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The slope and intercept are key characteristics of a line. They can be used to write the equation of the line and are very useful when graphing the line. This lesson will focus on writing equations in slope-intercept form and graphing them.

Catch-Up and Review

Here is a recommended reading before getting started with this lesson.

Challenge

Deriving the Equation for a Nonproportional Relationship

In a nonproportional relationship, the line passes through the point or the intercept. Use the Slope Formula to derive the equation for a nonproportional relationship.
Assume that the other point the line passes through is
Explore

Analyzing Graphs of Equations

The applet shows the graphs of different equations. Analyze these lines.
Graph of an Equation in the Form of y=mx+b
How do the coefficient of and the constant term affect position of the line? Try to find a relation between the line and its equation.
Discussion

Points Where a Line Crosses the Axes

The intercepts of a line provide information about the position of the line in the coordinate plane. They can be used to identify the equation of a line from known points or to graph a line from its equation.

Concept

Intercept

The intercept of a line is the coordinate of the point where the line crosses the axis. The intercept of a line is the coordinate of the point where the line crosses the axis. The intercept of an equation is also known as its initial value.

Graph of the line -1.5*x+3 with y-intercept at (0,b) and x-intercept at (a,0)
Pop Quiz

Finding the Intercepts of a Line

The applet shows the graphs of different lines. Identify the intercepts of these lines.

Finding the Intercepts of a Line
Discussion

Linear Equation

A linear equation is an equation with at least one linear term and any number of constants. No other types of terms may be included. Linear equations in one variable have the following form, where and are real numbers and

Linear equations in two variables have the form below, where and are real numbers and and

Assume that either or is zero. In this case, the equation above becomes a linear equation in one variable. The graph of a linear equation is a line.
Discussion

Slope-Intercept Form

A linear equation can be written in the following form called the slope-intercept form.

In this form, is the slope and is the 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.

The graph of the linear function y=2*x+1 with a slope of 2 (2 rise, 1 run) and a y-intercept at (0, 1)

This line has a slope of and a intercept of The equation of the line can be written in slope-intercept form using these values.

Pop Quiz

Identifying Slope-Intercept Form

The following applet generates linear equations. Those equations represent the relation between two variables and Determine whether the generated equation is in slope-intercept form.

Random generator creates equations written in different forms
Discussion

Graphing a Linear Equation in Slope-Intercept Form

A linear equation in slope-intercept form has the following form.
The slope and intercept are used to graph the equation. Consider the following function.
There are three steps to follow to graph it.
1
Plot the Intercept
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The intercept can be used to find the the first point the line passes through.
The intercept is Plot the point on a coordinate plane.
The point (0,-3) plotted on a coordinate plane
2
Use the Slope to Plot the Second Point
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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 Based on the equation, the slope is
This means that the rise is and the run is
From the first point move unit right and units up to plot the second point.
Two points (0,-3) and (1,-1) on the coordinate plane
3
Draw a Line Through the Points
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Finally, use a straightedge to draw a line through both points.

A line with the equation y=2x-3 passing through the points (1, -1) and (0, -3).

This line is the graph of

Example

Following the Fish Population

Jordan is investigating the fish population in Grassy Lake. The number of fish in the lake was counted as in

Fishes in a Lake.jpg

She finds out that the population increases by after each year. The following equation represents the fish population in the lake.
In this equation, is the number of fish in the lake after years.
a Graph the equation on a coordinate plane.
b Interpret what the slope and the intercept represent.

Answer

a
Graph of y=120x+60
b Slope: Increase in the fish population per year

intercept: Number of fish in the year

Hint

a Begin by identifying the slope and intercept.
b Interpret the given information based on the slope and intercept found in Part A.

Solution

a The equation that represents the fish population in the lake is in slope-intercept form. In this form, the slope and intercept are the important characteristics of the line of the equation. They determine the steepness and the position of the line on a coordinate plane. Begin by identifying these characteristics.
The slope is and the intercept is This means that the line crosses the axis at
Y-Intercept of y=120x+60

Next, move unit to the right and units up from the intercept to plot another point.

Y-Intercept of y=120x+60 and a second point at (1,180)

Finally, the graph of the equation can be competed by drawing a line passing through these two points.

Graph of y=120x+60

Note that the number of years passed or the number of fish cannot be negative. Therefore, only the first quadrant of the coordinate plane is shown.

b Recall the slope and the intercept of the line.
In this context, the slope represents the increase in the fish population per year. The intercept represents the number of fish in
Discussion

Writing the Equation of a Line in Slope-Intercept Form From a Graph

The intercept and the slope of a line must be found to write the equation of the graph of the line in slope-intercept form .
Consider the line shown as an example.
A line on a coordinate plane
There are four steps to writing the equation of this line.
1
Find the Intercept
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The intercept is the coordinate of the point where the line crosses the axis. This line intercepts the axis at which means that the intercept is

The y-intercept of the line is identified on the graph
2
Replace With the -Intercept
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The intercept can be substituted into the slope-intercept form equation for
3
Find the Slope
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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.
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.
Rise and run between the two chosen points are determined
For this line, the is and the is Substitute these values into the formula to calculate the slope of the line.
4
Replace With the Slope
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Finally, substitute in to the equation from Step to complete the equation.
The equation of the line in slope-intercept form is now complete.
Example

Chasing the Big Cats

Jordan turns her attention to the cheetah population over years.

Cheetah.jpg

She finds a graph on the Internet that shows the number of cheetahs over years since

Graph of Cheetah Population Over Years

Jordan wants to represent the graph algebraically to interpret it further.

a Write the equation of the line in slope-intercept form.
b Interpret what the slope and the intercept represent.
c Use the equation from Part A to estimate in which year cheetahs will go extinct if the current trend continues.

Answer

a
b Slope: Decrease in the cheetah population per year

intercept: Number of cheetahs in the year

c

Hint

a Begin by determining the slope and intercept of the line.
b How many cheetahs were there in How does the number of cheetahs change over the years?
c In the equation, represents the number of cheetahs after years.

Solution

a The intercept and the slope of the line need to be determined to write the equation of the line in slope-intercept form. Begin by remembering slope-intercept form of a line.
In this form, is the slope and is the intercept. The intercept of a line is the coordinate of the point where the line crosses the axis.
Graph of Cheetah Population Over Years
In the given graph, the line intercepts the axis at This means that the intercept of the line is Substitute for into the slope-intercept form of an equation.
The slope of a line is the ratio of the rise and run of the line.
Any two points on the line can be chosen to determine the rise and run. One point can be the one where the line intercepts the axis. The other point can be for simplicity.
Graph of Cheetah Population Over Years
The rise is and the run is Substitute these values into the ratio and calculate the slope.
The slope of the line is Finally, the equation of the line can be completed by substituting for
b Remember the equation from Part A.
The slope of the line is negative, which suggests that the number of cheetahs decreases by per year. The intercept represents that there were cheetahs in
c In the equation, represents the number of cheetahs after years. Substitute for into the equation and solve for to find after how many years cheetahs will go extinct, according to the graph.
The equation suggests that cheetahs will go extinct after years. The specific year can be found by adding to the starting year
This means that if the current trend continues, cheetahs will go extinct in
Discussion

Writing the Equation of a Line in Slope-Intercept Form Using Two Points

The slope and the intercept of a line must be known to write a linear equation in slope-intercept form.
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 and will be written.
1
Find the Slope
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Given two points on a line, the slope of the line can be found by using the Slope Formula. In this case, the coordinates and will be substituted in place of and respectively.
The slope of the line passing through the two points is
2
Replace With the Slope
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Now that the value of the slope is known, it can be substituted for in the slope-intercept form of an equation.
3
Find Using a Point
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Next, the intercept can be found by substituting either of the given points into the equation and solving for In the considered example, can be used. Substitute its coordinates into the equation from Step and solve for
Therefore, the intercept is
4
Write the Equation
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Lastly, the complete equation in slope-intercept form can be written by substituting the intercept into the equation from Step
The equation of the line in slope-intercept form is now complete.
Example

Reading the Research Rag

Jordan is interested in endangered animals. She reads a magazine to learn what can be done to save these animals.

Magazine Cover.jpg

Suppose Jordan reads the same number of pages each day. The following table shows the number of pages left in the magazine after days.

Number of Days
Number of Pages
Which of the following equations represents the information in the table?

Hint

Write an equation for the given situation in slope-intercept form using the Slope Formula.

Solution

The given equations are in slope-intercept form.
In this form, is the slope and is the intercept. These two characteristics need to be determined to write an equation in slope-intercept form. Begin by representing given information in terms of points
Number of Days
Number of Pages
Point
The slope of the relationship can be found by using the Slope Formula.
Use any two of the points to calculate the slope!
The slope of the relationship is
Next, any of the points can be substituted into the above equation to find the intercept of the relation.
The intercept of the relationship is Finally, the equation can be completed.
This equation can also be rearranged as follows.
Pop Quiz

Writing Equations in Slope-Intercept Form

Write an equation in slope-intercept form using the given line, two points, or equation.

Applet that generates a line, two of its points, or an equation
Pop Quiz

Determining the Slope and y-Intercept of a Line

Determine the slope or intercept the given line based on its graph, points, or its equation. Be careful — the equation may or may not be written in slope-intercept form!

Applet that randomly generates a graph of a line, two its points, or an equation
Closure

Equation of a Nonproportional Relationship

In a nonproportional relationship, the line passes through the point The coordinate of this point is the intercept of the line. The Slope Formula will be used to derive the equation for a nonproportional relationship.
Two points are needed to use the slope formula. If one of the points is let the other point be This point could be any point that the line passes through. Then, substitute these points into the formula and solve for