Graphs and Functions Presumed knowledge-linear functions 8 Sept. 2016 Work through the examples in the PPT Additional work: Text Ch1 pg 30/ 10, 12, 13, 14 IB Mathematics SL Graphs and Functions Presumed knowledge-linear functions From Thauvette, Beetz, Aldous

Review of Equations of Lines Objective: Using the Point-Slope Form to Write an Equation Recall that when the slope of a line and a point on the line are known, the equation of the line can also be found. To do this, we use the slope formula to write the slope of a line that passes through points (x, y) and (x1, y1). Beginning course details and/or books/materials needed for a class/project. We multiply both sides of this equation by to obtain This form is called the point-slope form of the equation of a line.

Point-Slope Form of the Equation of a Line A schedule design for optional periods of time/objectives.

Example 1 Write an equation of the line with slope –3 and containing the point (1, –5). Write the equation in slope-intercept form, y = mx + b Introductory notes.

Example 1 – Solution Because we know the slope and a point on the line, we use the Point-slope form with m = –3 and (x1, y1) = (1, -5). Objectives for instruction and expected results and/or skills developed from learning. In slope-intercept form, the equation is y = – 3x – 2

Practice 1 Write an equation of the line with slope –2 and containing the point (2, –4). Write the equation in slope-intercept form, y = mx + b Introductory notes. Answer: y = –2x

Example 2 Write an equation of the line through points (4, 0) and (–4, –5). Write the equation in standard form Ax + By = C. Solution: First we find the slope of the line. Relative vocabulary list. Next we make use of the point-slope form. We replace (x1, y1) by either (4, 0) or (–4, –5) in the point-slope form. We will choose the Point (4, 0). The line through (4, 0) with slope 5/8 is as follows. Continued on next slide

Example 2 – Solution Let’s multiply through by 8 so that the coefficients are integers and are less tedious to work with. A list of procedures and steps, or a lecture slide with media. Continued on next slide

Example 2 – Solution If we multiply both sides of –5x + 8y = –20 by –1, we have an Equivalent equation in standard form. Both –5x + 8y = –20 and 5x – 8y = 20 are acceptable. Example graph/chart.

Practice 2 Write an equation of the line through points (3, 0) and (–2, 4). Write the equation in standard form, Ax + By = C. Introductory notes. Answer: 4x + 5y = 12

Objective: Writing Equations of Vertical and Horizontal Lines A few special types of linear equations are those whose graphs are vertical and horizontal lines. Example 3. Example graph/chart. Solution: Recall that a horizontal line has an equation of the form y = c. Since the line contains the point (2, 3), the equation is y = 3

Practice 3 Write an equation of the horizontal line containing the point (–1, 6). Introductory notes. Answer: y = 6

Example 4 Write an equation of the line containing the point (2, 3) with undefined slope. Solution: Since the line has undefined slope, the line must be vertical. A vertical line has an equation of the form x = c, and since the line contains the point (2, 3), the equation is x = 2.

Practice 4 Write an equation of the line containing the point (4, 7) with Undefined slope. Introductory notes. Answer: x = 4

Objective: Writing Equations of Parallel and Perpendicular Lines Recall: Nonvertical parallel lines have the same slope and nonvertical perpendicular lines have slopes whose product is –1. Example 5 Conclusion to course, lecture, et al. Solution: Because the line we want to find is parallel to the line 2x + y = –6, the two lines must have equal slopes. So we first find the Slope of 2x + y = –6 by solving the equation for y to write it in the Form y = mx + b. Here y = –2x – 6, so the slope is –2.

Example 5 Now we use the point-slope form to write an equation of the line through (4, 4) with slope –2. Continued on next slide

Example 5 The equation, y = –2x – 6, and the new equation, y = –2x + 12 Have the same slope but different y-intercepts so their graphs are parallel, as shown. Also, the graph of y = –2x + 12 contains the point (4, 4), as desired.

Practice 5 Write an equation of the line containing the point (–1, 2) and parallel to the line 3x + y = 5. Write the equation in slope-intercept form y = mx + b Answer: y = –3x – 1

Example 6 Write an equation of the line containing the point (–2, 1) and perpendicular to the line 3x + 5y = 4. Write the equation in slope-intercept form, y = mx + b. Solution: First we find the slope of 3x + 5y = 4 by solving the equation for y. Continued on next slide

Example 6 Continued on next slide

Example 6 Continued on next slide

Example 6 The equation and the new equation Have negative reciprocal slopes, So their graphs are perpendicular. Also, the graph of contains the point (–2, 1), as desired.

Practice 6 Write an equation of the line containing the point (3, 4) and perpendicular to the line 2x + 4y = 5. Write the equation in slope-intercept form. Answer: y = 2x – 2

Objective: Using the Point-Slope Form in Applications Example 7. Southern Star Realty is an establish real estate company that has enjoyed constant growth in sales since 2001. In 2005 the company sold 250 houses, and in 2009 the company sold 330 houses. Use these figures to predict the number of houses this company will sell in 2013 Continued on next slide

Example 7 Solution: UNDERSTAND. Read and reread the problem. Then let x = the number of years after 2001 y = the number of houses sold in the year corresponding to x The information provided then gives the ordered pairs (4, 250) and (8, 330). To better visualize the sales of Southern Star Realty, we graph the line that passes through the points (4, 250) and (8, 330). Continued on next slide

Example 7 Continued on next slide

Example 7 2. TRANSLATE. We write an equation of the line that passes through the points (4, 250) and (8, 330). To do so, we first find the slope of the line. Then, using the point-slope form to write the equation, we have Continued on next slide

Example 7 3. SOLVE. To predict the number of houses sold in 2013, we use y = 20x + 170 and complete the ordered pair (12, ), since 2013 – 2001 = 12. 4. INTERPRET. CHECK: Verify that the point (12, 410) is a point on the line graphed in step 1. STATE: Southern Star Realty should expect to sell 410 houses in 2013.

Practice 7 Southern Regional is an established office product maintenance company that has enjoyed constant growth in new maintenance contracts since 2001. In 2003, the company obtained 15 new contracts, and in 2009, the company obtained 33 new contracts. Use these figures to predict the number of new contracts this company can expect in 2017. Answer: 57 new contracts

Practice 8 A rock is dropped from the top of a 400-foot building. After 1 Second, the rock is traveling 32 feet per second. After 3 seconds, the Rock is traveling 96 feet per second. Let y be the rate of descent And x be the number of seconds since the rock was dropped. Write a linear equation that relates time x to rate y. Use this equation to determine the rate of travel of the rock 4 seconds after it was dropped. Answer: (a) y = 32x; (b) 128 feet per second

Practice 9 The Whammo Company has learned that by pricing a newly released Frisbee at $6, sales will reach 2000 per day. Raising the price to $8 will cause the sales to fall to 1500 per day. Assume that the ratio of change in price to change in daily sales is constant, and let x be the price of the Frisbee and y be number of sales. Find the linear equation that models the price-sales relationship for this Frisbee. (b) Use this equation to predict the daily sales of Frisbees if the price is set at $7.50 Answer: (a) y = –250x + 3500; (b) 1625 Frisbees

Practice 10 In 2009, the average price of a new home sold in the United States Was $267,900. In 2004, the average price of a new home in the United States was $271,500. Let y be the average price of a new Home in the year x, where x = 0 represents the year 2004. Write a linear equation that models the average price of a new home in terms in the year x. (b) Use this equation to predict the average price of a new home in 2012. Answer: (a) y = –720x + 271,500; (b) $265,740

Practice 11 The number of people employed in the United States as registered Nurses was 2619 thousand in 2008. By 2018, this number is Expected to rise to 3200 thousand. Let y be the number of registered Nurses (in thousands) employed in the United States in the year x, Where x = 0 represents 2008. Write a linear equation that models the number of people (in thousands) employed as registered nurses in year x. (b) Use this equation to estimate the number of people who will be employed as registered in 2012. Answer: (a) y = 58.1x + 2619; (b) 2851.4 thousand

Finding equations of lines from graphs Find an equation of each line graphed in standard form. Answer: 2x + y = 3 Answer: 2x – 3y = –7

Summary

Concept Extension – Practice 12 The graph shows the cost function, C(x), and the revenue function, R(x), for a transistor manufacturer, where x is the number of units sold.

Concept Extension – Practice 13 A Water Supply company decides to change its rates to increase the user-pays component of water supply and it phases in the procedure. The new method included a fixed charge (possibly different for each property depending on its assessed value) and a rate of 45 cents per kilolitre for the first part used and 65 cents per kilolitre for the second part. In a particular bill, the charge for a quarter is $150 plus a charge for up to 50kl at the lower rate and the rest at the higher rate. If the water used is 63kl what will be the total bill?

Practice 13

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