Lesson 6 & 7 Unit 5

Objectives Be able to find equations for direct variation: y = kx Be able to find equations for inverse variation: y = k/x Be able to find equations for joint variation z = kxy Be able to apply models using variation Vocabulary: constant of proportionality, k = constant

HW Read p. 179 – 182 Do p. 182: 1 – 16 all; 17 – 27 odd Do p. 183: 29, 31, 32, 33, 40

Review 1.Is the equation below in Standard form, slope- intercept form or point-slope form? y = -3x – 11 slope-intercept form 2.What is the slope of the graph of the equation? slope or m = What are the coordinates of the y-intercept? (0, -11)

Review 4.Write the equation in slope-intercept form for the line that has a slope of 5 and a y-intercept with coordinates (0,0) y = 5x + 0 or y = 5x 5.What is the equation for a line with a slope of k and y-int of 0? y = kx + 0 or y = kx

New: Direct Variation Definition: “a mathematical model in which one quantity is a constant multiple of the other” 1.Simply put, this means the equation for direct variation is y = kx. The k is called the constant of proportionality. 2.The slope of the line is k; the y-int is (0,0) 3.Direct variation means as x increases, y also increases 4.Other expressions that mean direct variation are: a) y varies directly as x b) y is directly proportional to x c) y is proportional to x

New: Direct Variation 4. For example, if you earn $7.00 per hour at your job, the amount you earn varies directly with the number of hours worked (varies directly means as hours increase, the amount increases) Let y = the amount, x = the number of hours, and 7 is k or the constant of proportionality, then: y = 7x a) If you work 0 hours, you earn $0; point (0, 0) is on the graph of the line. b) If you work 8.5 hours, you earn $59.50; point (8.5, 59.5) is on the graph of the line

New: Direct Variation 5.How far you can drive a car depends or varies directly to the number of hours you drive. If d stands for distance and h stands for the hours you drive, what would the equation be? a) Since it is direct variation, then it has to be in the form of y = kx b) So, d = kh (continue on next slide)

New: Direct Variation The piece of information you do not have is k or the constant of proportionality. In this case, k would be the average speed. You can find k though if you know how far you drove and how long it took. If you drove 200 miles and it took 4.5 hours, then d = kh becomes 200 = k(4.5)  k = 200/4.5  k = 44.4 So, the equation is d = 44.4h

New: Direct Variation 6.Last example: Write the equation that expresses the statement “r varies directly or is proportional to t”if r = 4 when t = 12. a) Direct variation means y = kx b) So, for this example, r = kt c) To find k, we know (4, 12) is a point on the line; it makes the equation true d) So, 12 = 4k  k = 12/4  k =3 e) Finally, the equation is r = 3t

New: Inverse Variation 1.Similarly to direct variation, inverse variation has a consistent equation: y = k/x 2.With inverse variation, as x increases, y decreases 3.k is still the constant of proportionality 4.Other expressions that mean the same thing: a) y is inversely proportional to x b) y varies inversely as x

New: Inverse Variation 4.Example: write the equation if g is inversely proportional to h and g = 16 when h = 3 a) Inverse variation means y = k/x b) So, g = k/h c) To find k, substitute and solve: 16 = k/3  3*16 = k  k = 48 d) Finally, g = 48/h

New: Joint Variation Joint variation means that there are more than two quantities (such as x and y) related. In other words, there could be three quantities, such as x, y and z For example: a) if z is directly related to x and y, then z = kxy b) if z is directly related to x and inversely proportional to y, then z =(kx)/y

New: Joint Variation Remember: if a quantity is directly proportional to another quantity, then k is multiplied if a quantity is inversely related to another, then k is divided to find k, plug in the coordinates (x, y) for a point on the line