1. Summer Assignment Review
Graphing Linear Equations

2. Linear Equations: Multiple representations
Equations, tables of values, and graphs can all be used to describe a relationship between unknown values. Which ones we use depends on what we are given, and what our goals are. [Or of course, what the instructions tell us we must do to score points.]
Lets explore one relationship using each method of display.

3. The Relationship
Your Mom is pulling out all the stops in an effort to get you to work to your full potential in school. She offers you a deal. She will give you $20 if you have at least a B average for the first marking period, plus an extra $5 for every A you earn. Let’s assume that you are willing to put forth the effort (because that is all it really takes!) and you do have an average of at least a B. We can figure out how much money you can pocket each marking period.

4. The Equation
First we should define our unknowns (declare our variables)
We can write a word-equation describing the situation to guide ourselves into the actual equation
Lastly we can translate the situation into an algebraic equation.
y = The amount of money earned
x = The number of A’s earned
The amount of money earned = $20 for the average + $5 for each A
y = x
Or , to make it fit into slope intercept form…
y = 5x + 20

5. The Table Of Values
The table of values can be created if given the relationship, the graph, or the equation.
The table will only represent a selection of possible ‘solutions’.
If you have a hard time creating an equation, you can fill a table of values first in an attempt to better see the relationship.
You could then use that table of values to draw the graph, and use the graph to find the equation x y 20 1 25 2 30

6. The Graph
You can create the graph simply from a table of values. Each row represents a coordinate pair.
{ (0, 20), (1, 25), (2, 30) } 10 20 30 40 2 4 6 8 y x x y 20 1 25 2 30
Every point on the line shows a possible solution. (6,45) represents 6 A’s earning $45.

7. Slope
The slope of the line is a measurement that describes the steepness of a line.
If the line slants down from left to right
the slope is negative
If the line slants up from left to right
the slope is positive.
The closer the slope is to zero, the closer the line is to horizontal.
[horizontal lines have a slope of 0]
The closer the slope gets to ±∞, the closer the line is to vertical.
[vertical lines have an undefined slope]

8. X and Y intercepts The y-intercept. The x-intercept
The y-intercept is the value of y when x=0.
The y-intercept is the value of y when the line crosses the y-axis.
The y-intercept is the ‘b’ in slope intercept form
( y = mx + b)
The x-intercept
The x-intercept is the value of x when y=0.
The x-intercept is the value of x when the line crosses the y-axis.
The time we most often find the x-intercept is when we want to graph but are given an equation that is not easily transformed into slope intercept form.

9. Getting The slope and intercepts from an Equation
Slope can be identified from an equation only when the y variable is alone on one side of the equation. When y is alone, the slope is the coefficient of the x variable.
When the y variable is alone on one side of the equation, the y-intercept is the constant on the other side.
y = 5x + 2
Slope = 5
5-2x = y
Slope = -2
y = -3x + 2
y-int = 2
y = 5x
y-int = 0

10. Getting The slope and intercepts from an Equation
The y-intercept can always be found by setting x=0 and solving for y.
The x-intercept can always be found by setting y=0 and solving for x.
3x + 9y = 54
y-intercept : set x = 0
3(0)+9y = 54
y-intercept at (0,6)
9y = 54
y = 6
3x + 9y = 54
x-intercept : set y = 0
x-intercept at (18,0)
3x + 9(0) = 54
3x = 54
x = 18

11. Getting the Slope and Intercepts from a table of values
In a table of values, the y-intercept is the value of y where x=0.
In a table of values, the x-intercept is the value of x where y=0.
If you cant find these values you could use the data to write an equation and calculate the intercepts.
(18,0) X Y 18 3 5 6 -6 8
(0,6)

12. Getting the slope from a graph
You can calculate the slope by finding two accurate points on the graph and using the slope formula.
Slope m =
You can also determine the sign of the slope by sight and count the rise over run (the scales of the x and y axis are important here)x
What is the slope between the points A (3,-6) and B (-2,9) ? x y 5
From one point to the next you traveled up two units and right three.
Slope =

13. Getting the intercepts from the graph
The y-intercept is the value of y where the line crosses the y- axis
The x-intercept is the value of x where the line crosses the x-axis x y 5
The y-intercept (0,2)
The x-intercept (-3,0)

14. Practice
Given the following situation, create an equation. Using that equation create a table of values. Using the equation or the table of values to sketch the graph of the relationship.
Markies Bakery Charges $20 for the first 12 cupcakes in a special order. They then charge $0.75 for each additional cupcake. How much money might someone pay for their specialty cupcakes?

15. The Equation Declare the variables.
y = The total cost of the cupcake order
x = The additional cupcakes (after the first 12)
Write a sentence describing the relationship.
The cost of the cupcake order = $20 for the first 12 and $0.75 per additional cupcake
Translate the sentence into an algebraic equation.
y = x
Or ….

16. The Table of Values Our equation X Y -4 4 84 8
If possible, choose x values that will work easily with the equation. 17 20 23 26

17. The Graph y Chose your scale carefullyx y
Chose your scale carefully 32
Counting consistently on the x and y axis allows you to count boxes when using slope 24 16 8
The table of values contains points that would have to be estimated on this scale 8 16 24 32
We can determine slope and y-intercept from the equation.
Slope =
(0,20)
Plot the point (0,20) and use the slope to find additional points.