Lesson 3 Well, What Do You Know? Develop Understanding

Jump Start

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$x^{2} + 10 x + 24$
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$x^{2} + 5 x - 24$
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$x^{2} - 10 x - 24$
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$(x + 7) (x - 2)$
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$(x - 5) (x - 8)$

$x^{2} - 13 x + 40$
$(x + 6) (x + 4)$
$(x - 12) (x + 2)$

Learning Focus

Use patterns to efficiently graph quadratic functions from factored form.

How does the graph of a parabola relate to the equation of a quadratic function?

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

Open Up the Math: Launch, Explore, Discuss

This task is designed to help you to remember your work with parabolas from last year and to show what you know. For each of the following, find the indicated information.

1.

Given: $f (x) = - (x + 5) (x + 1)$

Type of function:

Vertex:

Line of Symmetry:

x-intercepts:

2.

Given: $f (x) = x^{2} - 2 x - 3$

Type of function:

Vertex:

Line of Symmetry:

$x$ -intercepts:

y-intercept:

3.

Given: $f (x) = x^{2} + x - 6$

$x$ -intercepts:

$y$ -intercept:

line of symmetry:

vertex:

factored form of the equation:

4.

Given: $f (x) = (x + 3) (x - 3)$

Standard form of the equation:

$x$ -intercepts:

$y$ -intercept:

Line of symmetry:

5.

Given: $f (x) = - (x^{2} - 3 x - 4)$

Factored form of the equation:

$x$ -intercepts:

$y$ -intercept:

vertex:

line of symmetry:

6.

How do you find the $y$ -intercept of a quadratic equation in standard form? In factored form?

7.

How do you find the $x$ -intercepts of a quadratic equation in factored form?

8.

How do you find the line of symmetry of a parabola when the equation is in factored form?

9.

How do you find the vertex when you know the line of symmetry?

10.

How do you find the standard form equation of a quadratic equation in factored form?

Ready for More?

See if you can find these functions:

a.

Two different quadratic functions that have $x$ -intercepts $(1, 0)$ and $(3, 0)$ .

b.

Two different quadratic functions that have a $y$ -intercept of $(0, - 3)$ .

c.

Two different quadratic functions that have a line of symmetry of $x = - 1$ .

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 parabola has a maximum if

Vocabulary

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

Lesson Summary

In this lesson, we reviewed how to graph quadratic functions in factored form and standard form. 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

1.

Fill in the table of values for each equation and then graph each function on the coordinate grid provided.

a.

$x$	$y = x$	$y = x - 4$	$y = 2 x$
$- 3$
$- 2$
$- 1$
$0$
$1$
$2$
$3$

b.

2.

How does the $- 4$ in the second equation change the graph in comparison to $y = x$ ?

3.

How does the $2$ in the third equation change the graph in comparison to $y = x$ ?

Write the equation of a line that is parallel to the given line and through the given point.

4.

$y = \frac{3}{5} x - 7$ point $(5, - 12)$

5.

$y = - \frac{2}{7} x + 1$ point $(- 14, 5)$