College Algebra K /DC Wednesday , 22 January 2014

1. College Algebra K/DCWednesday, 22 January 2014 • OBJECTIVETSW graph linear functions and apply linear functions to real-world applications. • TESTS are not graded. • ASSIGNMENT CHANGE • Sec. 2.4: pp. 225-226 (7-32 all) • 7-24 all, 29-32 all

2. Linear Functions 2.4 Graphing Linear Functions ▪ Standard Form Ax + By = C

3. Graphing a Linear Function in Slope-Intercept Form Graph . Give the domain and range. • Identify the slope. • Slope = Identify the y-intercept by finding f (0).

4. Graphing a Linear Function in Slope-Intercept Form Graph . Give the domain and range. Plot (0, 6) and another point using the slope, then join the points with a straight line. Domain: (–, ) Range: (–, )

5. Graphing a Horizontal Line Graph f(x) = 2. Give the domain and range. Since f(x) always equals 2, the value of y can never be 0. So, there is no x-intercept. The line is parallel to the x-axis. The y-intercept is 2. Functions whose graphs are horizontal lines are constant functions because their function values do not change. Domain: (–, ) Range: {2}

6. Graphing a Vertical Line Graph x = 5. Give the domain and range. Since x always equals 5, the value of x can never be 0. So, there is no y-intercept. The line is parallel to the y-axis. The relation is not a function. The x-intercept is 5. Domain: {5} Range: (–, )

7. Graphing a Line in Standard Form (Ax + By = C with C = 0) Graph 3x + 4y = 0. Give the domain and range. Find the intercepts. There is only one intercept, the origin, (0, 0). Choose any another point, say x = 4, to find a second point. 3(4) + 4y = 0 12 + 4y = 0 y = –3 (4, –3)

8. Graphing a Line in Standard Form (Ax + By = C with C = 0) Graph the points (0, 0) and (4, –3) and join with a straight line. Domain: (–, ) Range: (–, )

9. Assignment • Sec. 2.4: pp. 225-226 (7-24 all, 29-32 all) • Write each equation. • Due by the end of the period today (wire basket). 2-9

10. Graph each linear function. Identify any constant functions. Give the domain and range. 7)f (x) = x – 4 8)f (x) = –x + 4 9)f (x) = ½ x – 6 10)f (x) = 2/3x + 2 11) –4x + 3y = 9 12) 2x + 5y = 10 13) 3y – 4x = 0 14) 3x + 2y = 0 15)f (x) = 3x 16)f (x) = –2x17)f (x) = –4 18)f (x) = 3 Graph each vertical line. Give the domain and range of each relation. 19)x = 3 20)x = –4 21) 2x + 4 = 0 22) –3x + 6 = 0 23) –x + 5 = 0 24) 3 + x = 0 Graph each line. 29)y = 3x + 4 30)y = –2x + 3 31) 3x + 4y = 6 32) –2x + 5y = 10

11. Linear Functions 2.4 Average Rate of Change ▪ Linear Models 2-11

12. Finding Slopes with the Slope Formula the slope is undefined Find the slope of the line through the given points. (a) (–2, 4), (2, –6) (b) (–3, 8), (5, 8) (c) (–4, –10), (–4, 10)

13. Average Rate of Change The slope of a line is a ratio of the vertical changein y to the horizontal change in x. So, slope gives the average rate of change in y per unit change in x. If f is a linear function defined on [a, b], then

14. Interpreting Slope as Average Rate of Change In 1997, sales of VCRs numbered 16.7 million. In 2002, estimated sales of VCRs were 13.3 million. Find the average rate of change in VCR sales, in millions, per year. Graph as a line segment, and interpret the result. The average rate of change per year is = –0.68 million VCR

15. Interpreting Slope as Average Rate of Change In 1997, sales of VCRs numbered 16.7 million. In 2002, estimated sales of VCRs were 13.3 million. Find the average rate of change in VCR sales, in millions, per year. Graph as a line segment, and interpret the result. What? Increase or decrease? Interpretation Sales of VCRs decreased by an average of 0.68 million each year from 1997 to 2002. or Sales decreased by an average of 0.68 million VCRs each year from 1997 to 2002. Get in the habit of answering the interpretation part in complete sentences (with proper grammar and spelling.) How much? When? How often?

16. Writing Linear Cost, Revenue, and Profit Functions A linear cost function has the form where x represents the number of items produced, m represents the variablecost per item, and b represents the fixed cost, which is constant. A revenue function is given by where p is the price per itemand x is the number of units sold. A profit function is described by • The break-even point occurs when P(x) = 0. It indicates the number of units to sell to “break even” (neither lose money nor make money).

17. Writing Linear Cost, Revenue, and Profit Functions Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed cost is $2400, the variable cost per item is $120, and the item sells for $150. Write linear functions to model (a) cost, (b) revenue, and (c) profit. (a) Since the cost function is linear, it will have the form C(x) = mx + b with m = 120 and b = 2400. C(x) = 120x + 2400 (b) The revenue function is R(x) = px with p = 150. R(x) = 150x

18. Writing Linear Cost, Revenue, and Profit Functions 30x – 2400 > 0 30x > 2400 x > 80 (c) The profit is the difference between the revenue and the cost. P(x) = R(x) –C(x) = 150x – (120x + 2400) = 30x – 2400 (d) What is the minimum number of items the company must sell in order to make a profit? To make a profit, P(x) must be positive. Complete sentence. The company must sell at least 81 items to make a profit.

19. Assignment • Sec. 2.4: pp. 227-231 (65-68 all, 73-77 all, 88-92 all) • You do not have to write the problem. • Due on Friday, 24 January 2014.

20. Assignment: Sec. 2.4 pp. 227-231 (65-68 all, 73-77 all, 88-92 all)