Lesson 8 What Happens If…? Practice Understanding

Learning Focus

Compare the equation of an ellipse to the equation of other geometric figures.

Graph hyperbolas.

Write the equation of a hyperbola.

What figure is formed from the set of all points such that the differences of the distances to two given points is constant?

Open Up the Math: Launch, Explore, Discuss

After spending some time working with circles and ellipses, Maya notices that the equations are a lot alike. For example, here’s an equation of an ellipse and a circle:

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ , $x^{2} + y^{2} = 16$

1.

What are some of the similarities between the circle and the ellipse given in the equations above? What are some of the differences?

2.

Maya wonders what would happen if she took the equation of the circle and rearranged it so the right-hand side was $1$ , like the standard form of an ellipse. What would the equation of the circle become?

3.

After seeing this equation, Maya wonders if a circle is really an ellipse, or if an ellipse is really a circle. How would you answer this question?

4.

Maya looks at the equation of the ellipse and wonders what would happen if the “ $+$ ” in the equation were replaced with a “ $—$ ,” making the equation:

$\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ .

Without making any further calculations or graphing any points, predict whether or not the graph of this equation will be an ellipse. Using what you know about ellipses, explain your answer.

5.

Graph the equation from problem 4 to determine whether your prediction was correct. Be sure to use enough points to get a full picture of the figure.

6.

What are some of the features of the figure that you have graphed?

7.

Maya’s teacher tells her that the name of the figure represented in each of the two equations is a hyperbola. Maya wonders what would happen if the $x^{2}$ term in the equation were switched with the $y^{2}$ term, making the equation: $\frac{y^{2}}{9} - \frac{x^{2}}{16} = 1$ .

Graph this equation and compare it to the hyperbola that you graphed previously.

8.

What similarities and differences do you see between this hyperbola and the one that you graphed in problem 5?

One strategy that makes graphing the hyperbola from an equation more efficient is to notice that the square root of the numbers under the $x^{2}$ and $y^{2}$ terms can be used to make a rectangle and then to draw dotted lines through the diagonals that form the boundaries of the hyperbola.

Using this strategy to graph the equation: $\frac{y^{2}}{9} - \frac{x^{2}}{16} = 1$ , you would start by taking the square root of $9$ , which is $3$ , and going up and down $3$ units from the origin. Then you take the square root of $16$ , which is $4$ , and go left and right $4$ units from the origin. Make a rectangle with these points on the sides and draw the diagonals.

You will get this:

9.

So, Maya, the bold math adventurer, decides to try this graphing strategy with a new equation of a hyperbola. The standard form of the equation of a hyperbola centered at $(0, 0)$ is:

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (opens left and right).

$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$ (opens up and down).

Maya goes to work graphing the equation:

$\frac{x^{2}}{36} - \frac{y^{2}}{25} = 1$ .

Try it yourself on the graph, and see what you can come up with.

10.

Maya wonders what happens if the equation becomes:

$\frac{{(x - 1)}^{2}}{36} - \frac{{(y + 2)}^{2}}{25} = 1$

What is your prediction? Why?

11.

Write the equation of the hyperbola shown:

12.

What similarities and differences do you see between a hyperbola and an ellipse?

Ready for More?

Circles, ellipses, parabolas, and hyperbolas are all called conic sections because they can be thought of as slices of a double circular cone like the one shown below. Your challenge is to identify how “slicing” the cones with planes can form a circle, a parabola, an ellipse, and a hyperbola.

Takeaways

Definition of a hyperbola:

Vocabulary

asymptote
hyperbola
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about hyperbolas, the last of the conic sections. We learned that the graphs of hyperbolas can open up and down or left and right. The definition of a hyperbola is much like the definition of an ellipse, except that a point on a hyperbola is the difference between the distances to the foci and the graph of an ellipse is the sum of the distances to the foci. This makes the equation of a hyperbola like the equation of an ellipse, except the terms are subtracted rather than added.

Retrieval

1.

Identify each conic section.

a.

$4 x^{2} + 4 y^{2} - 2 x + 2 y - 20 = 0$

b.

$4 y^{2} + 3 x - 16 y + 19 = 0$

c.

$4 x^{2} + y^{2} + 8 x - 4 y + 4 = 0$

2.

Identify the conic section: $4 x^{2} - y^{2} = 8$ . Write the equation in standard form by completing the square.

If the conic 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.