1. Linear Functions
The output of function “f” when x is used as the input
Independent
Variable
Slope: the difference in “f” for consecutive values of x
y-intercept: The starting value

2. A relation where each input pairs with exactly one output
Function
A relation where each input pairs with exactly one output
Independent variable: the input to a function, often “x”
Dependent variable: the output of a function, often “y”
f(x) is a way to write “y” in an equation like y = mx + b

3. Any function that when graphed forms a non-vertical line
Linear Functions
Any function that when graphed
forms a non-vertical line
Note: A vertical line like x = 2 is still linear, but it is not a function, because any output, “y,” can be paired with it.
Key Properties:
Consecutive x’s have the same change in “y.” Here, each x, lowers y by 2
The value of y when x = 0 is the y-intercept. Here, it is y = 0 or (0, 0)

4. Display the inputs and outputs of a function
T - Tables
Display the inputs and outputs of a function
Chose a selection of x values (often these are given)
Evaluate the function for each of the x – values
Example: Complete the T-Table for f(x) = 2x + 7
for x = 2, 3, 4 and 5 x
f(x) = 2x + 7 2
2(2) + 7 = 11 3
2(3) + 7 = 13 4
2(4) + 7 = 15 5
2(5) + 7 = 17
T-Tables can be converted to ordered pairs by pairing the x’s with the function outputs
Here they are (2, 11), (3, 13), (4, 15)
and (5, 17)

5. Finding Slope and Y-Intercept
Slope = Rise / Run:
From a graph or table, pick any two points. Take the difference in the y’s and divide by the difference of the x’s.
Pick any 2 points: (3, 1390), (1, 1208)
Difference in y: y = = 182
Difference in x: x = 3 – 1 = 2
So, slope = 182 / 2 = 91
The y-intercept is the value of y, when x = 0.
Here, it is y = 1117, also written (0, 1117)

6. Finding Slope and Intercepts from an Equation
Intercepts: For any intercept, replace all other variables with “0,” then solve for the remaining variable.
Example: What is the x-intercept of 4x + 7y = 12?
Replace y with “0” x + 7(0) = 12
Solve for x: 4x = 12  x = 3 or (3,0)
Slope:
Standard Form: Ax + By = C Slope m = – A/B
Example: Find the slope of 4x + 7y = 12. m = – 4/7
Slope-Intercept Form: y = mx + b. The slope is m.

7. Finding the Equation from the Slope and a Point
Given a slope m and a point (h, k),
how can we find the equation?
Use the formula for a line: y – k = m(x – h)
Solve it for y
Example: Find the equation of the line of slope 2/3 going through (3, 4)
So, m = 2/3, h = 3, and k = 4.
y – 4 = 2/3 (x – 3)
y – 4 = 2/3 x – 2
y = 2/3 x – 2 + 4
y = 2/3 x + 4

8. Summary Defined Linear Functions and T-Tables
Finding Slopes and Intercepts
Finding the equation for a line, given the slope and a point on the line
Thank you!