Copyright © 2011 Pearson Education, Inc. Modeling Our World

Copyright © 2011 Pearson Education, Inc. Slide 9-3 Unit 9B Linear Modeling

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-4 Linear Functions A linear function has a constant rate of change and a straight-line graph.  The rate of change is equal to the slope of the graph.  The greater the rate of change, the steeper the slope.  Calculate the rate of change by finding the slope between any two points on the graph.

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-5 Finding the Slope of a Line To find the slope of a straight line, look at any two points and divide the change in the dependent variable by the change in the independent variable.

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-6 The Rate of Change Rule The rate of change rule allows us to calculate the change in the dependent variable from the change in the independent variable.

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-7 General Equation for a Linear Function dependent variable = initial value + (rate of change independent variable) Algebraic Equation of a Line In algebra, x is commonly used for the independent variable and y for the dependent variable. For a straight line, the slope is usually denoted by m and the initial value, or y-intercept, is denoted by b. With these symbols, the equation for a linear function becomes y = mx + b. Equations of Lines

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-8 Slope and Intercept For example, the equation y = 4x – 4 represents a straight line with a slope of 4 and a y-intercept of - 4. As shown to the right, the y-intercept is where the line crosses the y-axis.

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-9 Varying the Slope The figure to the right shows the effects of keeping the same y-intercept but changing the slope. A positive slope (m > 0) means the line rises to the right. A negative slope (m < 0) means the line falls to the right. A zero slope (m = 0) means a horizontal line.

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-10 Varying the Intercept The figure to the right shows the effects of changing the y-intercept for a set of lines that have the same slope. All the lines rise at the same rate, but cross the y-axis at different points.

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-11 Rain Depth Equation Use the function shown to the right to write an equation that describes the rain depth at any time after the storm began. Use the equation to find the rain depth 4 hours after the storm began. Since m = 0.5 and b = 0, the function is y = 0.5x. After 4 hours, the rain depth is (0.5)(4) = 2 inches.

9-B Copyright © 2011 Pearson Education, Inc. Slide 9-12 Step 1: Let x be the independent variable and y be the dependent variable. Find the change in each variable between the two given points, and use these changes to calculate the slope. Step 2: Substitute the slope, m, and the numerical values of x and y from either point into the equation y = mx + b and solve for the y-intercept, b. Step 3: Use the slope and the y-intercept to write the equation in the form y = mx + b. Creating a Linear Function from Two Data Points