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Transformations of Conics

Transformations of Conics. Pure Math 30. If the hyperbola x 2 - y 2 = -1 is stretched horizontally by a factor of 3 and vertically by a factor of ½, find the new equation. Solution: First convert equation to standard form by dividing by –1. x 2 - y 2 = -1 becomes -x 2 + y 2 = 1

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Transformations of Conics

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  1. Transformations of Conics Pure Math 30

  2. If the hyperbola x2 - y2 = -1 is stretched horizontally by a factor of 3 and vertically by a factor of ½, find the new equation. Solution: • First convert equation to standard form by dividing by –1.x2 - y2 = -1 becomes -x2 + y2 = 1 • Now apply the stretches. Because fractions in the denominator look incorrect, we convert by remembering that dividing by a fraction is the same as multiplying by the reciprocal. So the equation becomes

  3. Given state the conic and its horizontal and vertical stretches. • Remove the coefficients of each variable and take the square root • Reciprocate the square root of the coefficient and you have the stretches • Solution: The conic is an ellipse. • Horizontal coefficient is 4/25. Square root is 2/5. Horizontal stretch is 5/2. • Vertical coefficient is 49/9. Square root is 7/3. Vertical stretch is 3/7.

  4. State the transformations when the equation y = x2 becomes Solution: • Vertical stretch by a factor of 4 • Translations 2 units right and 2 units down.

  5. Given the ellipse , determine the new equation after a translation 3 units up and 7 units right. Solution: • Determine original center point (2, -4) • Apply translations to this point (2 + 7, -4 + 3) • The new center is (9, -1) Put this back into equation.

  6. The ellipse is stretched horizontally by a factor of ½ and vertically by a factor of 3. Determine the new equation. • Solution: Remove the stretches from the equation. H.s. is 3 and v.s. is 4. • Multiply by the new stretches and put these values back into equation. H.s. becomes 3 x ½ = 3/2. V.s. becomes 4 x 3 = 12 • New equation becomes

  7. A tunnel has a semi-elliptical cross section. The maximum height of the tunnel is 5.5 m, and the full tunnel width is 25 m. A truck in the right lane is 4.3 m tall, and will be 4 m away from the tunnel wall. Will the truck be able to get through the tunnel? • Solution: Draw a diagram. 4 m (8.5, ?) 5.5 m truck 12.5 m

  8. Solution cont. : • We can see the horizontal stretch is 12.5 and the vertical stretch is 5.5 to create the equation • Now sub in the given value of x (8.5) to calculate y. Since the height of the truck is taller than the tunnel the truck will not fit.

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