Graphing Quadratic Equations

This lesson will help students quickly graph a quadratic equation. It will also help them to understand the purpose of completing the square.

A lesson plan for grades 9–12 Mathematics

By Kathy Schadt

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• Learn more about mathematics.

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Learning outcomes

Students will sketch the graph of a quadratic equation, and put a quadratic equation in general graph form y=(x-h)^2 +k by completing the square.

Teacher planning

Time required for lesson

50 minutes

Materials/resources

classroom board
graph paper
pencils

student graphing calculators are a bonus, but not a must.

Technology resources

overhead graphing calculator

overhead

Pre-activities

Students should be able to complete the square before beginning this lesson.

Activities

Your warm-up on this day should be an activity on completing the square. If that was the homework from the night before, go over the homework and do a few problems. If completing the square wasn’t the homework, give the students these problems as a warm-up.

Complete the Square

1. 0=x^2+4x+3
2. 0=x^2+8x-5
3. 0=2x^2+12x+4
4. 0=x^2+5x+9

Go over the review problems slowly so that the students get a good review of completing the square.

Start the class with the graphing calculator on the overhead.

1. Graph the line y=x^2. You are going to leave this graph on your screen for the entire lesson.
2. Enter the graph of the parabola y=x^2 + 3Before you hit the graph key, ask the students to predict what they think will happen.Ask the students what they notice about the relationship between this graph and the previous graph. They should notice that the vertex moves to the point (0,3).
3. Now enter the equation y=x^2 -4Again ask for a prediction, then graph to confirm the prediction.
4. Delete the last 2 graphs, leaving y=x^2.
5. Enter y=(x+2)^2…Ask for a prediction.When you graph, you should notice that the vertex of your parabola moves to the point(-2,0).
6. Enter the graph y=(x-5)^2. Again ask for a prediction, then graph.
7. Now combine what they have learned by asking the students to predict the graph y=(x+1)^2 + 4. They should be able to tell you the vertex will be at the point (-1, 4).
8. Delete all the graphs except y=x^2.
9. Now give the students the graph y=-x^2. Ask them what they think might happen. Confirm with them that the graph flips to open down instead of up.
10. Give the students the parabola y=-(x+2)^2-5. Ask students to sketch this graph on their own. Look at the results, then graph on the overhad to show them the answer.
11. Now we are going to make the connection between completing the square and graphing a parabola. Give the students the equation y=x^2+4x+4. Show them the graph on the overhead calcuator. Look at the vertex of that graph. Where is it? It should be at the point (-2,0). Ask the students what equation of that parabola would look like in general graph form. They should come up with the answer y=(x+2)^2 based on the pattern you have shown them.
12. 12. Ask the students if anyone can find an algebraic method for transforming x^2+4x+4 into (x+2)^2. If no one can, help them make the connection: Show the students that by factoring the perfect square trinomial of y=x^2+4x+4 you get y=(x+2)^2
13. Ask them to complete the square of y=x^2+4x+5. They should get y=(x+2)^2+1Students should now know that this means the vertex of the equation is on the point (-2, 1).

Assessment

Now give the students some extra practice to do on their own. Ask them to do the following problems. The teacher should walk around and check the students work. When most students have completed the problems, ask some students to put the correct graphs on the board so all students can check their work.

Extra Practice

1. y=x^2+8
2. y=(x-5)^2
3. y=(x+1)^2-3
4. y=-(x+4)^2+1
5. y=x^2+18x+81
6. y=x^2+16x+10

Challenge Problem

y=3x^2+6x-2

Only give this problem to the students who are above the ability level of the rest of the class.

Supplemental information

None

Related websites

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Comments

Discovery lessons are very easy to do with the graphing calculator. I have found that students enjoy making predictions. It is also fun to let the students make up their own problems, and then enter them into the calculator for the entire class to see. Let them go crazy and use big numbers. The worst thing that can happen is you have to reset your calculator!