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Graphs of Linear Equations

Graph lines presented in ax+by = c form

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Practice Graphs of Linear Equations
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Graphs of Linear Equations

Dana is collecting information about caterpillars for science class. She’s comparing the lengths and widths of several caterpillars. Dana puts the data she has so far into a table. Dana is convinced there is a pattern. Can organize this information as a set of ordered pairs, graph it on a coordinate plane and write an equation that could model this?

 \begin{align*}x\end{align*} (width in cm) \begin{align*}y\end{align*} (length in cm) 2 2 3 4 4 6 5 8 6 10

In this concept, you will learn to graph linear functions on the coordinate plane.

Guidance

A linear function is a specific type of function. You may notice that the word “line” is part of the word “linear”. That fact can help you remember that when a linear function is graphed on a coordinate plane, its graph will be a straight line.

You can represent a function as a set of ordered pairs, through a table, and as an equation. You can also take the information in ordered pairs or in a table and represent a function as a graph.

Let’s look at an example.

The table of values below represents a function on a coordinate plane. On a coordinate plane, graph the linear function that is represented by the ordered pairs in the table below.

 \begin{align*}x\end{align*} \begin{align*}y\end{align*} -4 5 -2 3 0 1 2 -1 4 -3

You can represent the information in this table as a set of ordered pairs \begin{align*}\{(-4,5),(-2,3),(0,1),(2,-1),(4,-3)\}\end{align*}.

Plot those five points on the coordinate plane. Then, connect them as shown below.

Notice that the graph of this linear function is a straight line.

You can also graph a linear function if you are given an equation for that function. This will involve a few more steps. When you have an equation, you can use the equation to create a table. Then, plot several of the ordered pairs in the table and connect them with a line.

Here is another example.

The equation \begin{align*}y = 2x-1\end{align*} is a linear function. Graph that function on a coordinate plane.

First, use the equation to create a table and find several ordered pairs for the function. It is a good idea to use some negative \begin{align*}x\end{align*}-values, some positive \begin{align*}x\end{align*}-values and 0. For example, you can create a table to find the values of \begin{align*}y\end{align*} when \begin{align*}x\end{align*} is equal to -2, -1, 0, 1, and 2.

 \begin{align*}x\end{align*} \begin{align*}y\end{align*} \begin{align*}-2\end{align*} \begin{align*}-5\end{align*} \begin{align*}2(-2)-1=-5\end{align*} \begin{align*}-1\end{align*} \begin{align*}-3\end{align*} \begin{align*}2(-1)-1=-3\end{align*} \begin{align*}0\end{align*} \begin{align*}-1\end{align*} \begin{align*}2(0)-1=-1\end{align*} \begin{align*}1\end{align*} \begin{align*}1\end{align*} \begin{align*}2(1)-1=1\end{align*} \begin{align*}2\end{align*} \begin{align*}3\end{align*} \begin{align*}2(2)-1=3\end{align*}

The ordered pairs shown in the table are \begin{align*}(-2, -5), (-1, -3), (0, -1), (1, 1)\end{align*} and \begin{align*}(2, 3)\end{align*}.

Plot those five points on the coordinate plane. Then connect them as shown below.

Guided Practice

The table below represents inputs and outputs of a linear function. Can you represent this information as ordered pairs, figure out the equation for this function, and then graph the function?

 \begin{align*}x\end{align*} \begin{align*}y\end{align*} \begin{align*}1\end{align*} \begin{align*}5\end{align*} \begin{align*}2\end{align*} \begin{align*}10\end{align*} \begin{align*}3\end{align*} \begin{align*}15\end{align*} \begin{align*}4\end{align*} \begin{align*}20\end{align*}

You can extract information from the table and represent the same information as a set of ordered pairs. The \begin{align*}x\end{align*}-coordinate is the first value and the \begin{align*}y\end{align*}-coordinate is the second value.

Next, looking at the information in the table, you can see that when you multiply the \begin{align*}x\end{align*}-value by 5 you get the \begin{align*}y\end{align*}-value. The rule is multiply \begin{align*}x\end{align*} by 5 to get \begin{align*}y\end{align*}. You can write this as an equation.

You can graph plot the coordinates \begin{align*}\{(1,5),(2,10),(3,15),(4,20)\}\end{align*} and draw a line through them to see the graph.

Examples

Example 1

Is the function above increasing or decreasing?

Notice that as \begin{align*}x\end{align*} increases \begin{align*}y\end{align*} increases. Notice that every time you increase \begin{align*}x\end{align*} by 1, \begin{align*}y\end{align*} will always increase. In this case, \begin{align*}y\end{align*} increases by two every time \begin{align*}x\end{align*} increases by 1.

The answer is the function is increasing.

Example 2

In the point \begin{align*}(-3, 4)\end{align*} is the \begin{align*}x\end{align*}-value positive or negative?

The \begin{align*}x\end{align*}-value is the first value in the coordinate. It is a negative number.

The answer is the \begin{align*}x\end{align*}-value is negative.

Example 3

In \begin{align*}(-6, -7)\end{align*}, which value is \begin{align*}y\end{align*}-value?

The \begin{align*}y\end{align*}-value is the second value in a coordinate, and it is equal to -7.

The answer is the \begin{align*}y\end{align*}-value is -7.

Credit: Graham Wise

Remember Dana’s project comparing the lengths and widths of caterpillars?

She’s put the data collected so far in a table (shown below). Can you plot these points and write the equation that models this information?

 \begin{align*}x\end{align*} (width in cm) \begin{align*}y\end{align*} (length in cm) 2 2 3 4 4 6 5 8 6 10

First, represent this information as a set of ordered pairs so that you can plot the points \begin{align*}\{(2,2),(3,4),(4,6),(5,8),(6,10)\}\end{align*}.

Now, can you see a pattern in the table and then write the rule that describes it?

Notice that as \begin{align*}x\end{align*} increases by 1, \begin{align*}y\end{align*} increases by 2. So, you know that \begin{align*}2x\end{align*} is involved in the equation. But \begin{align*}y\end{align*} is not quite \begin{align*}2x\end{align*}. It is \begin{align*}2x -2\end{align*}.

So the equation that models this information is \begin{align*}2x -2\end{align*}.

Next, plot the points on the coordinate plane and draw a line through them. The graph is shown below.

Explore More

The information in the table represents points from a linear function. Plot the points in the table on a coordinate plane, and then draw a straight line through them to graph each function. Then identify the rule (equation) for the function.

1.

 Input Output 1 4 2 5 3 6 4 7

2.

 Input Output 2 4 3 6 4 8 5 10

3.

 Input Output 1 3 2 6 4 12 5 15

4.

 Input Output 9 7 7 5 5 3 3 1

5.

 Input Output 8 12 9 13 11 15 20 24

6.

 Input Output 3 21 4 28 6 42 8 56

7.

 Input Output 2 5 3 7 4 9 5 11

8.

 Input Output 4 7 5 9 6 11 8 15

9.

 Input Output 5 14 6 17 7 20 8 23

10.

 Input Output 4 16 5 20 6 24 8 32

Vocabulary Language: English

Cartesian Plane

Cartesian Plane

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.
Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Function Rule

Function Rule

A function rule describes how to convert an input value ($x$) into an output value ($y$) for a given function. An example of a function rule is $f(x) = x^2 + 3$.
Linear Function

Linear Function

A linear function is a relation between two variables that produces a straight line when graphed.
Slope

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$