5.9: Graphing Functions
- Page ID
- 129560
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After completing this module, you should be able to:
- Graph functions using intercepts.
- Compute slope.
- Graph functions using slope and -intercept.
- Graph horizontal and vertical lines.
- Interpret graphs of functions.
- Model applications using slope and -intercept.
In this section, we will expand our knowledge of graphing by graphing linear functions. There are many real-world scenarios that can be represented by graphs of linear functions. Imagine a chairlift going up at a ski resort. The journey a skier takes travelling up the chairlift could be represented as a linear function with a positive slope. The journey a skier takes down the slopes could be represented by a linear function with a negative slope.
Graphing Functions Using Intercepts
Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points. At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the
The table below lists where each of these lines crosses the - and -axis. Do you see a pattern? For each line, the -coordinate of the point where the line crosses the -axis is zero. The point where the line crosses the -axis has the form and is called the -intercept of the line. The -intercept occurs when is zero. In each line, the -coordinate of the point where the line crosses the -axis is zero. The point where the line crosses the -axis has the form and is called the -intercept of the line. The -intercept occurs when is zero.
Figure | The line crosses the at: | Ordered Pair for this Point | The line crosses the at: | Ordered Pair for This Point |
---|---|---|---|---|
Figure (a) | 3 | 6 | ||
Figure (b) | 4 | |||
Figure (c) | 5 | |||
Figure (d) | 0 | 0 | ||
General Figure |
Example 5.70
Finding - and -Intercepts
Find the -intercept and -intercept on the (a) and (b) graphs in Figure 5.73.
- Answer
In Figure 5.73, the graph crosses the
Your Turn 5.70
Example 5.71
Graphing a Function Using Intercepts
Find the intercepts of
- Answer
Let
to find they = 0 y = 0 -intercept, and letx x to find thex = 0 x = 0 -intercept.y y 2 x + y = 8 2 x + y = 8 2 x + y = 8 2 x + y = 8 To find the -intercept, letx x .y = 0 y = 0 2 x + 0 = 8 2 x + 0 = 8 To find the -intercept, lety y .x = 0 x = 0 2 ( 0 ) + y = 8 2 ( 0 ) + y = 8 Simplify. 2 x = 8 x = 4 2 x = 8 x = 4 Simplify. y = 8 y = 8 The -intercept is:x x ( 4 , 0 ) ( 4 , 0 ) The -intercept is:y y ( 0 , 8 ) ( 0 , 8 ) Plot the intercepts to get the graph in Figure 5.75.
Your Turn 5.71
Computing Slope
When graphing linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. In mathematics, the measure of the steepness of a line is called the slope of the line. To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices of the triangle and one side is horizontal and one side is vertical. Next, we measure or calculate the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run.
We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope (Figure 5.76) is a rate of change.
FORMULA
To calculate slope
where the rise measures the vertical change and the run measures the horizontal change.
The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of plumbing pipes, and the steepness of stairs are all applications of slope. As you ski or jog down a hill, you definitely experience slope.
Example 5.72
Finding the Slope from a Graph
Find the slope of the line shown in Figure 5.77.
- Answer
Step 1: Locate two points on the graph whose coordinates are integers, such as
Figure 5.78.( 0 , 5 ) Step 2: Count the rise; since it goes down, it is negative. The rise is −2.
Step 3: Count the run. The run is 3.
Step 4: Use the slope formula
substitute the values of the rise and run.m = rise run m = rise run m = − 2 3 m = − 2 3 The slope of the line is
.− 2 3 − 2 3 The solution is
decreases by 2 units asy y increases by 3 units.x x
Your Turn 5.72
Sometimes we will need to find the slope of a line between two points when we don’t have a graph to measure the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing. First, we need to introduce some algebraic notation.
We have seen that an ordered pair (
Let’s review how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points
On the graph, we count the rise of 3 and the run of 5. Notice on the graph that that (
We have shown that
FORMULA
To find the slope of the line between two points (
Example 5.73
Finding the Slope of the Line Using Points
Use the slope formula to find the slope of the line through the points (−2, −3) and (−7, 4).
- Answer
We’ll call (−2, −3) point 1 and (−7, 4) point 2.
Step 1: Use the slope formula:
m = y 2 − y 1 x 2 − x 1 m = y 2 − y 1 x 2 − x 1 Step 2: Substitute the values:
m = 4 − ( − 3 ) − 7 − ( − 2 ) m = 4 − ( − 3 ) − 7 − ( − 2 ) Step 3: Simplify:
m = 7 − 5 = − 7 5 m = 7 − 5 = − 7 5 Step 4: Verify the slope on the graph shown in Figure 5.81.
Your Turn 5.73
Graphing Functions Using Slope and y y -Intercept
We have graphed linear equations by plotting points and using intercepts. Once we see how an equation in slope-intercept form and its graph are related, we will have one more method we can use to graph lines. Review the graph of the equation
The vertical and horizontal lines in the graph show us the rise is 1 and the run is 2, respectively.
Substituting into the slope formula:
The
Look at the slope and
When a linear equation is solved for
Example 5.74
Finding the Slope and y y -Intercept of a Line
Identify the slope and
y = − 4 7 x − 2 y = − 4 7 x − 2 x + 3 y = 9 x + 3 y = 9
- Answer
- We compare our equation to the slope-intercept form of the equation.
Step 1: Write the slope-intercept form of the equation of the line.y = m x + b y = m x + b
Step 2: Write the equation of the line.y = − 4 7 x − 2 y = − 4 7 x − 2
Step 3: Identify the slope.m = − 4 7 m = − 4 7
Step 4: Identify the -intercept.y y y - intercept is ( 0 , − 2 ) y - intercept is ( 0 , − 2 ) - When an equation of a line is not given in slope-intercept form, our first step will be to solve the equation for
.y y
Step 1: Solve for .y y x + 3 y = 9 x + 3 y = 9
Step 2: Subtract from each side.x x 3 y = − x + 9 3 y = − x + 9
Step 3: Divide both sides by 3.3 y 3 = − x + 9 3 3 y 3 = − x + 9 3
Step 4: Simplify.y = − 1 3 x + 3 y = − 1 3 x + 3
Step 5: Write the slope-intercept form of the equation of the line.y = m x + b y = m x + b
Step 6: Write the equation of the line.y = − 1 3 x + 3 y = − 1 3 x + 3
Step 7: Identify the slope.m = − 1 3 m = − 1 3
Step 8: Identify the -intercept.y y .y - intercept is ( 0 , 3 ) y - intercept is ( 0 , 3 )
- We compare our equation to the slope-intercept form of the equation.
Your Turn 5.74
Example 5.75
Graphing the Slope and y y -Intercept
Graph the line of the equation
- Answer
The equation is in slope-intercept form
.y = m x + b y = m x + b y = − x + 4 . y = − x + 4 . Step 1: Identify the slope and
-intercept.y y
,m = − 1 m = − 1 -intercept isy y .( 0 , 4 ) ( 0 , 4 ) Step 2: Plot the
Figure 5.83).y - Identify the rise over the run.
m = − 1 = − 1 1 m = − 1 = − 1 1 - Count out the rise and run to mark the second point.
rise , run 1− 1 − 1
- Identify the rise over the run.
Your Turn 5.75
Graphing Horizontal and Vertical Lines
Some linear equations have only one variable. They may have just
( |
||
−3 | 1 | |
2 | ||
3 |
Plot the points from the table and connect them with a straight line (Figure 5.84). Notice that we have graphed a vertical line.
What is the slope? If we take the two points
Using the slope formula we get:
The slope is undefined since division by zero is undefined. We say that the slope of the vertical line
What if the equation has
( |
||
0 | 4 | |
2 | 4 | |
4 | 4 |
In Figure 5.85, we have graphed a horizontal line passing through the
What is the slope? If we take the two points
Example 5.76
Graphing A Vertical Line
Graph:
- Answer
The equation has only one variable,
Figure 5.86).x x = 2 x = 2 x x y y ( ,x x )y y 2 1 ( 2 , 1 ) ( 2 , 1 ) 2 2 ( 2 , 2 ) ( 2 , 2 ) 2 3 ( 2 , 3 ) ( 2 , 3 )
Your Turn 5.76
Example 5.77
Graphing A Horizontal Line
Graph:
- Answer
The equation
Figure 5.87).y = − 1 y = − 1 y = − 1 x x y y ( ,x x )y y 0 − 1 − 1 ( 0 , − 1 ) ( 0 , − 1 ) 3 − 1 − 1 ( 3 , − 1 ) ( 3 , − 1 ) − 3 − 3 − 1 − 1 ( − 3 , − 1 ) ( − 3 , − 1 )
Your Turn 5.77
The table below summarizes all the methods we have used to graph lines.
Interpreting Graphs of Functions
An important yet often overlooked area in algebra involves interpreting graphs. Oftentimes in math classes, students are given mathematical functions and can make graphs to represent them. But the interpretation of graphs is a more applicable skill to the real world. Being able to “read” a graph—understanding its domain and range, what the intercepts mean, and what the slope (or curve) means— that's a real-world skill.
Example 5.78
Interpreting a Graph
In Figure 5.88 the
- Answer
is the( 0 , 0 ) ( 0 , 0 ) - andx x -intercept and represents Juan at home before his bike ride. The distance from home is 0 miles and 0 minutes have passed.y y - In the first 30 minutes, the slope is
and indicates Juan is traveling 1 mile for every 5 minutes. Between 30 and 60 minutes, the slope is 0 and indicates that he’s not riding the bike (the distance is not increasing). Then between 60 and 90 minutes, the slope is1 5 1 5 again. Finally, after 90 minutes the slope is1 5 1 5 meaning Juan is getting 4 miles closer to home every 15 minutes.− 4 15 , − 4 15 , - Answers will vary. Juan left his house for a bike ride. After 30 minutes, he was 6 miles from home and he stopped for ice cream at his local ice-cream truck. He enjoyed his ice cream for 30 minutes. He then jumped back on his bike and rode to his friend’s house. He arrived there 30 minutes later. His friend’s house was 12 miles from his home. His friend was not home so he immediately turned around and quickly rode home in 45 minutes.
Your Turn 5.78
Modeling Applications Using Slope and y y -Intercept
Many real-world applications are modeled by linear equations. We will review a few applications here so you can understand how equations written in slope-intercept form relate to real-world situations. Usually when a linear equation model uses real-world data, different letters are used for the variables instead of using only
Example 5.79
Converting Temperature
The equation
- Find the Fahrenheit temperature for a Celsius temperature of 0°.
- Find the Fahrenheit temperature for a Celsius temperature of 20°.
- Interpret the slope and
-intercept of the equation.F F - Graph the equation.
- Answer
- Find the Fahrenheit temperature for a Celsius temperature of 0°.
Find whenF F .C = 0 C = 0 F = 9 5 ( 0 ) + 32 F = 9 5 ( 0 ) + 32 Simplify. F = 32 F = 32
- Find the Fahrenheit temperature for a Celsius temperature of 0°.
Find |
|
Simplify. | |
Simplify. |
Even though this equation uses
The slope
The
We will need to use a larger scale than our usual. Start at the
Your Turn 5.79
Example 5.80
Calculating Driving Costs
Sam drives a delivery van. The equation
- Find Sam’s cost for a week when he drives 0 miles.
- Find the cost for a week when he drives 250 miles.
- Interpret the slope and
-intercept of the equation.C C - Graph the equation.
- Answer
- Find Sam’s cost for a week when he drives 0 miles.
Find whenC C = 0.d d C = 0.5 ( 0 ) + 60 C = 0.5 ( 0 ) + 60 Simplify. C = 60 C = 60 Sam’s costs are $60 when he drives 0 miles.
- Find Sam’s cost for a week when he drives 0 miles.
Find |
|
Simplify. |
Sam’s costs are $185 when he drives 250 miles.
The slope, 0.5, means that the weekly cost,
We’ll need to use a larger scale than usual. Start at the
So to graph the next point go up 50 from the intercept of 60 and then to the right 100. The second point will be (100, 110).
Your Turn 5.80
Check Your Understanding
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Section 5.8 Exercises
The equation /**/P = 31 + 1.75w/**/ models the relation between the amount of Tuyet’s monthly water bill payment, /**/P/**/, in dollars, and the number of units of water, /**/w/**/, used.
Bruce drives his car for his job. The equation /**/R = 0.575m + 42/**/ models the relation between the amount in dollars, /**/R/**/, that he is reimbursed and the number of miles, /**/m/**/, he drives in one day.
Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation /**/S = 400 + 0.15c/**/ models the relation between her weekly salary, /**/S/**/, in dollars and the amount of her sales, /**/c/**/, in dollars.
Costa is planning a lunch banquet. The equation /**/C = 450 + 28g/**/ models the relation between the cost in dollars, /**/C/**/, of the banquet and the number of guests, /**/g/**/.