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Understanding Direct and Inverse Variation: Concepts and Examples

This guide explains the principles of direct and inverse variations in mathematics. Direct variation occurs when one variable increases or decreases in direct proportion to another variable, represented by the equation y = kx, where k is the constant of variation. This relationship necessitates a linear graph passing through the origin. In contrast, inverse variation describes how one variable increases as another decreases, illustrated through equations like y = k/x. Real-world examples are included, such as gas usage in vehicles and safety ascent rates in scuba diving, for practical understanding.

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Understanding Direct and Inverse Variation: Concepts and Examples

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  1. Direct Variation September 6

  2. X Y DIRECT VARIATION x is directly proportional to y x varies directly as y

  3. Direct Variation • y = kx • k is the constant of variation • the graph must go through the origin (0,0) and must be linear!! • Therefore it must be in y = kx form. The y-intercept is 0

  4. Example NonExample Direct Variation y = 3x y = .5x-1 y = 2/3x y = 5 y = 4 – 6x y = 11x y = 8.7x

  5. Direct Variation Ex 1)If x varies directly as y and x = 12 when y = 3, write an equation that relates x and y. Start with: y = kx Fill in x and y: 3 = k(12) Solve for k: Re-write equation with the k value: y = ¼ x

  6. Same problem, new ? Ex 1)If x varies directly as y and x = 12 when y = 3, find x when y = 10. y = ¼ x Fill in NEW x and y: 10 = ¼ (x) Solve for x: x = 40

  7. FIRST: what you are comparing NEXT: substitute your values correctly LAST: cross multiply to solve for missing variable. Another way to do the last ?

  8. 2) If y varies directly as x, and y = 28 when x = 7, find x when y = 52 write an equation that relates x and y. x = 13 The constant of variation is the reduced fraction. 4 What is the constant of variation? y = 4x

  9. 3) If y varies directly as the square of x, and y = 4 when x = 3, find y when x = 6 Use a proportion….. y = 16 write an equation that relates x and y.

  10. 4) A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles? 11.034 gallons

  11. 5) In scuba diving the time (t) it takes a diver to ascend safely to the surface varies directly with the depth (d) of the dive. It takes a minimum of 3 minutes from a safe ascent from 12 feet. Write an equation that relates depth (d) and time (t). Then determine the minimum time for a safe ascent from 1000 feet?

  12. 6) z varies directly with x and y. z = kxy Write the equation relating x, y and z if x = 2, y = -6 and z = 24.

  13. Inverse Variation 8.2

  14. X Y INVERSE or indirect VARIATION y is inversely proportional to x y varies inversely as x K is the constant of variation or constant of proportionality

  15. Inverse Variation • This is a NON-LINEAR function (it doesn’t look like y=mx+b) • It doesn’t even get close to (0, 0) • k is still the constant of variation

  16. t v Inverse Variation When you buy a car, as time (t) increases, the value (v) decreases. The constant of variation, k is the amount that it decreases. t is the age of the car. v is the value of the car.

  17. Write the model that represents this situation. 6) If y varies inversely as x and when y = 12, x = 10.

  18. 7)The intensity of a light “I” received from a source varies inversely with the distance “d” from the source. If the light intensity is 10 ft-candles at 21 feet, what is the light intensity at 12 feet? Write your equation first.

  19. Work with partners on the WS HW: finish WS 5 show all work!

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