Lesson 6 Stretching My Quads Practice Understanding

Learning Focus

Find patterns to efficiently graph quadratic functions from factored form.

What features of a parabola are highlighted in factored form? How can we use those features to graph a quadratic function?

How does factored form of a quadratic equation relate to solving quadratic equations?

Open Up the Math: Launch, Explore, Discuss

Now that we have the process of factoring in our mathematical bag of tricks, let’s see how factored form might be useful in analyzing quadratic functions. Use technology to graph each function and find the vertex, the line of symmetry, the $y$ -intercept, and the $x$ -intercepts. Be sure to properly write the intercepts as points. While you’re working, watch for generalizations that you could make about the features of a quadratic function in factored form.

1.

$y = (x - 1) (x + 3)$

Line of symmetry:

Vertex:

$x$ -intercepts:

$y$ -intercept:

2.

$f (x) = 2 (x - 2) (x - 6)$

Line of symmetry:

Vertex:

$x$ -intercepts:

$y$ -intercept:

3.

$g (x) = - x (x + 4)$

Line of symmetry:

Vertex:

$x$ -intercepts:

$y$ -intercept:

4.

Based on these examples, how can you use a quadratic function in factored form to:

a.

Find the line of symmetry of the parabola?

b.

Find the vertex of the parabola?

c.

Find the $x$ -intercepts of the parabola?

d.

Find the $y$ -intercept of the parabola?

e.

Find the direction of opening?

5.

If $f (x) = 2 (x - 2) (x - 6)$ , how do the $x$ -intercepts of $f (x)$ relate to the equation below?

$0 = 2 (x - 2) (x - 6)$

6.

If $f (x) = x^{2} + 11 x + 30$ , without using your calculator, find:

a.

The $x$ -intercepts of $f (x)$ .

b.

The solutions to the equation $0 = x^{2} + 11 x + 30$ .

Check your answers by graphing.

7.

If $f (x) = x^{2} + 5 x - 36$ , without using your calculator, find:

a.

The $x$ -intercepts of $f (x)$ .

b.

The solutions to the equation: $0 = x^{2} + 5 x - 36$ .

Check your answers by graphing.

8.

If $f (x) = 4 x^{2} + 4 x - 3$ , without using your calculator, find:

a.

The $x$ -intercepts of $f (x)$ .

b.

The solutions to the equation: $0 = 4 x^{2} + 4 x - 3$ .

Check your answers by graphing.

Ready for More?

Write three functions in factored form with a line of symmetry $x = 2$ . For each function, find the vertex and the intercepts.

Takeaways

When a quadratic function is in factored form $f (x) = a (x + m) (x + n)$ ,

The $x$ -intercepts can be found by
The $y$ -intercept can be found by
The line of symmetry can be found by
The vertex can be found by

Vocabulary

factored form of a quadratic function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned to use the factored form of a quadratic equation to graph parabolas. We learned to find the $x$ -intercepts from the factors, then find the line of symmetry between the $x$ -intercepts. Once we knew the line of symmetry, we could find the vertex. We observed that the $x$ -intercepts are solutions to the equation $f (x) = 0$ , which allows us to use factoring as a method for solving a quadratic equation.

Retrieval

Solve the equation.

1.

$2 x + 1 = 0$

2.

$x - 12 = 0$