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Graph figures from equations

Graph figures from equations

In this lesson, students learn basic coordinate geometry by graphing figures from equations and writing the equations of graphed figures.

​Objectives

  • Students will learn basic coordinate geometry by graphing figures from equations and writing the equations of graphed figures.
  • Students will learn the effects of parameter changes on graphs.

Learning outcomes

  • Students will graph circles based on a given center and radius.
  • Students will write the equation for figures shown in graphs.
  • Students will write the equation for circles with an identified center and radius.

Lesson procedure

​Introduction

In this exercise about coordinate geometry you will learn how to graph equations and discover the geometric figures that those equations represent. You will also learn how to write the equations for geometric figures for which you have certain information.

You will complete this exercise in two parts. First, using Student handout #1 you will graph a mystery geometric figure from an equation and watch what happens to the graph when you change the parameters of the equation. Next, using Student handout #2 you will graph more figures, work with equations that contain irrational numbers, and write the equations for circles with a given radius and center.

Give the students Student handout #1 (see link at right), which contains Activity 1. When they have completed the activity, provide them with Student handout #2 (see link at right). Make sure that students know how to access the Help document “How to Graph Equations in Microsoft Math.” Students who need more time to complete the other activities can skip Activity 4.

 

Student activity

Follow the steps below to guide your students through this lesson plan. See student handout links at right.

Activity 1: Graph the equation of a geometric figure and change its parameters on the graph.

  1. Graph EQ 1: x2+ y2=1. Be sure to select the proportional display button. What kind of figure do you get?
  2. On the same axis, graph EQ 2: x2+ y2=4. How is the graph you derived from equation 2 different from the graph from equation 1? What do 1 and 4 seem to represent on the graph? Graph x2 + y2= r2. Animate r from 0 to 2. What happens as r increases?
  3. Graph EQ 3: x2 + (y-1) 2=1. How is equation 3 different from equation 1? What does this number appear to determine? Graph x2 + (y-a) 2 = 1. Animate a from 0 to 2. What happens as a increases? Now animate a from -2- +2. What happens as a goes from 0 to -2?
  4. Graph EQ 4: (x -1) 2+ (y) 2=1. How is equation 4 different from equation 1? What does this number appear to determine? Graph (x-a)2 + (y) 2 = 1. Animate a from 0 to 2. What happens to the graph as a increases? Now animate a from -2- +2. What happens to the graph as a goes from 0 to -2?
  5. Graph (x+1) 2 + (y-2) 2=25. Before you graph, can you predict what the figure will be, what its size will be, and where it will be located? 

Activity 2: Graph more circles and write their equations.

  1. What is the radius of the circle with the equation (x+1) 2 + y2=100? What is the center of the circle? Graph to verify.
  2. Graph a circle that has a radius of 5 with a center of (-2, 2). What is its equation?

Activity 3: Work with irrational numbers.

  1. What is the radius of the circle with equation (x+2) 2 + y2=40? What is the center of the circle? Graph to verify.
  2. Graph a circle that has a radius of √5 with a center of (-2, 2). What is its equation?

Activity 4: Write the formula or equation for a circle.

(x-h) 2 + (y-k)^2 =r2 is the equation of a circle where (h,k) is the center of the circle and r is the radius.

  1. In the equation x2 + y2=1, what are h, k, and r?
  2. In the equation (x-2) 2 + (y+1) 2 = 9, what is h, k, and r?
  3. Write the equation for a circle with a center at (-3, 1) and a radius of 5. ​

Conclusion

  • Observe students as they complete each of the activities.
  • Evaluate each student’s work in the student handouts.
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