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

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

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)

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)

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 = 1390-1208 = 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)

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” 4x + 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.

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

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!