# Exploring Quadratic Graphs

**NC State Standard:**

Algebra 4.02; Integrated Math 4.02

**Objectives:
**

1. Students will explore and develop understanding of quadratic functions.

2. Students will understand how changing ‘a’, and ‘c’ of the equation

y = ax

^{2}+ bx + c changes the graph.

3. Students will become familiar with vocabulary associated with quadratic

equations.

**Launch:**

Warm‐up. On overhead.

On your graphing calculator please graph the following equations. Set your window

to Xmin = ‐10, Xmax = 10, Ymin = ‐10, and Ymax = 10.

y = x

y = x

^{2}

Draw a sketch of the equations and using sentences explain the similarities between

these two equations and the differences. Please use math terms that we have

learned such as linear and non‐linear.

**Transition:**

Review observations that the students had. Discuss how adding an exponent made

the graph non‐linear. After discussion and observations are complete, go over

objective for the day.

**Notes:**

1. Quadratic function: a function that can be written in the form y = ax

^{2}+ bx + c,

where a ≠ 0.

Examples:

a. y = 5x

^{2}

b. y = x

^{2}+ 4

c. y = 7x

^{2}+ 3x +1

2. Parabola: the graph of a quadratic function. It is a U‐shaped curve.

Examples: (graphed by hand and on calculator)

3. Vertex: highest or lowest point of parabola (bottom on
U shape) (label on

above graphs)

a. Minimum: the vertex when the parabola opens up.

b. Maximum: the vertex when the parabola opens down.

4. Questions?

Partner Activity: Discovering the quadratic function.

Students will work in partners to practice graphing quadratic functions by hand
and

on the calculator. Students will discover how the graph changes when ‘a’ and ‘c’

change.

**Individual Work:**

Students will write an essay summarizing what they learned from the partner

activity and what we learned in class today. This essay should include all the

vocabulary from today’s lesson. Questions to guide summaries are at the bottom
of

investigation worksheet. Rationale: Students need to practice with vocabulary
and

math discussion. This is a good opportunity for students to practice writing in
math.

**Closing:**

Quick review. Students will indicate the direction the parabola will open
according

to the ‘a’ term being negative or positive. Then we will switch to point in
which

direction the parabola will move (up or down) from y = x^{2} according to the c
term

that is added.

Your
name:_________________________________________________________

Partner’s name:_____________________________________________________________

In this lab you will discover how the shape of a quadratic equation changes when

the coefficients ‘a’ and ‘c’ change. Your goals in this lab are to discover
patterns and

trends. After this lab you will be responsible for writing a summary of your
results

so make sure you discuss parts that you do not understand with your partner.

Part I. The ‘a’ Coefficient.

Before you use the calculator lets make sure you can create these graphs by
hand.

1. Graph y = x^{2}. Show the values of x you picked and the corresponding y value.

Now you may use your graphing calculator.

2. Graph y = ‐x^{2}.

3. What are the differences between the two graphs from
question 1 and 2?

4. Remember that there is a 1 in front of x^{2} and a ‐1 in front of –x^{2} which

correspond to the a coefficient in the quadratic function. Try graphing the

following functions.

5. What kind of statement can you make about how the graph
changes

according to whether the ‘a’ coefficient is positive or negative?

6. When the ‘a’ term is negative, is the vertex the minimum point or the

maximum point?

7. Graph the following on your calculator at the same
time. Label which graph

is which equation in your sketch.

8. What do you notice about the width of the parabola as
the ‘a’ term changes?

If you are not sure try a few more examples.

9. Which would make a wider parabola y = 5x^{2} or y = 2x^{2}? _______________________

Why?

**Part II. The ‘c’ or constant coefficient.**

10. Graph y = x^{2} , y = x^{2} + 1 on the same graph. Label each equation on the
graph

that you sketch.

11. What happened to y = x^{2} when it became y = x^{2}+1?

12. Now graph y = x^{2} and y = x^{2} ‐ 2. Which direction did the constant term ‐2

move the vertex of the parabola?

13. Which vertex would be higher, the vertex on y = x^{2} + 5 or y = x^{2} + 6? Why?

14. How does the constant term move the parabola? Describe what positive

and negative values of ‘c’ do to the position of the vertex.

**Part III. Practice.**

15. Do p. 514 in your textbook problems 21 – 30.

**Summary Assignment:**

The following questions are to help guide your summary of
today’s lesson. You are

not supposed to simply answer these questions, but use these as a guide to write

one to two paragraphs about today’s lesson. Please remember that writing is an

important part of math and I expect proper punctuation and good sentences.

1. What does the graph of a quadratic function look like? What do we call it?

2. What happens to the graph as the ‘a’ coefficient changes from positive to

a negative? Talk about the vertex being a maximum or a minimum.

3. What happens to the graph with larger numbers as the ‘a’ coefficient?

Smaller numbers?

4. What happens to the graph as a constant term is added? How does the

graph move? How does the vertex move?