Introduction to Linear Functions PDF

Name: ____________________________________ Date: _________________________ INTRODUCTION TO LINEAR FUNCTIONS N-GEN MATH® ALGEBRA I One of the most important types of functions in math and...

Name: ____________________________________ Date: _________________________ INTRODUCTION TO LINEAR FUNCTIONS N-GEN MATH® ALGEBRA I One of the most important types of functions in math and science is known as a linear function. These functions are known as linear because their graphs consist of points that lie along a straight line. 2 2 Exercise #1: The function f  x   x  4 is a linear function. It can also be written as y  x  4. 3 3 (a) Find the value of f  9 . What point must lie on the graph of the function based on this calculation? (b) Fill in the table below for this function. Plot the points and connect with a straight-line. x 3 0 3 6 9 f  x  x, y  (c) In the table, what is the average rate of change (d) What is the function’s y-intercept? Circle it in between any two consecutive x-values? Circle the equation. this value in the equation and illustrate it on the graph. You should be generally familiar with the equation of a line from your last math course. We can now use this to define what a linear function is. LINEAR FUNCTIONS A linear function, , has the form or , where m is the line’s slope and b is the line’s y-intercept. The slope is defined as the average rate of change, or. Exercise #2: For the linear function y  5 x  2 , choose two x-values and show that the average rate of change between the two is equal to the slope of the line. ® N-GEN MATH® ALGEBRA I – UNIT 4 – LINEAR FUNCTIONS – LESSON 1 ® eMATHinstruction, RED HOOK, NY 12571, © 2022 It is important to be able to graph a linear function by setting up a table and plotting points. 1 Exercise #3: For the linear function y   x  7 answer the following. 2 (a) Graph the line on the grid provided by creating a table of values. Show your table below. (b) What is the slope of this line? Illustrate on the graph. (c) Is this function increasing or decreasing? Why does it behave this way? It is helpful to be able to graph lines using their slopes and y-intercepts without having to create tables to do so. Exercise #4: For each of the equations given below, write in the form of y  mx  b (if needed), identify the slope and y-intercept, and then graph and label the line on the grid. (a) y  2 x  3 (b) y   x  6 2 3 (c) y  x3 (d) y   x  7 5 2 ® N-GEN MATH® ALGEBRA I – UNIT 4 – LINEAR FUNCTIONS – LESSON 1 ® eMATHinstruction, RED HOOK, NY 12571, © 2022 Name: ____________________________________ Date: __________________ MORE WORK GRAPHING LINEAR FUNCTIONS (LINES) N-GEN MATH® ALGEBRA I The graphs of linear functions depend on their slope and their y-intercept. It is critical that you understand how these two parameters affect the line’s graph. Try the first exercise as a warm-up. Exercise #1: Four linear functions are graphed on the set of axes below. Write the number of the line beside each equation based on its slope and y-intercept. 2 (a) y   x  3 ______ 3 (b) y  x  5 ______ (c) y  2 x  7 ______ (d) y  2 x  3 ______ Recall that if a line is written in the form y  mx  b , then it is relatively easy to graph, especially if m and b are reasonably easy to work with. A quick review from the previous lesson. 3 Exercise #2: On the grid below, graph the equation y  x  4. First, identify its slope and y-intercept and then 2 graph the line on the grid provided. Slope: __________ y-intercept: __________ Exercise #3: Pick two sets of two coordinate points that lie on the line and find the average rate of change between the two points. What are both equal to? (a) First set of two points: (b) Second set of two points: ® N-GEN MATH® ALGEBRA I – UNIT 4 – LINEAR FUNCTIONS – LESSON 2 ® eMATHinstruction, RED HOOK, NY 12571, © 2022 Sometimes linear equations are not written in a form that makes it easy to determine the slope and the y-intercept. It is important to be able to rearrange these formulas in order to quickly identify these linear parameters. Exercise #4: Consider the linear equation given by 2 y  6 x  12. (a) Steps are shown below that rearrange this equation. Justify each step with a property of equality or a property of numbers. (1) 2 y  6 x  6 x  12  6 x _______________________________________________ (2) 2 y  6 x  12 _______________________________________________ 2 y 6 x  12 (3)  _______________________________________________ 2 2 6 x 12 (4) y   _______________________________________________ 2 2 y  3x  6 (b) Identify the slope and the y-intercept of this line: Exercise #5: Rearrange each of the following linear equations into y  mx  b form and identify the slope and the y-intercept. (a) 3 y  3 x  15 (b) 2 y  5 x  8 (c) x  3 y  6 (d) 6 x  4 y  20 ® N-GEN MATH® ALGEBRA I – UNIT 4 – LINEAR FUNCTIONS – LESSON 2 ® eMATHinstruction, RED HOOK, NY 12571, © 2022