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. Determine original center point (2, -4)
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, )
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 x ½ = 3/ V.s. becomes 4 x 3 = 12
New equation becomes

7. Solution: Draw a diagram.
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.