1. Objectives The student will be able to:
1. graph linear functions.
2. write equations in standard form.

2. Graphing Steps
Isolate the variable (solve for y).
Make a t-table. If the domain is not given, pick your own values.
Plot the points on a graph.
Connect the points.

3. Section 4.7 “Graph Linear Functions”
Function Notation-
a linear function written in the form y = mx + b where y is written as a function f.
x-coordinate
f(x) = mx + b
This is read
as ‘f of x’
slope
y-intercept
f(x) is another name for y.
It means “the value of f at x.”
g(x) or h(x) can also be used to name functions

4. What is the value of the function f(x) = 3x – 15 when x = -3?
Linear Functions
What is the value of the function f(x) = 3x – 15 when x = -3?
A B C D. 8
f(-3) = 3(-3) – 15
Simplify
f(-3) = -9 – 15
f(-3) = -24

5. Domain and Range
Domain = values of ‘x’ for which the function is defined.
Range = the values of f(x) where ‘x’ is in the domain of the function f.
The graph of a function f is the set of all points (x, f(x)).

6. Graphing a Function To graph a function:
(1) make a table by substituting into the function.
(2) plot the points from your table and connect the points with a line.
(3) identify the domain and range, (if restricted)

7. 3) Solve for y: x - 3y = 6 - x - x -3y = -x + 6 -3 -3 or Subtract x
Subtract x
Simplify
Divide both sides by -3 or

8. Review: Make a t-table If f(x) = 2x + 4, complete a table using the domain {-2, -1, 0, 1, 2}.
ordered pair -2
2(-2) + 4 = 0 (-2, 0)
2(-1) + 4 = 2 (-1, 2)
2(0) + 4 = 4 (0, 4)
2(1) + 4 = 6 (1, 6)
2(2) + 4 = 8 (2, 8) -1 1 2

9. Given the domain {-2, -1, 0, 1, 2}, graph 3x + y = 6
Solve for y: x + y = 6
Subtract 3x
- 3x x
y = -3x + 6
2. Make a table
ordered pair x
-3x + 6 -2 -1 1 2
-3(-2) + 6 = 12 (-2, 12)
-3(-1) + 6 = 9 (-1, 9)
-3(0) + 6 = 6 (0, 6)
-3(1) + 6 = 3 (1, 3)
-3(2) + 6 = 0 (2, 0)

10. Given the domain {-2, -1, 0, 1, 2}, graph 3x + y = 6
Plot the points (-2,12), (-1,9), (0,6), (1,3), (2,0)
Connect the points.
Bonus questions!
What is the x-intercept?
(2, 0)
What is the y-intercept?
(0, 6)
Does the line increase or decrease?
Decrease

11. Family of Functions
is a group of functions with similar characteristics. For example, functions that have the form f(x) = mx + b constitutes the family of linear functions.

12. Parent Linear Function
The most basic linear function in the family of all linear functions is called the PARENT LINEAR FUNCTION which is:
f(x) = x
f(x) = x x -5 -2 1 3
f(x)

13. Compare graphs with the graph f(x) = x.
Graph the function g(x) = x + 3, then compare it to the parent function f(x) = x.
f(x) = x
g(x) = x + 3
g(x) = x + 3
f(x) = x x
f(x) -5 -2 1 3 x
f(x) -5 -2 1 3 4 6
The graphs of g(x) and f(x) have the same slope of 1.

14. Compare graphs with the graph f(x) = x.
Graph the function h(x) = 2x, then compare it to the parent function f(x) = x.
f(x) = x
h(x) = 2x
h(x) = 2x
f(x) = x x
f(x) -5 -2 1 3 x
f(x) -3 -6 -2 -4 2 4 3 6
The graphs of h(x) and f(x) both have a y-int of 0.
The slope of h(x) is 2 and therefore is steeper than f(x) with a slope of 1.

15. Real-Life Functions
A cable company charges new customers $40 for installation and $60 per month for its service. The cost to the customer is given by the function f(x) = 60x +40 where x is the number of months of service. To attract new customers, the cable company reduces the installation fee to $5. A function for the cost with the reduced installation fee is g(x) = 60x + 5. Graph both functions. How is the graph of g related to the graph of f ?
The graphs of both functions are shown. Both functions have a slope of 60, so they are parallel. The y-intercept of the graph of g is 35 less than the graph of f. So, the graph of g is a vertical translation of the graph of f.

16. Standard Form Ax + By = C A, B, and C have to be integers
An equation is LINEAR (the graph is a straight line) if it can be written in standard form.
This form is useful for graphing (later on…).

17. Determine whether each equation is a linear equation.
4x = 7 + 2y
Can you write this in the form
Ax + By = C?
4x - 2y = 7
A = 4, B = -2, C = 7
This is linear!

18. Determine whether each equation is a linear equation.
2) 2x2 - y = 7
Can you write it in standard form?
NO - it has an exponent!
Not linear
3) x = 12
x + 0y = 12
A = 1, B = 0, C = 12
Linear

19. Here’s the cheat sheet! An equation that is linear does NOT contain the following:
Variables in the denominator
Variables with exponents
Variables multiplied with other variables.
xy = 12

20. Is this equation linear?
Yes No
Standard Form
x – 4y = 3

21. Is this equation linear?
Yes No
Exponents are not allowed!

22. Is this equation linear? y = -3
Yes No
Standard Form
0x + y = -3