Activity Set 3.3PREP PPTX

Visual Algebra for Teachers

Chapter 3

REAL NUMBERS AND QUADRATIC FUNCTIONS

Visual Algebra for Teachers

Activity Set 3.3

Algebra Pieces andQuadratic Functions

Visual Algebra for Teachers

PURPOSETo learn: to use algebra pieces to find the key features of

quadratic graphs such as y-intercepts, x-intercepts, turning points and to find points of intersections with other quadratic and linear graphs.

to connect quadratic algebra piece work with the corresponding symbolic steps.

about the factored form of a quadratic function and its relationship to the x-intercepts of a parabola.

Black and red tiles, white and opposite white n-strips and black and red x-squares

Graphing calculator with table functions (recommended)

MATERIALS

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INTRODUCTION

Quadratic Function (general form)Since a quadratic function is a polynomial of degree two, we can think of all quadratic functions as having a form:

where the coefficients a, b and c are real numbers and a is nonzero.

2y ax bx c

Quadratic Features / Algebra PiecesFinding Important Features of Quadratic Functions with Algebra Piece Models

You may have noticed that all of the important features of the parabolas graphed in Activity Set 3.2 were easy to find.

For many quadratic functions, the important features are not so obvious.

Let’s consider the extended tile sequence with rectangle tile figures from Activity Set 3.2

Quadratic Features/Algebra Pieces

It turns out we can use the xth figure of this sequence to easily determine the y-intercept (this is always true), and in this case, the x-intercepts.

Once we have found the x-intercepts, it is relatively easy to find the turning point of any quadratic function.

Function and x-interceptsThe x-intercepts on any graph are the points where the graph crosses the x-axis and the y-value is 0.

We determined the symbolic formula by looking at the edge sets of the xth figure of this sequence.

( ) ( 1)g x x x

x-Intercepts from Edge SetsWe can also use these edge sets to find the x-intercepts.

The first step for finding the x-intercepts of any function is to set the function equal to zero.

In this case:

For this function, this is:

( ) 0g x

( 1) 0x x

Zero ArraysIn Chapter 1 we learned that an array of black and red tiles has net value 0, only if at least one of the edge sets of the array also has net value 0.

The xth figure of g(x) is rectangular, and therefore, it is an array.

The array representing g(x) can only be equal to 0 (have net value 0) if one or both edge sets has net value 0.

Zero Arrays and g(x)Therefore, to find the x-intercepts of , we set each edge set = 0 and solve

This yields x = 0 and x = -1 as the x values for the x-intercepts of

( )g x

( ) ( 1) 0

0 or

1 0 1

g x x x

x

x x

( )g x

g(x) all togetherx-intercepts are x = 0 and x = 1y-intercept is (0, 0) [no B/R pieces in xth figure]

Must Be Array = 0You can ONLY find x-intercepts (and solve equations) using the edge sets of a rectangle if you have an array set equal to zero.

For example, if an algebra piece model is organized as:

you would first need to move all of the pieces to one side of the equal sign before proceeding. In this case, this would look like:

2 6 7x x

2 6 7 0x x

Factored Form of a QuadraticSo far we see g(x) can be written in two ways:

g(x) =x2 +1

g(x) =x(x+1)

The second method is called the Factored Form of g(x)

The ideas of factoring a quadratic function and finding the x-intercepts of a quadratic function go hand and hand. Can you see why?

Non-Rectangular xth Figures Not all xth figures are rectangular.

However, one can often add zero pairs of white and opposite white x-strips to create a rectangular shape and we will explore this idea in this activity set.

Try to figure this out before going on to the next slide.Analyze the extended sequence, sketch the xth figure and give the symbolic formula for the sequence simplified into a y = ax2 + bx + c form.

Question #1a (prep demo)

Use algebra pieces to construct the xth figure before going on to the next slide.

12 reds + the figure number squared + opposite the figure number

Question #1a (prep demo)

xth figure

Question #1a (prep demo)

2( ) 12f x x x

What is the y-intercept for y = f(x)?

How can you look at an extended tile sequence and quickly tell the y-intercept?

Question #1b

Try to figure this out before going on to the next slide. Use algebra pieces to model setting the xth figure of f(x) equal to 0.

The xth figure of f(x) is not rectangular.

Add zero pairs (one pair at a time) of white and opposite white x-strips to the xth figure of f(x) until you can form a set of algebra pieces with the same net value as f(x) that can be made into a rectangle.

Lay out the edges of the rectangle and sketch the edge and rectangle model.

Question #1c (prep demo)

Add 3 sets of zero pair white / opposite white strips and arrange in a rectangle. Use this to determine the factored form of the function.

Question #1cd (prep demo)

2 12

( 4)( 3)

y x x

x x

e) Use the edge sets from part c (and the factored form of f(x)) to determine the x-intercepts of f(x). Show your work.

f) Use the x-intercepts from part d. to determine the turning point for f(x) . Show your work.

g) Plot the x-intercepts, the y-intercept and the turning point for y = f(x) , if necessary, plot a few more coordinate pairs and then sketch the entire graph.

h) What is the range of y = f(x)?

Questions #1efgh

You are now ready for:

PREP QUIZ 3.3

See Moodle

Visual Algebra for Teachers