Lesson 1 Quick It’s Quadratic Develop Understanding

Learning Focus

Find patterns in the equations and graphs of quadratic functions.

How can the graph of a quadratic function be predicted from the equation?

Open Up the Math: Launch, Explore, Discuss

In Unit 6, we used quadratic functions to model real situations. Although we saw a variety of quadratic functions, we never fully considered the most basic quadratic function, which is $f (x) = x^{2}$ . In this lesson, we will analyze $f (x) = x^{2}$ and many other related quadratic functions to see if we can discover some features about the graphs of quadratic functions.

1.

Start with $f (x) = x^{2}$ .

a.

Make a table of values that includes both positive and negative values for $x .$

b.

Explain how the table demonstrates that $f (x) = x^{2}$ is a quadratic function.

2.

Graph $f (x) = x^{2}$ .

3.

Answer the following questions about $f (x) = x^{2}$ .

a.

What is the domain of $f (x) = x^{2}$ ?

b.

What is the range of $f (x) = x^{2}$ ?

c.

On what interval(s) is $f (x)$ increasing?

d.

On what interval(s) is $f (x)$ decreasing?

e.

Is $f (x)$ symmetric? If so, what is the equation of the line of symmetry?

f.

Does $f (x)$ have a maximum or minimum? If so, at what point?

g.

What are the $x$ -intercept(s) of $f (x)$ ?

h.

What is the $y$ -intercept of $f (x)$ ?

$f (x) = x^{2}$ is often called “the parent function” for quadratic functions because all the other graphs of quadratic functions are related. The standard form of the equation of a quadratic function is $y = a x^{2} + b x + c$ . You can see that the parent function is part of the standard form, but it is possible to add terms if they are linear or constant. The $x^{2}$ term might also have a coefficient other than $1$ . We are going to use technology to help us discover relationships that will allow us to predict features of the graphs, given a quadratic equation in standard form.

4.

Let’s get started by graphing these three quadratic functions in the same viewing window.

$f (x) = x^{2} + 2 x$ , $g (x) = 2 x^{2} - 3 x - 5$ , and $h (x) = - x^{2} - 2 x + 4$

a.

What features appear to be the same on all quadratic functions?

b.

Write your own equations for two more quadratic functions:

c.

Use technology to graph your equations. Do they have the same features you noticed in the previous three graphs? Modify your observations about common features of quadratic functions, if necessary.

d.

What possible differences occur among quadratic functions?

You probably noticed that the graphs of quadratic functions all have similar shapes. This shape is called a parabola. You may also have noticed that sometimes the parabola has a maximum, and sometimes the parabola has a minimum. The maximum or minimum point on the parabola is called the vertex.

5.

Graph these six functions and see if you can come up with a strategy for determining if the vertex of the parabola will be a maximum or a minimum.

$f (x) = x^{2} + 2 x - 1 g (x) = - x^{2} + 2 x - 1 h (x) = x^{2} - 2 x - 1$

$j (x) = x^{2} - 2 x - 1 k (x) = 2 x^{2} - 3 x + 1 l (x) = - 2 x^{2} - 3 x + 1$

Test your strategy with two graphs of your own. Write your equations here:

Modify your strategy, if necessary, and write it here:

6.

Another possible difference in parabolas is where the $y$ -intercept is located. Graph these three quadratic equations and find the $y$ -intercept for each.

$f (x) = x^{2} + 5 x + 3 g (x) = - 2 x^{2} - 4 x - 5 h (x) = x^{2} - 3 x + 10$ Compare the vertex to the equation and then write your strategy for quickly determining the $y$ -intercept of a quadratic function.

Explain why your strategy works.

7.

Now let’s put it all together and use our strategies without technology. The equation of a quadratic function and the vertex of the parabola are given. Answer the questions about the functions.

a.

$f (x) = x^{2} + 6 x + 7$ ; vertex: $(- 3, - 2)$

Is the vertex a maximum or a minimum?

What is the line of symmetry of $f (x)$ ?

What is the $y$ -intercept of $f (x)$ ?

What is the domain of $f (x)$ ?

What is the range of $f (x)$ ?

On what interval is $f (x)$ increasing?

On what interval is $f (x)$ decreasing?

b.

$f (x) = - 2 x^{2} + 4 x + 3$ ; vertex: $(1, 5)$

Is the vertex a maximum or a minimum?

What is the line of symmetry of $f (x)$ ?

What is the $y$ -intercept of $f (x)$ ?

What is the domain of $f (x)$ ?

What is the range of $f (x)$ ?

On what interval is $f (x)$ increasing?

On what interval is $f (x)$ decreasing?

Ready for More?

These equations emerged in some of our previous lessons. Without using technology, determine in which graph(s) the vertex is a minimum and in which graph(s) the vertex is a maximum.

a.

$f (n) = n (n + 2)$

b.

$f (x) = x (36 - x)$

Takeaways

Features of the Graphs of Quadratic Functions:

Adding Notation, Vocabulary, and Conventions

Parabola:

Vertex:

Line of symmetry:

Vocabulary

line of symmetry
parabola
vertex
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we identified common features of quadratic functions such as the vertex, line of symmetry, and the shape of the graph, which is called a parabola. We also learned how to efficiently find the domain and range of a quadratic function and determine if the graph of the function opens upward or downward.

Retrieval

1.

Given the functions: $f (x) = - 2 x + 7$ and $g (x) = 5 x - 16$ .

Find the function for $h (x)$ if $h (x) = f (x) + g (x)$ .

2.

The graph shows the functions $p (x)$ and $q (x)$ . Add $r (x)$ to the graph based on knowing $r (x)$ is equal to the sum of $p (x)$ and $q (x)$ , $r (x) = p (x) + q (x)$ .

3.

Rewrite the expression in equivalent form without the square root.

$\sqrt{25 x^{4} y^{2}}$