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3. Linear Functions
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Algebra 1

3.03 Slope-intercept form

Lesson
Practice

Slope-intercept form

As explored in the previous lesson, transformations on f\left(x\right)=x allow us to shift, dilate, and reflect linear functions. We can write the equations of lines in slope-intercept form:

\displaystyle y=mx+b
\bm{m}
slope
\bm{b}
y-intercept

The main advantage of slope-intercept form is that we can easily identify two key features: the slope and the y-intercept directly from the equation.

Exploration

Consider the following representations

A table with 5 rows and 2 columns titled x and f of x. The data is as follows: 0, 2; 1, 3.5; 2, 5; 8, 14, and 15, 24.5.
Table
A line in quadrant one of the coordinate plane. Labeled Depth in inches along the y-axis and Time in hours along the x-axis. The line passes through (0,2), (1,3.5), (2,5), (8,14), (13, 21.5).
Graph
A text saying The initial depth of snow is 2 inches. The snow falls at a rate of 1.5 inches per hour.
Description
An equation. f of x equals 1.5 x plus 2.
Equation
  1. What do the table, description, graph, and equation representations have in common?
  2. What is different about each representation?

Given information about a linear function, we can write the equation in slope-intercept form.

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\,\\\,Consider this graph of a leaky faucet that drips at a rate of 4 cups per hour. It has already dripped 15 cups of water.

Let x represent the number of hours and y represent the number of cups of water that have leaked from the faucet.

First, identify or solve for the slope, which represents the m in y=mx+b. In the graph, we can use a slope triangle to see the slope is \dfrac{4}{1}. We can also see the rate of change in the context is 4.

Next, identify or solve for the y-intercept, which represents the b in y=mx+b.

In the graph, the y-intercept is at (0, 15), so b=15. From the context, we can see the initial value of the leaked water is 15 cups.

Lastly, rewrite y=mx+b, substituting the solved values of m and b. Since we have found m = 4 and b=15, we can write the equation: y = 4x+15

Slope-intercept form is especially helpful when we want to graph a linear function. The graph of a line represents the set of points that satisfies the equation of a line.

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\,\\\,Consider the equation y = 2x-1.

First, identify the value of b, and plot this as the y-intercept. Since the y-intercept is -1, plot a point at (0,-1).

Then, identify the value of m, and the value of the rise and the run. From the y-intercept, use the rise and run to create slope triangles to plot additional points.

Each point on the graph is a solution of the equation, so every point satisfies y=2x-1.

Examples

Example 1

Consider the following graph of a line:

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a

What is the slope of the line shown in the graph?

Worked Solution
Create a strategy

The slope of a line is equal to the vertical 'rise' divided the by horizontal 'run'. It is the ratio of the vertical change to the horizontal change.

Apply the idea

Considering two points from the graph: (0,1) and (5,0), from (0,1) requires moving 1 unit down and 5 units to the right.

The slope is \dfrac{-1}{5} or -\dfrac{1}{5}.

b

What is the y-value of the y-intercept of the line shown in the graph?

Worked Solution
Create a strategy

The y-intercept is the point where the line intersects the y-axis.

Apply the idea
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Looking at the graph, the line intersects the y-axis at point (0,1). Thus, the y-value of the y-intercept is y=1.

c

Write the equation of the line in slope-intercept form.

Worked Solution
Create a strategy

Substitute the values of the slope and y-intercept in the slope-intercept form of the equation of a line.

Apply the idea
\displaystyle y\displaystyle =\displaystyle mx+bStart with the slope-intercept form
\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{5}x+1Substitute m=-\dfrac{1}{5} and b=1
Reflect and check

We can check our equation using points from the graph. Since each point must be a solution to our equation, choose any point on the line.

For this example, choose (5, 0).

\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{5}x+1Substitute x=5 and y=0
\displaystyle 0\displaystyle =\displaystyle -\dfrac{1}{5}\cdot 5+1Simplify
\displaystyle 0\displaystyle =\displaystyle 0

Since it is true that 0=0, this point satisfies our equation.

d

Find the domain and range of the line.

Worked Solution
Create a strategy

The domain of a line in the graph represents all possible x-values that the line covers.

The range of a line in the graph represents all possible y-values that the line covers.

Apply the idea

Domain: -\infty < x < \infty

Range: -\infty < y < \infty

Reflect and check

We can see from the graph that the line extends without any breaks or endpoints along the x and y-axis, confirming that the domain includes all real numbers.

The domain in interval notation is:

(-\infty,\,\infty)

The domain in set notation is:

\{x\,|\,x \in \Reals\}

The range in interval notation is:

(-\infty,\,\infty)

The range in set notation is:

\{ y\,|\,y \in \Reals \}

Example 2

Find the equation in slope-intercept form for:

a

A line with a slope of \dfrac{1}{2} and a y-intercept of -3.

Worked Solution
Create a strategy

Subtitute the values of the slope and y-intercept in the slope-intercept form

Apply the idea
\displaystyle y\displaystyle =\displaystyle mx+bStart with the slope-intercept form
\displaystyle y\displaystyle =\displaystyle \dfrac{1}{2}x-3Substitute m=\dfrac{1}{2} and b=-3
b

A line with a slope of -3 and passes through the point (2,3).

Worked Solution
Create a strategy

Subtitute the slope and point in the slope-intercept form, and solve for the y-intercept before rewriting.

Apply the idea
\displaystyle y\displaystyle =\displaystyle mx+bStart with the slope-intercept form
\displaystyle 3\displaystyle =\displaystyle -3(2)+bSubstitute y=3,\,m=-3,\, and x=2
\displaystyle 3\displaystyle =\displaystyle -6+bEvaluate the multiplication
\displaystyle 9\displaystyle =\displaystyle bAdd 6 to both sides
\displaystyle y\displaystyle =\displaystyle -3x+9Rewrite using slope-intercept form
c

A line that passes through the points (-4, 5) and (8,8).

Worked Solution
Create a strategy

Subtitute the given points into the slope formula to find slope. Use the slope and one point to solve for the y-intercept before rewriting in slope-intercept form.

Apply the idea
\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Start with the slope formula
\displaystyle m\displaystyle =\displaystyle \dfrac{8-5}{8-(-4)}Subtitute (x_1,y_1) = (5,-4) and (x_2,y_2)=(8,8)
\displaystyle m\displaystyle =\displaystyle \dfrac{3}{12}Evaluate
\displaystyle m\displaystyle =\displaystyle \dfrac{1}{4}Simplify
\displaystyle 8\displaystyle =\displaystyle \dfrac{1}{4}(8) + bChoose (x_1,y_1) or (x_2,y_2) to subsitute with m into y=mx+b
\displaystyle 8\displaystyle =\displaystyle 2+bEvaluate the multiplication
\displaystyle 6\displaystyle =\displaystyle bSubtract 2 from both sides
\displaystyle y\displaystyle =\displaystyle \dfrac{1}{4}x+6Rewrite using slope-intercept form

Example 3

Write each of the following equations in slope-intercept form.

a

-4y=12-8x

Worked Solution
Create a strategy

Slope-intercept form is y=mx+b, where m is slope and b is the y-intercept.

Apply the idea

Each term will be divided by -4 and simplified.

\displaystyle -4y\displaystyle =\displaystyle 12-8xOriginal equation
\displaystyle \frac{-4y}{-4}\displaystyle =\displaystyle \frac{12}{-4}+ \frac{-8x}{-4}Divide each term by -4
\displaystyle y\displaystyle =\displaystyle 2x-3Simplify and reorder terms

The slope-intercept form of -4y=12-8x is y=2x-3.

Reflect and check

Slope is the simplified ratio of \dfrac{\text{vertical change}}{\text{horizontal change}}. By looking at the graph, we could see a change of \dfrac{-8}{-4} by counting down and left between points. This rate of change matches the simplified ratio of \dfrac{2}{1}, or a slope of 2.

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b

7x-14y=-28

Worked Solution
Create a strategy

Isolate the y and rearrange terms to write in slope-intercept form, y=mx+b.

Apply the idea

The x term will be moved first then divide by the coefficient of y.

\displaystyle 7x-14y\displaystyle =\displaystyle -28Original equation
\displaystyle 7x-14y-7x\displaystyle =\displaystyle -28-7xSubtract 7x from each side
\displaystyle -14y\displaystyle =\displaystyle -7x-28Write with x term before constant
\displaystyle \frac{-14y}{-14}\displaystyle =\displaystyle \frac{-7x}{-14}+\frac{-28}{-14}Divide each term by -14
\displaystyle y\displaystyle =\displaystyle \frac{1}{2}x+2Simplify

The slope-intercept form of 7x-14y=-28 is y=\dfrac{1}{2}x+2.

Reflect and check

Other forms of equations will create the same graphs as slope-intercept form, but may highlight different information.

By creating a table of values and graphing, we see that 7x-14y=-28 has the same slope and y-intercept as y=\dfrac{1}{2}x+2.

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c

y+2=-\dfrac{2}{3}(x-6)

Worked Solution
Create a strategy

Slope-intercept form is y=mx+b. Distribute any values outside parentheses and rearrange terms to isolate y.

Apply the idea

We will first distribute the -\dfrac{2}{3} then isolate the y to write the equation in slope-intercept form.

\displaystyle y+2\displaystyle =\displaystyle -\frac{2}{3}(x-6)Original equation
\displaystyle y+2\displaystyle =\displaystyle -\frac{2}{3}x+4Distribute -\dfrac{2}{3} and multiply
\displaystyle y+2-2\displaystyle =\displaystyle -\frac{2}{3}x+4-2Subtract 2
\displaystyle y\displaystyle =\displaystyle -\frac{2}{3}x+2Simplify

The slope-intercept form of y+2=-\dfrac{2}{3}(x-6) is y=-\dfrac{2}{3}x+2.

Example 4

A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of water.

The table of values shows the linear relationship of the amount of water remaining in the tub, y, after x minutes.

\text{Time in minutes } \left(x\right)0123
\text{Water remaining in gallons } \left(y\right)30282624
a

Determine the linear equation in slope-intercept form that represents this situation.

Worked Solution
Create a strategy

We can pick two points to calculate the rate of change for the slope. Then we can recognize that the y-intercept is given in the table of values.

Apply the idea

Find the slope using the values \left(0,\,30\right) and \left(1,\,28\right):

\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Slope formula
\displaystyle m\displaystyle =\displaystyle \dfrac{28-30}{1-0}Substitute \left(x_1,y_1\right)=\left(1,30\right) and \left(x_2,y_2\right)=\left(1,28\right)
\displaystyle m\displaystyle =\displaystyle -2Evaluate

Notice that the initial value, or y-intercept is given in the table as \left(0,\,30\right).

Writing in the form y=mx+b the equation that represents this situation is y=-2x+30.

Reflect and check

If we had not noticed that the y-intercept was given, we could have substituted in any pair of values for x and y, and solved for b.

b

Draw the graph of this linear relationship with a clearly labeled scale. Only show the viable solutions.

Worked Solution
Create a strategy

We can't have a negative time (x \geq 0) and we should end the graph when the tub is empty (y=0). To plan our graph, we need to find when the tub is empty.

Apply the idea

To find when the tub is empty:

\displaystyle y\displaystyle =\displaystyle -2x+30Original equation
\displaystyle 0\displaystyle =\displaystyle -2x+30Substitute y=0
\displaystyle -30\displaystyle =\displaystyle -2xSubtraction property of equality
\displaystyle 15\displaystyle =\displaystyle xDivision property of equality

This tells us that we have the restriction that 0 \leq x\leq 15 and 0 \leq y\leq 30, which will help us choose the appropriate axes and scale.

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\text{Time in minutes}
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We know that the bathtub begins with 30 gallons which is represented by the y-intercept at (0,30) and that the slope of the linear equation is -2 which means that the tub loses 2 gallons of water every minute until there is no water in the tub at 15 minutes.

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Reflect and check

We can also graph the slope by using the idea of \dfrac{\text{rise}}{\text{run}}. Since the slope is -2, we can write it as a fraction \dfrac{-2}{1} and identify that the change in y-values (or rise) is -2 and that the change in x-values (or run) is 1.

c

Find the domain and range.

Worked Solution
Create a strategy

The domain of a function represents all possible x-values.

The range of a function represents all possible y-values.

Consider values that may fit the pattern but do not fit the context.

Apply the idea

Domain: 0 \leq x \leq 15

Notice that even though negative x-values could fit the pattern, it does not make sense with the context to have a negative amount of time.

Range: 0 \leq y \leq 30

In the context, water is never added so it can never be above the starting amount of 30 gallons. Continuing the pattern, the y-value will never drop below empty, or 0 gallons.

d

Describe how the graph would change if, instead, there were initially 40 gallons of water in the tub, and it emptied at 2.5 gallons per minute.

Worked Solution
Create a strategy

The initial value is represented by the y-intercept on the graph. The rate of change is represented by the slope of the graph. We should consider how these are changing from the original question.

Apply the idea

The original question had an initial value of 30 gallons, and the new scenario has an initial value of 40 gallons. This means the new y-intercept will be higher on the y-axis.

The original question had a rate of change of -2 as it was decreasing at 2 gallons per minute. The new scenario has a rate of change of -2.5 as it is emptying at 2.5 gallons per minute. This means the slope will be steeper in the new scenario.

We can see this on the graph:

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Reflect and check

Notice that despite starting with 10 extra gallons of water, the tub with 40 gallons of water only takes 1 more minute to empty than the 30-gallon tub, because it is emptying at a faster rate. This is reflected in the graph as a steeper slope. The second function is decreasing at a greater rate than the first.

Example 5

Which of the following has the higher y-intercept?

A
The line with a slope of 4 that crosses the y-axis at (0,\,6).
B
The line given by the equation y=x+4.
Worked Solution
Create a strategy

For an equation of the form y=mx+b, b is the value of the y-intercept.

Apply the idea

In option A, the y-intercept is 6.

In option B, the equation is of the form y=mx+b where b=4. So the y-intercept is 4

Since 6 \gt 4, option A has the higher y-intercept.

Idea summary

The slope-intercept form of a line is:

\displaystyle y=mx+b
\bm{m}
slope
\bm{b}
y-intercept

Slope-intercept form is useful when we know, or want to know the slope of the line and the y-intercept of the line.

Outcomes

A.F.1

The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.1a

Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.

A.F.1c

Write equivalent algebraic forms of linear functions, including slope-intercept form, standard form, and point-slope form, and analyze and interpret the information revealed by each form.

A.F.1di

Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: i) given the graph of a line;

A.F.1dii

Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: ii) given two points on the line whose coordinates are integers;

A.F.1diii

Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: iii) given the slope and a point on the line whose coordinates are integers;

A.F.1f

Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.

A.F.1g

For any value, x, in the domain of f, determine f(x), and determine x given any value f(x) in the range of f, given an algebraic or graphical representation of a linear function.

A.F.1h

Compare and contrast the characteristics of linear functions represented algebraically, graphically, in tables, and in contextual situations.

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