# 4.08 Graphing lines

Lesson

## Slope and y-intercept

We have already learned about slope and $y$y-intercept.

#### Exploration

Use the applet below to refresh your memory on slope and $y$y-intercept and try to answer the following questions:

1. How do the slope and $y$y-intercept impact the graph of a line?
2. What part of the equation represents the slope? What about he $y$y-intercept?

### Slope

The applet above highlights that the $m$m value affects the steepness of the line, or 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 farther the value of $m$m is from $0$0 the steeper the line.

### Y-intercept

We also found that the $b$b value affects the $y$y-intercept, or where the line crosses the $y$y-axis.

• If $b$b is positive then the line crosses the $y$y-axis above the origin.
• If $b$b is negative then the line crosses the $y$y-axis below the origin.
• If $b$b is $0$0, or not in the equation at all, then the line crosses the $y$y-axis at the origin.

Did you know?

In the equation $y=mx+b$y=mx+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.

## Graphing from slope-intercept form

The Slope-Intercept Form

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

y=m*x+b

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

To graph any line you only need two points that are on the line.  When we are given an equation in slope-intercept form, we are given one point (the $y$y-intercept) and the ability to find a second point (using the slope), so we are all set!

1. Start by plotting the $y$y-intercept from the equation
2. Identify the rise and run from $m$m in the equation. If the slope is given as an integer, remember that we can write $\frac{\text{rise }}{\text{run }}=\frac{m}{1}$rise run =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.

#### Worked example

##### Question 1

Graph the line with equation $y=-2x+4$y=2x+4.

Think: The slope is $-2$2 and the $y$y-intercept is $4$4. How do we put that information on the graph?

Do:

 Graph the $y$y-intercept of $4$4 by plotting the point $\left(0,4\right)$(0,4) Use the slope to plot a second point. $\text{slope }$slope  $=$= $\frac{\text{rise }}{\text{run }}$rise run ​ $=$= $\frac{-2}{1}$−21​ From the $y$y-intercept, move down 2 units and to the right 1 unit. Connect the points and extend the line

Our graphs may not always be in this form so we may need to rearrange the equation to isolate the variable $y$y  (that means $y$y is on one side of the equation and everything else is on the other side).

## Equations of horizontal and vertical lines

Recall that lines can also be either horizontal or vertical.  These types of lines will not look like slope-intercept form.  However, they do follow another type of special pattern.

Horizontal lines are the set of all points with a fixed $y$y value.  They are parallel to the $x$x-axis and have equations of the form $y=a$y=a, where $a$a is a real number.  Recall that horizontal lines have a slope of zero.

The horizontal line $y=2$y=2

Reflect: What happens to the equation $y=mx+b$y=mx+b if you substitute $0$0 for $m$m?  How does this relate to the equation of a horizontal line?

Vertical lines are the set of all points with a fixed $x$x value.  They are parallel to the $y$y-axis and have equations of the form $x=a$x=a, where $a$a is a real number.  Recall that vertical lines have a slope that is undefined.

The vertical line $x=1$x=1

#### Practice questions

##### Question 2

State the slope and $y$y-intercept of the equation, $y=8x-1$y=8x1

1.  Slope $\editable{}$ $y$y - intercept $\editable{}$

##### Question 3

State the slope and $y$y-intercept of the equation $y=-2x$y=2x

1.  Slope $\editable{}$ $y$y - intercept $\editable{}$

##### Question 4

Consider the equation $y=-8-\frac{2x}{3}$y=82x3.

1. State the slope of the line.

2. State the value of the $y$y at the $y$y-intercept.

##### Question 5

Consider the following graph of a line.

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

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

3. What is the equation of the line? Write your answer in slope-intercept form.

##### Question 6

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

##### Question 7

Plot the line $x=4$x=4 on the coordinate plane.