2.18 Slope-intercept form

Guiding questions

1. How do the slope and $y$y-intercept work? How do they impact the graph of a line?
2. Discuss with a partner how to change the equation of a line to fit a pair of given data points.

The values of $m$m and $b$b mean specific things. Explore for yourself what these values do by exploring on this interactive.

Let's have a quick recap of what we know about straight lines on the $xy$xy-plane so far.

• They have a slope, a measure of how steep the line is.
• They can be increasing (positive slope) or decreasing (negative slope).
• They can be horizontal (zero slope).
• They can be vertical (slope is undefined).
• They have an $x$x intercept, a $y$y intercept, or both an $x$x and a $y$y intercept.
• The slope can be calculated using $\frac{\text{rise }}{\text{run }}$rise run ​ or $\frac{y_2-y_1}{x_2-x_1}$y2​−y1​x2​−x1​​.
• They have an equation of the form $y=mx+b$y=mx+b.

Slope

So what you will have found is that the $m$m value affects the slope.

• If $m<0$m<0, the slope is negative and the line is decreasing
• if $m>0$m>0, the slope is positive and the line is increasing
• if $m=0$m=0 the slope is $0$0 and the line is horizontal
• Also, the larger the value of $m$m the steeper the line
• In practical situations, the slope can be interpreted as the rate of change.

Y-Intercept

We also found that the $b$b value affects the $y$y-intercept.  

• If $b$b is positive then the line is vertically translated (moved) up.
• If $b$b is negative then the line is vertically translated (moved) down.
• In practical situations, the y-intercept can be interpreted as the initial value.

Creating equations from information given about the line

To create an equation of the form $y=mx+b$y=mx+b, we need 2 pieces of information, the slope and $y$y-intercept. If we know the slope and the $y$y-intercept, we can instantly write down the equation. However, we may not be told this information explicitly, so will need to read it from a graph or calculate them given two points.

Practice questions

Question 1

Question 2

Graphing from slope-intercept form

To graph any liner relationship you only need two points that are on the line.  You can use any two points from a table of values, or substitute in any two values of $x$x into the equation and solve for corresponding $y$y-value to create your own two points.  Often, using the y-intercept and slope is the easiest ways to sketch the line.

Sketch from the slope and the y-intercept

When we are given an equation in slope-intercept form, we are basically given one point and the ability to find a second, so we are all set!

1. Start by plotting the $y$y-intercept from the equation
2. Identify the rise and run from the $m$m in the equation. If the slope is given as an integer, remember that we can write $m$m as $\frac{m}{1}$m1​. 
3. Put your pencil on the y-intercept and count "rise" squares up or down and then "run" squares to the right. Repeat as needed.
4. Connect the two or more points to form a line, extending beyond the two points.

Here is a little more detail on step 2.

For a slope of $4$4, move $1$1 unit across and $4$4 units up.

For a slope of $-3$−3, move $1$1 unit across and $3$3 units down.

For a slope of $\frac{1}{2}$12​, move $1$1 unit across and $\frac{1}{2}$12​ unit up.

For example, plot the line with equation $y=-2x+4$y=−2x+4, hence slope of $-2$−2 and $y$y-intercept of $4$4.  

Start with the point, ($y$y intercept of $4$4)

Step out the slope, (-$2$2 means $2$2 units down)

Draw the line

Practice questions

Question 3

Question 4

Question 5