Lesson 9 Lining Up Quadratics Solidify 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 the factored form of a quadratic equation relate to graphing a parabola?

Open Up the Math: Launch, Explore, Discuss

Use technology to graph each function and find the vertex, the $y$ -intercept, and the $x$ -intercepts. Be sure to properly write the intercepts as points.

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 vertical stretch?

Now it’s time to try your strategy! Factor each of the functions, use the strategy that you found in problem 4 to find the vertex, intercepts, and line of symmetry, and graph each parabola without technology. Check your work with technology, and if your graph is wrong, go back and examine each step of your work to diagnose the problem.

5.

$f (x) = x^{2} - 2 x - 8$

Factored form of the function:

Line of symmetry:

Vertex:

$x$ -intercepts:

$y$ -intercept:

Vertical stretch:

6.

$f (x) = - x^{2} - 6 x - 5$

Factored form of the function:

Line of symmetry:

Vertex:

$x$ -intercepts:

$y$ -intercept:

Vertical stretch:

7.

$f (x) = 2 x^{2} - 12 x + 16$

Factored form of the function:

Line of symmetry:

Vertex:

$x$ -intercepts:

$y$ -intercept:

Vertical stretch:

8.

$f (x) = - 2 x^{2} - 5 x + 3$

Factored form of the function:

Line of symmetry:

Vertex:

$x$ -intercepts:

$y$ -intercept:

Vertical stretch:

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

The vertical stretch and reflection can be found by

Vocabulary

factored form
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 several patterns that helped to make factored form an efficient way to graph quadratics.

Retrieval

Multiply each expression, and write your answer as a trinomial.

1.

$(2 x + 7) (x + 5)$

2.

$(3 x - 2) (x + 4)$

3.

$(3 x + 1) (2 x - 9)$

Given the vertex form of a quadratic function, identify the vertex, intercepts, and vertical stretch of the parabola.

4.

$y = {(x + 5)}^{2} - 4$

a.

Vertex:

b.

$x$ -intercept(s):

c.

$y$ -intercept:

d.

Stretch: