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U SING D IRECT AND I NVERSE V ARIATION

y. =. k ,. x. U SING D IRECT AND I NVERSE V ARIATION. D IRECT V ARIATION. The variables x and y vary directly if, for a constant k,. or y = kx,. k  0. k. =. y ,. x. U SING D IRECT AND I NVERSE V ARIATION. I NDIRECT V ARIATION.

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U SING D IRECT AND I NVERSE V ARIATION

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  1. y = k, x USING DIRECT AND INVERSE VARIATION DIRECT VARIATION The variables x and y vary directly if,for a constant k, or y = kx, k  0.

  2. k = y, x USING DIRECT AND INVERSE VARIATION INDIRECT VARIATION The variables x and y vary inversely, if for a constant k, or xy = k, k  0.

  3. MODELS FOR DIRECT AND INVERSE VARIATION k y = x USING DIRECT AND INVERSE VARIATION DIRECT VARIATION INVERSE VARIATION y = kx k > 0 k > 0

  4. Using Direct and Inverse Variation y = k x 4 = k 2 y An equation that relates x and y is = 2, or y = 2x. x When x is 2, y is 4. Find an equation that relates x and y in each case. x and y vary directly SOLUTION Write direct variation model. Substitute2for x and4for y. 2 = k Simplify.

  5. Using Direct and Inverse Variation 8 An equation that relates x and y is xy = 8, or y = . x When x is 2, y is 4. Find an equation that relates x and y in each case. x and y vary inversely SOLUTION xy= k Write inverse variation model. (2)(4) = k Substitute2for x and4for y. 8 = k Simplify.

  6. Comparing Direct and Inverse Variation 8 Make a table using y = 2x and y = . x x 1 2 3 4 y = 2x 2 4 6 8 8 8 8 4 2 y= x 3 Compare the direct variation model and the inverse variation model you just found using x = 1, 2, 3, and 4. SOLUTION Direct Variation:k > 0. Asxincreases by 1, yincreases by 2. Inverse Variation:k > 0. Asxdoubles (from 1 to 2), yis halved (from 8 to 4).

  7. Comparing Direct and Inverse Variation Inverse y= 8 x Compare the direct variation model and the inverse variation model you just found using x = 1, 2, 3, and 4. SOLUTION Plot the points and then connect the points with a smooth curve. Direct Variation: the graph for this model is a line passing through the origin. Direct y=2x Inverse Variation: The graph for this model is a curve that gets closer and closer to the x-axis as x increases and closer and closer to the y-axis as x gets close to 0.

  8. Writing and Using a Model USING DIRECT AND INVERSE VARIATION IN REAL LIFE BICYCLINGA bicyclist tips the bicycle when making turn. The angle B of the bicycle from the vertical direction is called the banking angle. banking angle, B

  9. Writing and Using a Model r turning radius Banking angle (degrees) Turning Radius banking angle, B BICYCLINGThe graph below shows a model for the relationship between the banking angle and the turning radius for a bicycle traveling at a particular speed. For the values shown, the banking angle B and the turning radius r vary inversely.

  10. Writing and Using a Model r turning radius Banking angle (degrees) banking angle, B Turning Radius Find an inverse variation model that relates B and r. Use the model to find the banking angle for a turning radius of 5 feet. Use the graph to describe how the banking angle changes as the turning radius gets smaller.

  11. Writing and Using a Model k B= r Banking angle (degrees) k 32= 3.5 112= k Turning Radius 112 The model isB = , whereBis in degrees andris in feet. r Find an inverse variation model that relates B and r. SOLUTION From the graph, you can see that B = 32° when r = 3.5 feet. Write direct variation model. Substitute 32for B and3.5 for r. Solve for k.

  12. Writing and Using a Model Banking angle (degrees) Turning Radius 112 = 22.4 B= 5 Use the model to find the banking angle for a turning radius of 5 feet. SOLUTION Substitute 5 for rin the model you just found. When the turning radius is 5 feet, the banking angle is about 22°.

  13. Writing and Using a Model Use the graph to describe how the banking angle changes as the turning radius gets smaller. SOLUTION As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles. As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles. As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles. As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles. Notice that the increase in the banking angle becomes more rapid when the turning radius is small.

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