# 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**

## 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.
- 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.
- 0=x^2+4x+3
- 0=x^2+8x-5
- 0=2x^2+12x+4
- 0=x^2+5x+9
- Go over the review problems slowly so that the students get a good review of completing the square.

## Activities

Start the class with the graphing calculator on the overhead.

- Graph the line y=x^2. You are going to leave this graph on your screen for the entire lesson.
- Enter the graph of the parabola y=x^2 + 3. Before 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).
- Now enter the equation y=x^2 -4. Again ask for a prediction, then graph to confirm the prediction.
- Delete the last 2 graphs, leaving y=x^2.
- 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).
- Enter the graph y=(x-5)^2. Again ask for a prediction, then graph.
- 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).
- Delete all the graphs except y=x^2.
- 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.
- 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.
- 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 calculator. 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.
- 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.
- Ask them to complete the square of y=x^2+4x+5. They should get y=(x+2)^2+1. Students 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

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

### 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!

- Common Core State Standards
- Mathematics (2010)
#### High School: Algebra

- Reasoning with Equations & Inequalities
- ALG.REI.4Solve quadratic equations in one variable. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Solve...

- Reasoning with Equations & Inequalities
#### High School: Functions

- Interpreting Functions
- FUN.IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph square root,...
- FUN.IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and...

- Interpreting Functions

## North Carolina curriculum alignment

### Mathematics (2004)

#### Grade 9–12 — Algebra 1

**Goal 4**: Algebra - The learner will use relations and functions to solve problems.**Objective 4.02**: Graph, factor, and evaluate quadratic functions to solve problems.

#### Grade 9–12 — Algebra 2

**Goal 2**: Algebra - The learner will use relations and functions to solve problems.**Objective 2.02**: Use quadratic functions and inequalities to model and solve problems; justify results.- Solve using tables, graphs, and algebraic properties.
- Interpret the constants and coefficients in the context of the problem.

#### Grade 9–12 — Integrated Mathematics 2

**Goal 4**: Algebra - The learner will use relations and functions to solve problems.**Objective 4.02**: Use quadratic functions to model and solve problems; justify results.- Solve using tables, graphs, and algebraic properties.
- Interpret the constants and coefficients in the context of the problem.

#### Grade 9–12 — Technical Mathematics 2

**Goal 2**: Algebra - The learner will use relations and functions to solve problems.**Objective 2.01**: Use quadratic equations to model and solve problems; justify results.- Solve using tables, graphs, and algebraic properties.
- Interpret the constants and coefficients in the context of the problem.