We have seen that we can identify characteristics (slope and the y-intercept) of linear functions seen in tables and graphs, and that values can be substituted into equations to create a table needed to plot the function on a coordinate plane. Now we will explore their equations more closely.

Exploration

Drag the m and b sliders and observe how they change the graph.

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How do different values of m affect the graph?
How do different values of b affect the graph?

Linear functions can be written in a form that uses the slope as a coefficient of x since it is a rate of change and the y-value of the y-intercept as a constant. The slope-intercept form of an equation is

\displaystyle y=mx+b

\bm{m}

is the slope

\bm{b}

is the y-intercept

The value of m affects the steepness of the line, or the slope.

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Each line shown has the same y-intercept of 1, but different slopes.

The absolute value of the slope changes the steepness of the line, while the sign of the slope changes the direction the line slopes.

Slopes with larger absolute values appear steeper.

The value of b affects the y-intercept, or where the line crosses the y-axis.

A graph with one solid line and 2 dashed lines. Ask your teacher for more information.

The solid line shown is y = x. The line passes through the y-axis at the origin, \left(0,\,0\right), so b is originally 0.

If the line shifts up 2 units, the slope is still 1 but the y-intercept is now 2, so the equation of the new line is y=x+2.

If the line shifts down 3 units, the slope is still 1 but the y-intercept is now -3, so the equation of the new line is y=x-3.

These 'shifts' are called vertical translations.

The slope, m, and the y-intercept, b, can be used not only to write equations but also to graph a line. We use the following steps to graph in slope-intercept form:

Plot b from the equation as the y-intercept.
Identify the vertical change and horizontal change from m in the equation. If the slope is written as an integer, we can write m =\dfrac{m}{1}.
Starting at the y-intercept, use the slope to count the vertical and horizontal change and plot a point where you end up.
Draw a straight line through the points, extending past the points to fill the coordinate plane.

Note that we are able to reverse both directions indicated by the slope in order to graph points to the left of the y-intercept.

Examples

Example 1

Consider the equation y=-4x+5.

State the slope and y-intercept of the equation.

Worked Solution

Create a strategy

We can use the general slope-intercept form: y=mx+b, where m is the slope and b is the y-intercept.

Apply the idea

The equation y=-4x+5 is in the form y=mx+b:

\displaystyle m	\displaystyle =	\displaystyle -4	Identify the slope
\displaystyle b	\displaystyle =	\displaystyle 5	Identify the y-intercept

Complete the table of values for the given equation:

x	-1	0	1	2
y

Worked Solution

Create a strategy

Substitute each x-value from the table into the given equation and evaluate to find y.

Apply the idea

For x=-1:

\displaystyle y	\displaystyle =	\displaystyle -4\cdot (-1)+5	Substitute -1 for x
	\displaystyle =	\displaystyle 9	Evaluate

Similarly, if we substitute the other values of x, (x=0,\, x=1,\, x=2), into y=-4x+5, we get:

x	-1	0	1	2
y	9	5	1	-3

Example 2

Consider the following graph of a line:

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

Worked Solution

Create a strategy

The slope is the ratio of vertical change to horizontal change or \dfrac{\text{change in }y}{\text{change in }x}.

Choose two points on the graph and draw a slope triangle to help find the vertical and horizontal change.

Apply the idea

Choosing the points: (-5,\,2) and (0,\,1) we see that to get from (-5,\,2) to (0,\,1) we need to move 1 unit down and 5 units to the right.

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The vertical change is -1 and the horizontal change is +5.

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

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.

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+b	Slope-intercept form
\displaystyle y	\displaystyle =	\displaystyle -\dfrac{1}{5}x+1	Substitute the values of the slope and y-intercept

Example 3

Graph the line y=3x+2 using its slope and y-intercept.

Worked Solution

Create a strategy

Determine the slope, y-intercept, and a starting point using the slope-intercept form of the line.

Apply the idea

The slope is 3, and the y-intercept is 2.

This means that one of the points on y=3x+2 is (0,\,2). We can plot this point and move across 1 and up 3 to get the next point:

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Then we can draw a line through these two points:

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Reflect and check

A table also could be used to graph the line by substituting chosen values of x in to the equation y=3x+2 and determining the corresponding y-values before graphing.

For the x-values

x	-2	-1	0	1
y

\displaystyle 3(-2)+ 2	\displaystyle =	\displaystyle -4	Substituting -2
\displaystyle 3(-1)+ 2	\displaystyle =	\displaystyle -1	Substituting -1
\displaystyle 3(0)+ 2	\displaystyle =	\displaystyle 2	Substituting 0
\displaystyle 3(1)+ 2	\displaystyle =	\displaystyle 5	Substituting 1

x	-2	-1	0	1
y	-4	-1	2	5

The creates a table of values that can be graphed as ordered pairs. This would create the same line graphed by the slope and y-intercept.

Example 4

Given the table of a linear function:

x	-4	-2	0	2	4
y	-2	1	4	7	10

Find the slope.

Worked Solution

Create a strategy

The slope of a table is written as a simplified ratio of the change in y to the change in x.

Apply the idea

The x-values of \left\{-4,-2,0,2,4\right\} have a change of +2.

The y-values of \left\{-2,1,4,7,10\right\} have a change of +3.

The slope of the table is \dfrac{3}{2}.

Identify the y-intercept of the table and write the equation of the line represented by the table in slope-intercept form.

Worked Solution

Create a strategy

The y-intercept is the ordered pair \left(0,\,y\right) in a table.

The slope from part (a) will be used as m and the y-intercept of the table will be used as b in the slope-intercept form y=mx+b.

Apply the idea

The point (0,4) is in the table, so b=4, and the slope from part (a) means m=\dfrac{3}{2}.

The equation of the line represented by the table is y=\dfrac{3}{2}x+4.

Reflect and check

Graphing the points from the table would also let us find the equation of the line since we could use the graph to find the slope and y-intercept.

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We see a y-intercept at (0,4) and a change of 3 squares up and 2 squares right, making the slope \dfrac{3}{2}. This would give the same values of m and b to write the equation y=\dfrac{3}{2}x+4.

Example 5

The graph of y=-\dfrac{1}{3}x is shown.

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The graph will be shifted 4 units down. Write the new equation of the line after the translation.

Worked Solution

Create a strategy

If a line only shifts up or down, the y-intercept of the graph will change, but the slope will remain the same. In an equation, keep m the same, but find the new b.

Apply the idea

Moving 4 units down corresponds to -4 being added to b. The equation y=-\dfrac{1}{3}x has a b of 0, making the full original slope-intercept form y = -\dfrac{1}{3}x + 0.

\displaystyle y	\displaystyle =	\displaystyle -\frac{1}{3}x + 0 - 4
\displaystyle y	\displaystyle =	\displaystyle -\dfrac{1}{3}x-4

The new equation of the line will be y = -\dfrac{1}{3}x - 4.

Graph the equation from part (a).

Worked Solution

Create a strategy

Use the m and b from the new slope-intercept form to graph the y-intercept and use slope to find additional points.

Apply the idea

The slope-intercept form is y=-\dfrac{1}{3}x-4, so the line will cross the y-axis at \left(0,\,-4\right) and move 1 unit down and 3 units right between points.

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Reflect and check

Another way to graph the new equation would be to make a table of points from the original graph of y=-\dfrac{1}{3}x and subtract 4 from each y-value. The table below shows pairs from y=-\dfrac{1}{3}x and subtracting 4 from each y-value.

x	-6	-3	0	3	6
y	2-4	1-4	0-4	-1-4	-2-4

After simplifying the y-values, the new table matches the points seen in the graph.

x	-6	-3	0	3	6
y	-2	-3	-4	-5	-6

Idea summary

A linear equation is said to be in slope-intercept form when it is expressed as

\displaystyle y=mx+b

\bm{m}

is the slope.

\bm{b}

is the y-intercept.

The y-intercept will shift a line above the x-axis if b is positive and below the x-axis if b is negative.

When given an equation in slope-intercept form, we can graph the line by plotting the y-intercept as the first point. Then we can use the slope to find additional points before drawing our line.

Outcomes

8.PFA.3

The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

8.PFA.3a

Determine how adding a constant (b) to the equation of a proportional relationship y = mx will translate the line on a graph.

8.PFA.3b

Describe key characteristics of linear functions including slope (m), y-intercept (b), and independent and dependent variables.

8.PFA.3c

Graph a linear function given a table, equation, or a situation in context.

8.PFA.3d

Create a table of values for a linear function given a graph, equation in the form of y = mx + b, or context.

8.PFA.3e

Write an equation of a linear function in the form y = mx + b, given a graph, table, or a situation in context.

8.PFA.3f

Create a context for a linear function given a graph, table, or equation in the form y = mx + b.

3.06 Slope-intercept form

Ideas

Slope-intercept form

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Outcomes

8.PFA.3

8.PFA.3a

8.PFA.3b

8.PFA.3c

8.PFA.3d

8.PFA.3e

8.PFA.3f

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