Lesson 5 Getting Centered Solidify Understanding

Jump Start

Find the missing term that makes each expression a perfect square trinomial, and then write an equivalent expression. (You may use a diagram to help.)

a.

$x^{2} + 6 x \underset{―}{}$

b.

$y^{2} + 4 y \underset{―}{}$

c.

$x^{2} - 10 x \underset{―}{}$

Learning Focus

Write and graph the equation of a circle.

Find the center and radius of a circle in general form.

Do translations work on circles like they do on functions?

How are different forms of the equation of a circle related?

Open Up the Math: Launch, Explore, Discuss

Malik’s family has decided to put in a new sprinkler system in their yard. Malik has volunteered to plan and draw the system out on a grid. Sprinklers are available at the hardware store that spray a full circle with a maximum radius of $15 feet$ . The sprinklers can be adjusted so that they spray a smaller radius. Malik needs to be sure that the entire yard gets watered, which he knows will require that some of the circular water patterns overlap. He gets out a piece of graph paper and begins with a scale diagram of the yard. In this diagram, the length of the side of each square represents $5 feet$ .

1.

As he begins to think about locating sprinklers on the lawn, his parents tell him to try to cover the whole lawn with the fewest number of sprinklers possible so that they can save some money. The equation of the first circle that Malik draws to represent the area watered by the sprinkler is:

${(x + 25)}^{2} + {(y + 20)}^{2} = 225$

Draw this circle on the diagram using a compass.
Help Malik lay out a possible configuration for the sprinkling system by adding two more sprinklers, and the area that they cover, to the diagram that includes the first sprinkler pattern that you drew.
Find the equation of each of the circles that you have drawn.

Malik wrote the equation of one of the circles and, just because he likes messing with the algebra, he did this:

Original equation:

${(x - 3)}^{2} + {(y + 2)}^{2} = 225$

$x^{2} - 6 x + 9 + y^{2} + 4 y + 4 = 225$ $x^{2} + y^{2} - 6 x + 4 y - 212 = 0$

Malik thought, “That’s pretty cool. It’s like a different form of the equation. I guess that there could be different forms of the equation of a circle like there are different forms of the equation of a parabola or the equation of a line.” He showed his equation to his sister, Sapana, and she thought he was nuts.

Sapana said, “That’s a crazy equation. I can’t even tell where the center is or the length of the radius anymore.” Malik said, “Now it’s like a puzzle for you. I’ll give you an equation in the new form. I’ll bet you can’t figure out where the center is.”

Sapana said, “Of course I can. I’ll just do the same thing you did, but work backwards.”

2.

Malik gave Sapana this equation of a circle:

$x^{2} + y^{2} - 4 x + 10 y + 20 = 0$

Help Sapana find the center and the length of the radius of the circle.

3.

Sapana said, “OK. I made one for you. What’s the center and length of the radius for this circle?”

$x^{2} + y^{2} + 6 x - 14 y - 42 = 0$

4.

Sapana said, “I still don’t know why this form of the equation might be useful. When we had different forms for other equations like lines and parabolas, each of the various forms highlighted different features of the relationship.” Why might this form of the equation of a circle be useful?

$x^{2} + y^{2} + A x + B y + C = 0$

Ready for More?

How might you use the general form of the equation of a circle, $x^{2} + y^{2} + A x + B y + C = 0$ , to find the equation of the circle that contains the following points: $p (2, 1)$ , $q (0, 5)$ , $r (- 1, 2)$ ?

Try it. See if you can find values for $A$ , $B$ , and $C$ to write the equation of the circle.

Takeaways

General form of the equation of a circle:

Standard form of the equation of a circle:

Changing from general form to standard form:

Example: $x^{2} + y^{2} - 2 x - 6 y + 5 = 0$	Description of steps:

Vocabulary

circle: equation in standard form; equation in general form
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned to write equations of circles in both standard form and general form. We used the process of completing the square to change an equation from standard form to general form.

Retrieval

1.

The graph of $f (x) = \sqrt[3]{x}$ is translated to the right $5$ units and down $7$ units and undergoes a vertical stretch by a factor of $9$ . Write the equation for the transformed function.

Solve for $x$ .

2.

$\frac{3}{7} x - 8 = 13$

3.

$x^{2} - 11 x = 42$