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4.04 Slope-intercept form

Lesson

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 $\frac{y_2-y_1}{x_2-x_1}$y2y1x2x1.
  • 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.

 

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. 

 

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

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

  1. Give your answer in slope-intercept form.

Question 2

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

  1. Slope $\editable{}$
    $y$y-value of the $y$y-intercept $\editable{}$

 

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

The Slope-Intercept Form

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


$y=mx+b$y=mx+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.

Practice questions

Question 3

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 4

Consider the following graph of a line.

Loading Graph...

  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 5

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

  1. Loading Graph...

Outcomes

I.F.IF.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior.

I.F.IF.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

I.F.IF.7.a

Graph linear functions and show intercepts.

I.F.LE.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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