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

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.

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

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. -24 B. -6 C. -2 D. 8 f(-3) = 3(-3) – 15 Simplify f(-3) = -9 – 15 f(-3) = -24

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)).

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)

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

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

Given the domain {-2, -1, 0, 1, 2}, graph 3x + y = 6 Solve for y: 3x + y = 6 Subtract 3x - 3x - 3x 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)

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

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.

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)

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.

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.

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.

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…).

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!

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

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

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

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

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