Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope I. Linear Functions 1. A linear equation is an equation whose graph is a straight line. 2. A linear equation in standard form: Ax +By=C ex: 4x 6y=12 3. A linear equation in slope-intercept form: y = mx + b, slope m; y-intercept (0,b) 4. A linear function is written as: f(x) = mx + b II. Graphing Linear Equations using Intercepts A. The x-intercept of graph is the point where the graph intersects the x-axis. It is the point (x, 0). B. The y-intercept of graph is the point where the graph intersects the y-axis. It is the point (0, y). y x C. To graph a line using intercepts: 1. Find the x-intercept by letting y = 0 and solving for x. 2. Find the y-intercept by letting x = 0 and solving for y. 3. Find a checkpoint by choosing a value for x (or y) and solving for y (or x). 4. Plot the points you found above. Draw a straight line through them. Example: Use intercepts and a checkpoint to graph. 4x 6y = 12
III. Slope of a line A. The slope of a line measures the steepness of a line. The slope is thought of as the change in y ratio that compares the change in y to the change in x. change in x = rise run B. If (x 1, y 1 ) and (x 2, y 2 ) are two points, the slope of the line through them is: slope = m = This is known as the. C. The slope tells us the direction of the line: a. positive: line rises from left to right b. negative: line falls from left to right c. zero: line is horizontal d. undefined: line is vertical Examples: Find the slope of the line passing through the pair of points or state that the slope is undefined. Then indicate whether the line rises, falls, is horizontal or is vertical. 1) (2, 4) and ( 1,3) 2) (5, 1) and (8, 4) 3) (2, 4) and (5,4) Blitzer 2.4 Page 2 of 7
IV. Graphing Linear Equations using the Slope and y-intercept A. To graph an equation using the slope and y-intercept, the equation must be written in slope-intercept form: y = mx + b or f(x)=mx+b B. To graph an equation using the slope and y-intercept: 1. Plot the y-intercept, (0,b). 2. Use the slope to obtain other points. 3. Draw a line through the points. Examples: Give the slope and y-intercept of each line whose equation is given. Then graph it. 1) y = 3x 4 2) f x ( ) = 2 3 x + 5 m= m= y-intercept: (, ) y-intercept: (, ) Examples: a) Rewrite the equation in slope-intercept form. b) Give the slope and y-intercept. c) Graph the equation. 1) 2x y = 5 Blitzer 2.4 Page 3 of 7
2) 3x + 4y = 12 V. Graphing Horizontal and Vertical Lines A. If an equation has only one variable, then its graph is either a horizontal line or a vertical line. B. Consider the equation, x = 2. a. What ordered pairs are solutions to this equation? b. Plot those points and draw a line through them. c. What does the line look like? We can generalize and say: The graph of x = # is a line going through the point ( #, 0). C. If we did the above work for y = 2, we would see that we get a line. So we can generalize and say: The graph of y = # is a line going through the point ( 0, #). Examples: 1) x = 4 2) 2y = 6 3) f(x)= 2 Blitzer 2.4 Page 4 of 7
VI. Slope as a Rate of Change Recall: the slope is the ratio that compares the change in y to the change in x. It describes how fast y is changing with respect to x. Think of the rate of change as the slope with units. Examples: Find the slope of each model. Then describe what this means in terms of the rate of change. 1) The function B(x)=3.2x + 472 models the number of bachelor s degrees, in thousands, awarded to men x years after 1980. m = change in y change in # of bachelor's degrees = = change in x change in # of years 2) The function f(x)= 0.28x + 1.7 models the percentage of US taxpayers who were audited by the IRS, f(x), x years after 1996. 3) The function f(x)=1.48x + 120.6 models the female US population f(x), in millions, x years after 1985. Blitzer 2.4 Page 5 of 7
VI. Modeling Data using the slope-intercept form of the equation 1) The following graph is a scatter plot that shows the median household income for Californians. Also shown is a line that passes through or near the points. Median Household Income CA 63,000 61,000 59,000 57,000 55,000 53,000 51,000 50,200 61,021 59,948 56,645 53,629 51,189 49,000 (0, 50220) 47,000 45,000 0 1 2 3 4 5 Number of years after 2003 (5, 61021) a) According to the graph, what is the y-intercept of the line? Describe what this represents in this situation. b) Use the coordinates of the points shown to compute the slope of the line. Describe what this means in terms of the rate of change. c) Use the y-intercept from part (a) and the slope from part (b) to write a linear function that models the median household income for Californians, C(x), x years after 2003. d) Use the function of part (c) to predict the median household income for Californians in 2009. Blitzer 2.4 Page 6 of 7
2) The following graph is a scatter plot that shows the number of Americans, in millions, living below the poverty level for various years. Also shown is a line that passes through or near the points. Number below poverty (in millions) 38 37 36 35 34 33 (0, 33) 32.9 34.6 35.9 37 37 (5, 37.0) 32 0 1 2 3 4 5 Number of years after 2000 a) According to the graph, what is the y-intercept of the line? Describe what this represents in this situation. b) Use the coordinates of the points shown to compute the slope of the line. Describe what this means in terms of the rate of change. c) Use the y-intercept from part (a) and the slope from part (b) to write a linear function that models the number of Americans, P(x), living below the poverty level x years after 2000. d) Use the function of part (c) to predict the number of Americans living in poverty in 2010. Blitzer 2.4 Page 7 of 7