field trip. Is there a common cost in each?

Solution to Real – Life Example

To determine the intercept, we must first begin with the two equations.

The bowling trip used the equation \begin{align*}y = 3g + 2\end{align*}.

The Omni trip used the equation \begin{align*}y=5x+2\end{align*}

You might notice right away that the 2 is common in both. We can check and see if this is indeed the intercept by graphing both equations. Here is the graph.

The 2.00 fee for shoes or ticket service fee is the common factor between both trips. ## Vocabulary Here are the vocabulary words that are found in this lesson. \begin{align*}x\end{align*} – intercept the point where a line crosses the \begin{align*}x\end{align*} – axis. It will always have the coordinates \begin{align*}(x, 0)\end{align*}. \begin{align*}y\end{align*} – intercept the point where a line crosses the \begin{align*}y\end{align*} – axis. It will always have the coordinates \begin{align*}(y, 0)\end{align*}. ## Time to Practice Directions: Determine the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercepts of each equation. There will be two answers for each equation. 1. \begin{align*}3x+4y=12\end{align*} 2. \begin{align*}6x + 2y = 12\end{align*} 3. \begin{align*}4x + 5y = 20\end{align*} 4. \begin{align*}4x + 2y = 8\end{align*} 5. \begin{align*}3x + 5y = 15\end{align*} 6. \begin{align*}-2x + 3y = -6\end{align*} 7. \begin{align*}-3x + y = 9\end{align*} 8. \begin{align*}-2x - 2y = 6\end{align*} 9. \begin{align*}7x + 3y = 21\end{align*} 10. \begin{align*}2x + 9y = 36\end{align*} Directions: Look at each graph and identify the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} – intercept of each equation. Each graph will have two answers. Directions: Solve the following problems. 1. Tickets to a hockey game cost60 for the front section and $30 for the rear. A manager wants to take his employees but cannot spend more than$300. How many employees can he take to the front or rear sections?
2. The ASB at a school will be purchasing drinks that they will sell at a school dance. They plan to have 200 drinks available. Bottled water costs $.25 per bottle and soda costs$.35 per can. What is the minimum and maximum that they might spend on the drinks?

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