1. Linear Equations, Tables, and Graphs

2. A Linear Function is… From a Graph:
A linear function is a function (or rule) that has a constant rate of change
From a Graph:
Linear Functions will graph as one straight line

3. A Linear Function is… From an Equation:
A linear function is a function (or rule) that has a constant rate of change
From an Equation:
The equation will not have exponents greater than 1 on the variables
The equation will not have a variable in the denominator of a fraction
Examples of linear functions:
f(x) = 2x + 1
f(x) = -1/2x – 5
y = -4x – 2.8
Examples of non-linear functions:
y = 3x2 + 2x + 5
f(x) = -5x3
y = 8/x

4. A Linear Function is… From a Table:
A linear function is a function (or rule) that has a constant rate of change
From a Table:
The rate of change will be the same every time
Constant rate of change because the ys change by +12 for each change in 1 x
Constant rate of change because the ys change by +4 for each change in 1 x +1
+12 +4 +1
slope
slope
+12
+12 = 12 +4 +1 +1
+4 = 4 +1 +1
+.5 +6
It’s even ok here because it went up by 6 for .5 of an x, therefore will go up a full 12 for a full 1 x +4 +1

5. Slope Refresher: Slope = rate of change = steepness of a line
Slope is a fraction of the change in y over change in x.
Slope is represented by the letter m
Vertical line has NO slope (it’s undefined)
Horizontal line has a slope zero (it’s 0)
change in y
change in x
rise
run or

6. Formula for slope:

7. Find the slope of a line passing through points (-2,4) and (3,6)
To Find Slope (rate of change) from Points:
Find the slope of a line passing through points (-2,4) and (3,6)
(x1,y1)
(x2,y2)
1. Assign the first point as (x1,y1) and assign the second point as (x1,y1)
2. Use slope formula
6 - 4 2
3. Plug values in and simplify
m = =
3 - -2 5

8. Find slope from a graph. down 3 -3 over 2 2 “Count it out”
Find at least 2 clear points on the line. Start on the left
Count how many units up or down
Count how many units over to the right
“Count it out”
down 3
over 2
down 3 -3
A slope of -3/2 means that for every time y goes down 3, x moves over 2 to the right
over 2 2

9. Be Careful – What if each grid line isn’t just one unit??
Always check the intervals on the graph! Each line doesn’t have to be only one unit!
To find the slope, start with one point and count how many units it goes up and count how many units it goes over. Make this a fraction and reduce if necessary.
Over 2
Up 15
This line goes up 15 units and over 2 units.
The slope is 15/2 = 7.5

10. Make basic graphs from a function…
Graph the function f(x) = 2x – 3.
Start by making a table of values.
Graph the ordered pairs.
Connect the dots. x
f(x) -2 -1 1 2 -7 -5 -3 -1 1

11. Write a function from a graph.
Step 1: Find where the line will cross the y-axis.
(0, 2)
Step 2: Is the line going up or down?
Up = positive slope
Down = negative slope
Step 3: Calculate the slope.
Find two points on the line.
Count the number of spaces up or down (numerator).
Count the number of spaces to the right (denominator).
Reduce the fraction = this is the slope of the line.
Step 4: Write the function:
f(x) = mx + b
f(x) = (slope)(x) + (where it crosses the y-axis)
f(x) = 2x + 2 2 1
2 = 2 1

12. Write a function from a graph.
Step 1: Find where the line will cross the y-axis.
(0, 0)
Step 2: Is the line going up or down?
Up = positive slope
Down = negative slope
Step 3: Calculate the slope.
Find two points on the line.
Count the number of spaces up or down (numerator).
Count the number of spaces to the right (denominator).
Reduce the fraction = this is the slope of the line.
Step 4: Write the function:
f(x) = mx + b
f(x) = (slope)(x) + (where it crosses the y-axis)
f(x) = -1/3x 1 3 -1 3

13. Change in miles Change in time(hours)
RATE OF CHANGE in a real life situation:
At 3:00 p.m. a car leaves the city. By 5:00 it has traveled 90 miles. Find its average speed.
Change in miles
Change in time(hours)
Notice you have two pairs:
(3,0) and (5,90)
Where x is time and y is miles.

14. Renting a Moving Van…
A rental company charges a flat fee of $30 to rent a moving van plus $0.25 per mile. Write an equation in slope-intercept form modeling this situation.
Start with the format f(x) = mx + b.
What is the repeating amount in this problem?
Substitute that number in for “m.”
f(x) = 0.25x + b
What is the one time charge?
Substitute that number in for “b.”
f(x) = 0.25x + 30
This is the equation that models this situation.

15. Renting a Moving Van…
Use the equation you wrote about the moving van and complete table.
f(x) = 0.25x + 30
Miles (x) 25 50 75
100
Cost (y)
36.25
42.50
48.75
55.00
y = 0.25(25) + 30
y =
y = 36.25
y = 0.25(50) + 30
y =
y = 42.50
y = 0.25(75) + 30
y =
y = 48.75
y = 0.25(100) + 30
y =
y = 55.00

16. Speeding Rabbit From a Word Problem: D = rt
Suppose a rabbit travels 256 feet in 4 seconds. Assume that this is a constant speed.
Situations with a constant rate of change will be linear functions.
Write a linear equation in two variables to represent the situation.
In science you learned the relationship between distance and time. What is that equation?
D = rt
What values were given in this story that we could substitute into this formula?
Distance = 256 feet
Time = 4 seconds
We can solve for “r” rate.

17. D = rt 256 = r(4) Plug in the values we know for D and r
256 = r(4) Solve the equation by dividing both sides by 4.
64 = r
r = 64 feet/second (that is his rate/constant speed)
The rabbit’s personal equation to determine how far he could go in a certain amount of time is: d = 64t.

18. Use the equation to make predictions about the distance the rabbit traveled over various intervals of time.
How can an equation help make predictions about distance the rabbit traveled in various amounts of time?
We could make a table of values.
The x column would correspond to time the rabbit ran (the input).
The y column would correspond to the distance the rabbit ran AFTER the time had passed (the output).
We could substitute (plug in) the times into the rabbit’s formula D = 64t.
Time (t)
Distance (D) 64 1 2
128 3
192 4
256 5
320
The equation (function) allows us to PREDICT the distance the rabbit has traveled after any duration of time.