Lesson 8 What Happens If…? Practice Understanding

Ready

Classify each of the equations based on the shape of its graph.

1.

$\frac{x^{2}}{25} + \frac{y^{2}}{12} = 1$

2.

$\frac{x^{2}}{4} - \frac{y^{2}}{16} = 1$

3.

$\frac{x^{2}}{49} + \frac{y^{2}}{49} = 1$

4.

$x^{2} = 16 + y$

5.

$9 x^{2} = 36 + 4 y^{2}$

6.

$9 x^{2} = 36 - 9 y^{2}$

Set

Use the information provided for each hyperbola to find its equation, and sketch the graph of each hyperbola.

7.

Foci are located at $(\sqrt{41}, 0)$ and $(- \sqrt{41}, 0)$ .

Vertices are located at $(- 4, 0)$ and $(4, 0)$ .

Equation:

Asymptotes:

8.

Foci are located at $(0, \sqrt{13})$ and $(0, - \sqrt{13})$ .

Vertices are located at $(0, 3)$ and $(0, - 3)$ .

Equation:

Asymptotes:

9.

Foci are located at $(\sqrt{85}, 0)$ and $(- \sqrt{85}, 0)$ .

Asymptotes are $y = \frac{6}{7} x$ and $y = - \frac{6}{7} x$ .

Equation:

Vertices:

For problems 10 and 11, rewrite the equation to highlight important distances between features of the hyperbola and use them to find the asymptotes, the vertices, and foci, and graph the hyperbola.

10.

$4 x^{2} - 16 y^{2} = 64$

11.

$12 x^{2} - 3 y^{2} = 48$

Use the information provided for each hyperbola to find its equation, asymptotes, and to sketch the graph of each hyperbola.

12.

Vertices located at $(- 1, - 2)$ and $(7, - 2)$

Foci located at $(8, - 2)$ and $(- 2, - 2)$

Equation:

Asymptotes:

13.

Foci located at $(3, 5 + \sqrt{13})$ and $(3, 5 - \sqrt{13})$

Vertices located at $(3, 3)$ and $(3, 7)$

Equation:

Asymptotes:

Go

Classify each equation as a parabola, circle, ellipse, or hyperbola, and then identify the desired features. (Hint: You may rewrite the equation in a different form to highlight the features.)

If the equation is:

a parabola, identify the vertex, focus, and directrix.
a circle, identify the center and the radius.
an ellipse, identify the center and the radius for the horizontal and vertical axes.
a hyperbola, write the equations of the asymptotes.

14.

$x^{2} - 4 x + y^{2} + 6 y = 1$

15.

$16 x^{2} - 9 y^{2} - 72 y - 288 = 0$

16.

$y^{2} - 4 y + 4 x + 4 = 0$

17.

$4 x^{2} + y^{2} + 16 x - 6 y + 9 = 16$