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 (Microsoft Office Word document, 50 KB), which contains Activity 1. When they have completed the activity, provide them with Student Handout #2 (Microsoft Office Word document, 34 KB). 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.

Teacher Tips

  • Cause and effect

    Ask your students to graph the following two linear equations on the same coordinate plane: y= x+1, and y= 2x+1. Ask students to write what happens as the coefficient of x changes. What is the coefficient called?

  • From paper to purpose

    Circles are used in everyday life in fields such as architecture and design. Ask students to suggest other ways in which circle shapes are used.

Student activity

Follow the steps below to guide your students through this lesson plan.

Note teachers: Please download the student activity handouts located in the sidebar under Software and Materials Needed, for additional details about the main activities for this lesson plan.

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

Note students: If you need help graphing equations, graphing on a new set of axes, or animating the parameters on a graph, read the document "How to Graph an Equation in Microsoft Math" that your teacher has saved to your classroom computer.

  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? 

Note students: For the following activities, if you need help graphing equations, graphing on a new set of axes, or animating the parameters on a graph, read the document "How to Graph an Equation in Microsoft Math" that your teacher has saved to your classroom computer.

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 is 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.