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High School Core Standards - Algebra I Assessment Anchors

4.07 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-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

A1.2.1.2.1

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

A1.2.1.2.2

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

A1.2.2.1.3a

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.

A1.2.2.1.3b

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.

A1.2.2.1.3c

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.

A1.2.2.1.4

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

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