20S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Real Number System Lesson: LMP-L5 Direct Variation Direct Variation Learning Outcome B-4 LMP-L5 Objectives: To analyze problems using the perspective of direct variation. y = kx

20S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Real Number System Lesson: LMP-L5 Direct Variation Direct variation Two variable quantities that have the same rate, or ratio, regardless of the values of the variables, have direct variation. For example, if a student's rate of pay was $6 per hour, then the total pay varies directly as the number of hours worked. The table below illustrates this. The variables p and t are related by the equation p = 6t The relation between p and t can therefore be graphed, and shows a straight line which goes through the origin. The graph’s equation is of the form y = mx + b where b = 0. If the y-intercept is zero, i.e. b =0, the relation is called a direct variation. Theory – Direct Variation Defined

20S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Real Number System Lesson: LMP-L5 Direct Variation A direct variation is a straight line function defined by an equation of the form, y = kx, where k ≠ 0. In the direct variation p =6t, p is said to vary directly as t. You can also say that p is directly proportional to t. The number 6 is called the constant of variation or the constant of proportionality. As you can see, direct variation is very much like a linear equation in the form of y = mx + b, but where the y-intercept or 'b' is zero. In the context of linear equations, m is the slope. In the context of direct variations, k is the constant of variation. k and m talk about the same thing. (k = m) Theory – Constant of Variation (k)

20S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Real Number System Lesson: LMP-L5 Direct Variation We want to be able to take a set of data that relates two quantities and determine if they vary directly. If they do, we identify the constant of variation, write the equation (y = kx), and then use the equation to extrapolate or interpolate further information about either of the quantities. Theory – Constant of Variation (k)

20S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Real Number System Lesson: LMP-L5 Direct Variation Jimmy has observed that the distance to a thunderstorm can be estimated by counting the number of seconds between a flash of lightning and the sound of the thunder. With further investigation, he obtains the following information: Example 1 a. Complete the pattern shown in the chart up to 21 seconds. b. Graph the information using t as the independent variable and d as the dependent variable. (By hand or using Winstats) c. What is the equation for this direct proportion? What is the constant of proportionality? d. Estimate the distance if the time is 10 seconds, 20 seconds, 30 seconds.

20S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Real Number System Lesson: LMP-L5 Direct Variation In an experiment involving pollution control, a team of scientists measure the amount of carbon monoxide (CO) produced by a car engine over different periods of time. The amount of CO is measured in litres (L). Example 2 a. Graph the information using time (t), as the independent variable and Amount (A), of CO as the dependent variable. b. What is the equation for this direct proportion? What is the constant of proportionality? c. Estimate the amount of CO produced by the engine in 15 s, 115 s.