1.3 Linear Functions
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1 1.3 Linear Functions A function of the form f(x) = ax + b, where a and b are constants, is called a linear function. The graph of f(x) = ax + b or y = mx + b, is a line, with slope a (or m), and y intercept b or (0, b). Aug 30 12:02 PM 1
2 Example: Calculate the slope of the line between the points ( 2, 7) and (3, 4). Aug 30 12:03 PM 2
3 Calculate the slope of the line between the points ( 7, 5) and (1, 9). Calculate the slope of the line between the points ( 3, 4) and ( 3, 7). Aug 30 12:04 PM 3
4 We can also find the slope of a line from its equation or its graph. Find the slope of each of the following linear equations. Aug 30 12:05 PM 4
5 Determine the slope of the line given by the following equation: y = 4x 5 Determine the slope of the line given by the following equation: 2x + 6y = 4 A. 2 B. 1/3 C. 3 D. 1/3 Sep 1 12:29 PM 5
6 Find the slope of each line whose graph is shown below. Aug 30 12:05 PM 6
7 State the slope of the line whose graph is shown below. Aug 30 12:06 PM 7
8 The expression rate of change means the same thing as slope. i.e. Rate of change means how the y values are changing with respect to the x values. A line may have positive slope, negative slope, zero slope, or undefined slope. However, a vertical line is not a function. Determine the rate of change of the line passing through the points (1, 5) and (3, 2). Determine the rate of change of the line whose equation is y = 2x + 7. Aug 30 12:07 PM 8
9 A linear function whose slope is 0 is called a constant function. Examples of constant functions: Aug 30 12:08 PM 9
10 Which graph below shows a positive rate of change? Aug 30 12:09 PM 10
11 The y intercept is the point where the graph intersects the y axis, and thus its x coordinate is 0. To find the y intercept of a linear function, let x = 0 and solve for y. Ex: Find the y intercept of each equation below. A. y = 3x + 5 B. 2x 4y = 12 Aug 30 12:10 PM 11
12 Which graph below shows a positive y intercept? Determine the slope and the y intercept of the equation below. Aug 30 12:11 PM 12
13 The x intercept is the point where the graph intersects the x axis, and thus its y coordinate is 0. To find the x intercept of a linear function, let y = 0 and solve for x. Ex: Find the x intercept of each equation below. A. y = 3x + 5 B. 2x 4y = 12 Aug 30 12:13 PM 13
14 To graph a line, you can use either the slope and y intercept, plot ordered pairs, or use the x intercept and y intercept. Graph each line below using the slope and y intercept. 1) y = 2x 4 2) y = 1/3x + 1 Aug 30 12:14 PM 14
15 Graph each line below using the x intercept and y intercept. Aug 30 12:15 PM 15
16 Determine the x intercept and the y intercept of the equation below. Which of the following is the graph of the equation above? Aug 30 12:16 PM 16
17 The percentage of Fortune Global 500 firms that actively recruited workers on the Internet from 1998 through 2000 can be modeled by the following function, where x is the number of years after The rate of change for this function is an increase of 26.5 percent of firms per year Aug 30 12:16 PM 17
18 The units for the rate of change are the units of y per unit of x. Note that if the slope is positive, then the rate of change is described as an increase; if the slope is negative, then the rate of change is described as a decrease. Examples: 1) If y = x represents the cost (in dollars) from the production of x hats, then the units of y are dollars and the units of x are hats. Therefore the units for the rate of change are dollars per hat. (Increase or decrease?) 2) (In this case, V takes the place of y and t takes the place of x.) If V = 300,000 10,000t represents the value (in dollars) of a property after t years, then the units of V are dollars and the units of t are years. Therefore the units for the rate of change are dollars per year. (Increase or decrease?) Aug 30 12:19 PM 18
19 A) B) C) D) Jan 19 12:15 PM 19
20 If y = x represents the cost (in dollars) from the production of x shirts, then determine the rate of change for the function, including units. In this context, the rate of change is also referred to as the marginal cost. A. $3000 per shirt B. $5 per shirt C. 5 shirts per dollar D shirts per dollar If P = 40x 5000 represents the profit (in dollars) from the production and sale of x bicycles, then determine the rate of change for the function, including units. In this context, the rate of change is also referred to as the marginal profit. A. $5000 per bicycle B. $5000 per bicycle C. 40 bicycles per dollar D. $40 per bicycle Jan 19 11:49 AM 20
21 If the value of a brand new house is $200,000, and its value after 10 years is $150,000, determine the rate of change in the value of the house per year. Include the units. If the value of a brand new car is $40,000, and its value after 5 years is $25,000, determine the rate of change in the value of the car per year. Include the units. A. decrease of $15,000 per year B. increase of $3,000 per year C D. decrease of $3000 per year Aug 30 12:21 PM 21
22 Jan 19 12:08 PM 22
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