4. Linear Functions

Lesson

- How do the slope and $y$
`y`-intercept work? How do they impact the graph of a line? - 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 $xy$`x``y`-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 $\frac{y_2-y_1}{x_2-x_1}$
`y`2−`y`1`x`2−`x`1. - They have an equation of the form $y=mx+b$
`y`=`m``x`+`b`.

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.

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.

Did you know?

In the equation $y=mx+b$`y`=`m``x`+`b`, the terms $m$`m` and $b$`b` are called parameters. A parameter is a placeholder for a value that indicates certain characteristics of a function, such as its slope or $y$`y`-intercept.

To create an equation of the form $y=mx+b$`y`=`m``x`+`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.

Find the equation of a line which has a slope of $-5$−5 and crosses the $y$`y`-axis at $7$7.

Give your answer in slope-intercept form.

State the slope and $y$`y`-value of the $y$`y`-intercept of the equation, $y=2x+3$`y`=2`x`+3

**Slope**$\editable{}$ **$y$**`y`-intercept$\editable{}$

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.

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!

- Start by plotting the $y$
`y`-intercept from the equation - 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 $\frac{m}{1}$`m`1. - Put your pencil on the y-intercept and count "rise" squares up or down and then "run" squares to the right. Repeat as needed.
- 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`=−2`x`+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 |

The Slope-Intercept Form

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

$y=mx+b$`y`=`m``x`+`b`

where $m$`m` is the slope and $b$`b` is the $y$`y`-intercept

Our equations may not always be given in this form so we may need to rearrange the equation to solve for the variable y.

Consider the equation $y=-8-\frac{2x}{3}$`y`=−8−2`x`3.

State the slope of the line.

State the value of the $y$

`y`at the $y$`y`-intercept.

Consider the following graph of a line.

Loading Graph...

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

What is the $y$

`y`value of the $y$`y`-intercept of the line shown in the graph?What is the equation of the line? Write your answer in slope-intercept form.

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

- Loading Graph...

Create, interpret, and/or use the equation, graph, or table of a linear function.

Translate from one representation of a linear function to another (i.e., graph, table, and equation).

Write or identify a linear equation when given • the graph of the line Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

Write or identify a linear equation when given • two points on the line, or Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

Write or identify a linear equation when given • the slope and a point on the line. Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

Determine the slope and/or y-intercept represented by a linear equation or graph.