12.5 Direct Variation y = kx

12.5 Direct Variation Vocabulary Linear function written as y = kx Constant of Variation = k (really the slope) A non-zero constant of the equation If y=kx, divide both sides by x y/x = kx/x, then simplify So, y/x = k , and the ratio of y/x should give the same answer of “k” for all points on the line. The line will always go through the origin (0,0).

12.5 Direct Variation Example 1a Does the data set show direct variation? Method 1: k=y/x, so try all pairs of numbers to check if you get all the same answers. 39/7=5.571, 41/8=5.125, 43/9=4.778, 44/10=4.400, 45/11=4.091 Different answers=NOT Direct Variation SHOE SIZES X=US Size 7 8 9 10 11 Y=Europe 39 41 43 44 45

12.5 Direct Variation Example 1b Method 2: Cross Multiply Ratios Multiply ratios across the equal sign from top to bottom and bottom to top. Choose 2 ratios at a time. If cross multiply answers don’t match then ratios aren’t equal. 39 = 45 315 7 11 429 Method 3: Graph is not linear either.

12.5 Direct Variation Example 2 x=Input Power 6 8 12 20 28 y=Sound Out 4.5 6 9 15 21 Direct Variation is a Linear equation: y = kx Write ratios as Sound Out / Input Power 4.5/6 = 6/8 = 9/12 = 15/20 = 21/28 = 3/4 Ratios are constant and include point (0,0), so it is Direct Variation where y = kx. Function may be written as: y = (3/4)x y is the sound out and x is the input power.

12.5 Direct Variation Example 3 Find the equation of direct variation when: y is 48 when x is 3. Find k first for y = kx. Since k = y/x, then k = 48/3 = 16, so y = 16 x

12.5 Direct Variation Example 4 Find the equation of direct variation when: y is 15 when x is 10. Find k first for y = kx. Since k = y/x, then k = 15/10 = 3/2, so y = 3/2 x

12.5 Direct Variation Example 5a Train Travel by Minutes Time (min) 10 20 30 40 Distance (miles) 25 50 75 100 Write ratios as mile / min and compare them. If they match, function is direct variation. If they do not match, function is variable. What is the answer?

12.5 Direct Variation Example 5b The mile/minute fractions are equal as shown: 25/10 = 50/20 = 75/30 = 100/40 = 5/2 Ratios are constant and include the point (0,0) so the function is Direct Variation. Function may be written as: y = (5/2)x

New: Car Stopping Distance Car stopping distance is the sum of Reaction Distance (travel before applying brake) and Brake Distance (travel after applying brakes). Is there a Direct Variation between either of these data sets and speed? Find the equation. Speed 15 35 55 75 Reaction 33 77 121 165 Braking 11 59+ 144 268

Reaction Distance vs. Speed Write ratios as reaction / speed and compare: 33/15 = 77/35 = 121/55 = 165/75 = 11/5 = 2.2 Ratios are constant and include point (0,0), so the function is Direct Variation. Function may be written as: y = 2.2x x is the speed and y is the reaction distance.

Braking Distance vs. Speed Write ratios as braking / speed and compare: 11/15 = 0.73 59/35 = 1.69 If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios. Try p657 #5-8 all

Review Notes Direct Variation is a linear equation written as: y = kx k is non-zero and called Constant of Variation Divide both sides of the equation by x and get: y/x = k k (which is really the slope) should be the same for all points on the straight line equation, just take y and divide by x for each.

Assignment Try p657 #5-8 all