Section 9.8: Linear Functions

Definitions Def: A linear function is a function whose graph is a line in the coordinate plane. What makes a function have a straight line as its graph? Changes in the input and the corresponding change in output are always in the same ratio. Def: This common ratio is called the slope of the line. slope= change in y change in x Def: The 𝑦-intercept is the y-coordinate where the line crosses the y-axis.

Equations of Linear Functions Slope-intercept form: 𝑦=𝑚𝑥+𝑏 where 𝑚 is the slope and 𝑏 is the 𝑦-intercept Point-slope form: 𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1 where 𝑚 is the slope and ( 𝑥 1 , 𝑦 1 ) is any point on the line

Example Problem Ex 1: Find an equation to represent the following scenario: Buying donuts at Krispy Kreme costs you $7 for the first dozen donuts and 50 cents for each additional donut.

Common Linear Relationships Proportional Relationships Ex 2: A latte at Starbucks is made from 2 parts espresso to 3 parts milk. If 𝑥 ounces of espresso are used, how many ounces of milk should be used?

Common Linear Relationships Arithmetic Sequences Recall: Nth term of an arithmetic sequence is (increase amt) ∙ N = (0th entry) Ex 3: Consider the sequence 2, 5, 8, 11, 14…. Write a linear function so that when 𝑥 is a whole number, the output 𝑦 is the 𝑥th entry in the sequence.