4. Writing and Graphing Equations in Slope-Intercept Form
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Chapter 2
4.
Writing and Graphing Equations in Slope-Intercept Form
The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line tells you rise over run, or how much the y value changes for every one unit change in the x value. The y-intercept is the point where the line crosses the y-axis, and it tells you the value of y when x is zero.
Show less Show more expand_moreTo graph an equation in slope-intercept form, first find the y-intercept. This is the point where the line crosses the y-axis, and it is easy to find because it is the constant term in the equation. Once you have the y-intercept, you can start plotting points on the line. To do this, choose any value for x and plug it into the equation to find the corresponding value for y. Then, plot the point (x, y) on the graph.
Repeat this process for several different values of x, and you will start to see a line emerge. The more points you plot, the more accurate your graph will be.
The slope of a line can also be found by plotting two points on the line and measuring the rise over run between them. The rise is the change in the y value between the two points, and the run is the change in the x value between the two points. Once you have the rise and run, you can calculate the slope by dividing the rise by the run.
The slope-intercept form of a linear equation is a very useful way to represent lines. It is easy to understand and use, and it makes it easy to graph lines. If you are ever unsure how to graph a line, the slope-intercept form is a great way to get started. Here are some additional tips for graphing equations in slope-intercept form: If the slope is positive, the line will slant upwards from left to right. If the slope is negative, the line will slant downwards from left to right. The steeper the slope, the more quickly the line will slant. If the y-intercept is positive, the line will cross the y-axis above the origin. If the y-intercept is negative, the line will cross the y-axis below the origin.
| Student Learning Objectives: |
|---|
|
| | 15 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Writing and Graphing Equations in Slope-Intercept Form
Slide of 15
There are several forms in which linear equations can be written. A type called the slope-intercept form will be presented and analyzed in this lesson. It will also be shown how to graph an equation written in such form.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Analyzing the Graphs of Linear Functions
Analyze the graphs of different linear functions. Do these lines have anything in common?
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Explore
Finding Similarities Between Lines
Examine the graphs of different linear functions. Are there any similarities between these graphs?
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Discussion
Slope-Intercept Form of a Linear Equation
When trying to find similarities between lines, the first group of lines all have the same y-intercept, while the second group of lines have the same slope. These two characteristics can be used to write an equation of any line.
Concept
Slope-Intercept Form
A linear equation or linear function can be written in the following form called the slope-intercept form.
y=mx+b
In this form, m is the slope and b is the y-intercept. These are the general characteristics or parameters of the line. They determine the steepness and the position of the line on the coordinate plane. Consider the following graph.
This line has a slope of 2 and a y-intercept of 1. The equation of the line can be written in slope-intercept form using these values.
y= mx+ b ⇓ y= 2x+ 1
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Pop Quiz
Identifying Slope-Intercept Form
Consider the given linear equation that represents the relation between two variables x and y. Determine whether the equation is written in slope-intercept form.
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Discussion
Graphing a Linear Function in Slope-Intercept Form
A linear equation or linear function in slope-intercept form has the following form. y=mx+b The slope m and y-intercept b are used to graph the equation. Consider the following function. y=2x-3 There are three steps to follow to graph it.
1
Plot the y-Intercept
expand_more The y-intercept b can be used to find the the first point the line passes through. y=2x-3 ⇔ y= 2x+( - 3)
The y-intercept b is - 3. Plot the point (0, - 3) on a coordinate plane.
2
Use the Slope to Plot the Second Point
expand_more There should be at least two points to draw a line. The second point can be plotted on the coordinate plane by using the slope m. Based on the equation, the slope is 2. y= 2x+( - 3)
This means that the rise is 2 and the run is 1. m=rise/run ⇔ 2=2/1
From the first point (0,- 3), move 1 unit right and 2 units up to plot the second point.
3
Draw a Line Through the Points
expand_more 
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Example
Graphing an Equation and Using the Graph to Solve Problems
Tiffaniqua is driving from her home in New York to visit her sister, who lives in Springfield, Missouri. On the first day of the trip, she traveled 250 miles to Washington DC to pick up her friend Maya.
Together they continued traveling 350 miles per day. The number of miles that Tiffaniqua traveled is represented by the following linear equation. m=350d+250 Here, m is the total number of miles traveled and d is the number of days after Tiffaniqua picked up her friend.
a
Graph the equation on a coordinate plane.
b
Use the graph to find on which day Tiffaniqua traveled a total of 1200 miles.
Answer
a 
b
On the second day of traveling together, the third day of traveling for Tiffaniqua, Tiffaniqua traveled a total of 1200 miles.
Hint
a
Determine the slope and the y-intercept using the given equation and use their values to plot two points on a coordinate plane.
b
Find the x-coordinate of the point on the line with the y-coordinate of 1200.
Solution
a
The equation that represents the situation is given in slope-intercept form. Therefore, the slope and the y-intercept can be used to graph it. To begin, identify their values by comparing the equation with the general equation of a line written in slope-intercept form.
y= mx+ b m= 350d+ 250 As seen above, the slope is 350 and the y-intercept is 250. Now, the y-intercept can be used to find the first point. Since the value of the y-intercept is 250, plot (0,250) on a coordinate plane.
Next, by using the slope, the second point on the line can be determined. Since the slope is 350, move 1 unit right and 350 units up from the first point, then plot the new point.
Finally, by drawing the line through the two plotted points, the graph of the equation can be completed.
Note that since m and d represent the number of miles and days, respectively, they cannot have negative values. This is why the line is only drawn in the first quadrant and is actually a ray.
b
To determine on which day Tiffaniqua passed the mark of 1200 traveled miles, find that value on the y-axis and then identify the corresponding point on the line.
Next, determine the x-coordinate of that point on the line.
From the graph it can be concluded that Tiffaniqua passed the mark of 1200 miles on the second day of traveling together with Maya. Since on the first day Tiffaniqua traveled 250 miles alone to pick up Maya, the second day of traveling together with Maya is her third day of travel in total.
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Example
Finding and Graphing an Equation of a Real-Life Problem
Tiffaniqua's car broke down right after she arrived at her sister's house, so Tiffaniqua decided to rent a new car while visiting her sister. The rental company charges a one-time insurance fee of $20 and an additional $4 per hour.
a
Write an equation in slope-intercept form for the total cost c of renting a car for h hours.
b
Graph the equation.
c
Using the graph, determine how much Tiffaniqua would have to pay for 7 hours of the rental.
Answer
a
c=4h+20
b
Graph:
c
$ 48
Hint
a
Recall what an equation in slope-intercept form looks like. What is the total cost c of renting a car for h hours?
b
First, use the y-intercept and the slope to locate two points that lie on the line.
c
Locate the point on the line where the x-coordinate is 7. What is its y-coordinate?
Solution
a
Start by recalling what an equation in slope-intercept form looks like.
y=mx+b Here, m is the slope and b is the y-intercept. The hourly rate of the renting company is $ 4, so by multiplying that value by the number of hours h, the total hourly amount of renting the car for h hours can be calculated. 4h Furthermore, by adding a one-time insurance fee of $ 20, the total cost c of renting the car can be found. c= 4h+ 20
b
In order to graph the equation found, the slope and the y-intercept will be used. First, plot the y-intercept on a coordinate plane. The y-intercept is 20, which corresponds to the point (0, 20).
Next, the slope 4 will be used to locate a second point. In order to plot this point, move 1 unit right and 4 units up. Points on the line can also be plotted by using multiples of the slope — in this case, for example, 3 units right and 3* 4=12 units up.
Finally, draw a line through these two points. Note that since c and h represent the cost and number of hours the car is rented, respectively, they can only have non-negative values. This means that the line should only be graphed in the first quadrant.
c
To find the cost of renting a car for 7 hours, find 7 on the x-axis and then move vertically to the corresponding point on the line.
Now move horizontally to the y-axis to identify the y-coordinate of this point.
The y-coordinate is 48, which means that when renting a car for 7 hours, Tiffaniqua will have to pay a total of $48.
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Discussion
Writing the Equation of a Line in Slope-Intercept Form From a Graph
The y-intercept b and the slope m of a line must be found to write the equation of the graph of the line in slope-intercept form . y=mx+b Consider the line shown as an example.
There are four steps to writing the equation of this line.
1
Find the y-Intercept
expand_more The y-intercept is the y-coordinate of the point where the line crosses the y-axis. This line intercepts the y-axis at (0,- 4), which means that the y-intercept is - 4.
2
Replace b With the y-Intercept
expand_more The y-intercept can be substituted into the slope-intercept form equation for b.
y=mx+ b ⇓ y=mx - 4
3
Find the Slope
expand_more Next, the slope of the line must be determined. The slope of a line is the ratio of the rise and run of the line.
m=rise/run
The rise is the vertical distance between two points and the run is the horizontal distance. Any two points on the line can be used to find the slope.
For this line, the rise is 6 and the run is 2. Substitute these values into the formula to calculate the slope of the line. m=6/2 ⇒ m=3
4
Replace m With the Slope
expand_more Finally, substitute m= 3 in to the equation from Step 2 to complete the equation.
y= mx-4 ⇓ y= 3x-4
The equation of the line in slope-intercept form is now complete.
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Example
Using a Graph to Write an Equation in Slope-Intercept Form
The evening after Tiffaniqua and Maya arrived at Tiffaniqua's sister's house, the girls decided to pass the evening by putting together puzzles. The number of the remaining puzzle pieces as the girls complete the puzzle is shown in the following graph.
On the diagram, p represents the number of puzzle pieces and t represents time spent completing the puzzle in minutes.
a
Write the equation of the line in slope-intercept form.
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b
How many pieces are in the puzzle?
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c
How many pieces per minute did the girls put together?
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Hint
a
Identify the y-intercept of the line. Then choose a second point on the line and find the slope by analyzing the rise and run between the two points.
b
Which point on the line describes the moment, when the girls have not yet started completing the puzzle?
c
The speed at which the girls put the puzzle together is represented by the slope.
Solution
a
The equation of the line should be written in slope-intercept form, which means that it should have the following form.
y=mx+b However, in this case, instead of x and y, the variables will be t and p, respectively. Use the given graph to determine the values of the slope m and the y-intercept b. To find the y-intercept, locate the point where the line intercepts the y-axis.
The coordinates of the y-intercept are (0,500), so b is equal to 500. This value can be substituted for b in the general equation. p&=mt+ b & ⇓ p&=mt+ 500 To determine the value of the slope, locate a second point on the line and analyze the rise and run between the two points.
As can be seen, for each 10 units to the right, the line goes 100 units down. By dividing both values by 10, it is obtained that for each 1 unit to the right, the line goes 10 units down. 10 units right and 100 units down ⇕ 1 unit right and10 units down The line goes down as it moves to the right, so it has a negative slope. Therefore, m is - 10. By substituting this value for m, the equation of the line can be completed. p&= mt+500 & ⇓ p&= - 10t+500
b 
Note that the point where t=0 is also the y-intercept. In Part A, the y-intercept was found to be 500, so there are 500 puzzles in the game.
To determine how many pieces the puzzle has, think of what the graph represents. The line shows the number of remaining pieces as the girls spend more time putting the puzzle together. Therefore, the point where t=0 represents the total number of puzzle pieces, as the girls have not yet begun putting the puzzle together.
c
The speed at which the girls put the puzzle together is represented by the slope of the line. In Part A, the slope was found to be - 10. This means that the number of remaining puzzle pieces decreased by 10 every minute the girls worked on the puzzle. In other words, the girls completed 10 pieces per minute.
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Discussion
Writing the Equation of a Line in Slope-Intercept Form Using Two Points
The slope m and the y-intercept b of a line must be known to write a linear equation in slope-intercept form. y=mx+b When only two points on the line are known, the following four-step method can be used. For example, the equation of the line that passes through the points (- 4,1) and (8,4) will be written.
1
Find the Slope
expand_more Given two points on a line, the slope of the line can be found by using the Slope Formula. In this case, the coordinates ( - 4, 1) and ( 8, 4) will be substituted in place of (x_1,y_1) and (x_2,y_2), respectively.
m = y_2-y_1/x_2-x_1
SubstitutePoints
Substitute ( - 4, 1) & ( 8,4)
m=4- 1/8-( - 4)
SubNeg
a-(- b)=a+b
m=4-1/8+4
AddSubTerms
Add and subtract terms
m=3/12
CalcQuot
Calculate quotient
m=0.25
The slope m of the line passing through the two points is 0.25.
2
Replace m With the Slope
expand_more Now that the value of the slope is known, it can be substituted for m in the slope-intercept form of an equation.
y= mx+b ⇓ y= 0.25x+b
3
Find b Using a Point
expand_more Next, the y-intercept can be found by substituting either of the given points into the equation and solving for b. In the considered example, (8,4) can be used. Substitute its coordinates into the equation from Step 2 and solve for b.
y=0.25x+b
SubstituteII
x= 8, y= 4
4=0.25( 8)+b
Multiply
Multiply
4=2+b
SubEqn
LHS-2=RHS-2
2=b
RearrangeEqn
Rearrange equation
b=2
Therefore, the y-intercept is 2.
4
Write the Equation
expand_more Lastly, the complete equation in slope-intercept form can be written by substituting the y-intercept into the equation from Step 2. y=0.25x+ b ⇓ y=0.25x+ 2
The equation of the line in slope-intercept form is now complete.
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Example
Using Two Points to Write an Equation and Graph a Line
Tiffaniqua really wants to make a dress that she has been dreaming about for a long time. In her sister's town, there is a very famous fabric store, so she decided to go there for the necessary fabric.
After choosing the perfect fabric, she asked the shop assistant how much 4 and 6 yards of the fabric would cost. The shop assistant told her that the costs would be $3 and $4.50, respectively. (4,3) and (6,4.5) Tiffaniqua realizes that she may want to make a matching accessory, so she might need more fabric. The cost of the fabric will help her decide which, if any, accessory she wants to make.
a
Help Tiffaniqua determine the price for however much fabric she wants to buy by finding the equation of the line. Write the equation in slope-intercept form.
b
Graph the equation.
c
Verify that the points written above lie on the line.
Answer
a
y=0.75x
b
Graph:
c
From the diagram, it can be seen that the points (4,3) and (6,4.5) lie on the line.
Hint
a
Use the Slope Formula to find the slope of the line. Then substitute the coordinates of either of the given points into the equation with the slope.
b
Plot the y-intercept on a coordinate plane. Then use the slope to plot another point on the line.
c
Plot the given points and see if they lie on the line.
Solution
a
To write the equation in slope-intercept form, the slope and the y-intercept of the line should be known. First, find the slope by substituting the coordinates of the two given points into the Slope Formula.
Now that the slope is known, 0.75 can be substituted for m into the equation. y= mx+b ⇓ y= 0.75x+b Next, by substituting either of the two given points into the partly completed equation, the y-intercept b can be found. For example, substitute (4,3) and solve for b.
The y-intercept of the line is 0. Now the equation can be completed. y=0.75x+ 0 ⇕ y=0.75x
b
In order to graph an equation in slope-intercept form, the slope and the y-intercept can be used. First, plot the y-intercept, which in this case is 0, on a coordinate plane.
Next, use the slope of 0.75 to plot the second point that lies on the line. Note that 0.75 can be rewritten as 34. Therefore, by moving 4 units right and 3 units up, the second point can be located.
Finally, draw a line through the two points to obtain the graph of the equation.
Note that since it is not possible to buy a negative number of yards of fabric, the line should only be drawn for non-negative values of x.
c
To verify that the given points lie on the line, plot them on the coordinate plane and see if they are on the line. Recall that the coordinates of the points are (4,3) and (6,4.5).
As can be seen, the points indeed lie on the line.
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Pop Quiz
Practice Writing Equations in Slope-Intercept Form
Given a line, two of its points, or its equation in either standard or point-slope form, write an equation in slope-intercept form.
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Pop Quiz
Identifying the Slope and Y-Intercept of a Line
Determine the slope or y-intercept of a line given its graph, two of its points, or its equation, which may or may not be written in slope-intercept form.
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Closure
Analyzing Two Lines With the Same Slope
In this lesson, writing and graphing linear equations written in slope-intercept form were studied. However, in every case a line or an equation was analyzed individually. Now, consider two lines with the same slope on one coordinate plane. What can be said about these pairs of lines?
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Writing and Graphing Equations in Slope-Intercept Form
Exercise 2.1
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We are asked to find the equation of the line that goes through the given points (0,b) and (1,b+m). Recall the form of a linear equation in slope-intercept form. y=slope* x+y-intercept The y-intercept of any line is the point at which the line crosses the y-axis. Its x-coordinate is always 0. We are given the point (0,b). Since its x-coordinate is 0, we can conclude that b is the y-intercept. y=slope* x+ b Now, let's use the Slope Formula and the given points to find the slope.
Now we have enough information to write a complete equation of the line. y= mx+ b
Let's substitute the point (-1,b-m) into the equation from Part A and check whether it still holds true.
We obtained a true statement as b-m will always equal b-m. Therefore, we know that the point lies on the line no matter the values of b and m.
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Solution
The line y = 3x + m passes through the points (-3,6) and (2b,b). Determine b.
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We are given an equation of a line with an unknown m. To find it, we will substitute the first given point (-3,6) into the equation and solve it for m.
Therefore, the equation of the line is y = 3x + 15. Now, we can substitute the coordinates of the second given point (2b,b) and solve the equation for b.
Therefore, b=-3.
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Solution
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{"type":"choice","form":{"alts":["Yes, during the 26th week.","Yes, during the 27th week.","Yes, during the 28th week.","No."],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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Since the weekly deposit is constantly $120, we can model the described situation with a linear equation. y=mx+b In this form, m is the slope and b is the y-intercept. We know that the initial deposit is $4000. Therefore, the equation's constant is b= 4000. Also, we know that the weekly deposit is $120, which means the slope is m= 120. d(t)= 120t+ 4000
Similarly, we can model Emily's expenses by a linear equation. The first week's expenses are $500. Since the first week is represented by t=0, the equation's constant is b= 500. Additionally, we know that the weekly expenses after the first week are $250, which means the slope is m= 250.
w(t)= 250t+ 500
To find B(t)=d(t)-w(t), we have to substitute d(t) and w(t) with the corresponding expressions and simplify.
Each week, d(t) represents the amount of money in the account and w(t) represents the amount of money taken out of the account. This means that B(t) is the amount of money left in the account after t weeks.
To determine whether Emily will run out of money and if so, when, we need to find when B(t)=0.
B(t) equals 0 when t is about 27. This means that during the 27th week, Emily will run out of money.
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Solution
Consider three points (1,6), (-5,3), and (7,p) on the same line. What is the value of p?
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We are given three points lying on the same line, (3,7), (- 6,1), and (7,p) and asked to find the value of p. To do so, we need to find the equation of the line which passes through the first two points. Then, we will use the x-coordinate 7 of the third point to find its y-coordinate p.
Writing the Equation of the Line
We will write the equation of the line in its slope-intercept form. y=mx+b Here, m is the slope and b the y-intercept. Let's start by finding the slope using the Slope Formula.
The slope of the line is 12. We can partially write its equation. y=1/2x+b To find the y-intercept b, we can substitute either of the given points into this equation. Let's use (1,6).
We can now write the equation of the line. y=1/2x+11/2
Finding the Value of p
To find the value of p, we will use the fact that the line also passes through the point (7,p). Let's substitute 7 and p for x and y, respectively, and solve for p.
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Solution
Writing and Graphing Equations in Slope-Intercept Form
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