Quadratic and polynomial functions Lesson 4 Source : Lial, Hungerford and Holcomb (2007), Mathematics with Applications, 9th edition, Pearson Prentice.

Quadratic and polynomial functions Lesson 4

Source :Lial, Hungerford and Holcomb (2007), Mathematics with Applications, 9th edition, Pearson Prentice Hall, ISBN 0-321-44947-9 (Chapter 3 pp.163-178)

Part 1: Quadratic Functions

• Properties of Quadratic function• Graphing Quadratic Function• Vertex form

Example:

f(x) = 3x2 + 3x + 5 g(x) = 5 x2

Quadratic Function Properties

• The graph of a quadratic function is a parabola—a U shaped graph that opens either upward or downward.

• A parabola opens upward if its leading coefficient a is positive and opens downward if a is negative.

• The highest point on a parabola that opens downward and the lowest point on a parabola that opens upward is called the vertex. (The graph of a parabola changes shape at the vertex.)

• The vertical line passing through the vertex is called the axis of symmetry.

• The leading coefficient a controls the width of the parabola. Larger values of |a| result in a narrower parabola, and smaller values of |a| result in a wider parabola.

2f(x) = ax +bx + c

Leading Coefficient

Examples of different parabolas

Quadratic Functions

The graph of a quadratic function is a parabola.

A parabola can open up or down.

If the parabola opens up, the lowest point is called the vertex.

If the parabola opens down, the vertex is the highest point.

NOTE: if the parabola opened left or right it would not be a function!

y

x

Vertex

Vertex

y = ax2 + bx + c

The parabola will open down when the a value is negative.

The parabola will open up when the a value is positive.

Standard Formy

x

The standard form of a quadratic function is

a > 0

a < 0

y

x

Line of Symmet

ry

Line of Symmetry

Parabolas have a symmetric property to them.

If we drew a line down the middle of the parabola, we could fold the parabola in half.

We call this line the line of symmetry.

The line of symmetry ALWAYS passes through the vertex.

Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.

Find the line of symmetry of y = 3x2 –

18x + 7

Finding the Line of Symmetry

When a quadratic function is in standard form

The equation of the line of symmetry is

y = ax2 + bx + c,

2ba

x

For example…

Using the formula…

This is best read as …

the opposite of b divided by the quantity of 2 times a.

182 3

x 186

3

Thus, the line of symmetry is x = 3.

Finding the Vertex

But we know the line of symmetry always goes through the vertex.

Thus, the line of symmetry gives us the x – coordinate of the vertex.

To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.

STEP 1: Find the line of symmetry

STEP 2: Plug the x – value into the original equation to find the y value.

y = –2x2 + 8x –3

8 8 22 2( 2) 4ba

x

y = –2(2)2 + 8(2) –3

y = –2(4)+ 8(2) –3

y = –8+ 16 –3

y = 5

Therefore, the vertex is (2 , 5)

Example

Use the graph of the quadratic function shown to determine the sign of the leading coefficient, its vertex, and the equation of the axis of symmetry.

Leading coefficient: The graph opens downward, so the leading coefficient a is negative.

Vertex: The vertex is the highest point on the graph and is located at (1, 3).

Axis of symmetry: Vertical line through the vertex with equation x = 1.

Graphing Quadratic Function: Method 1

Graphing Quadratic Function: Method 1

Graphing Quadratic Function: Method 2

Graphing Quadratic Function: Method 2

The standard form of a quadratic function is given by

y = ax2 + bx + c

STEP 1: Find the line of symmetry

STEP 2: Find the vertex

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

Graphing Quadratic Function: Method 3

STEP 1: Find the line of symmetry

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

( )4

12 2 2

bx

a

-= = =

y

x

Thus the line of symmetry is x = 1

Graphing Quadratic Function: Method 3

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

STEP 2: Find the vertex

y

x

( ) ( )22 1 4 1 1 3y= - - =-

Thus the vertex is (1 ,–3).

Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.

Graphing Quadratic Function: Method 3

5

–1

Let's Graph ONE! Try …

y = 2x2 – 4x – 1

( ) ( )22 3 4 3 1 5y= - - =

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

y

x

( ) ( )22 2 4 2 1 1y= - - =-

3

2

yx

Graphing Quadratic Function: Method 3

The quadratic function f(x) =ax2 + bx + c can be written in an alternate form that relies on the vertex (h, k).

Example: f(x) = 3(x - 4)2 + 6 is in vertex form with vertex (h, k) = (4, 6).

What is the vertex of the parabola given by

f(x) = 7(x + 2)2 – 9 ? ?

Vertex = (-2,-9)

ExampleWrite the formula f(x) = x2 + 10x + 23 in vertex

form by completing the square.

223 10y x x

2 10 23y x x Given formulaGiven formula

Subtract 23 from each Subtract 23 from each side.side.

Add (10/2)Add (10/2)22 = 25 to both = 25 to both sides.sides.

Factor perfect square Factor perfect square trinomial.trinomial.

Subtract 2 form both Subtract 2 form both sides..sides..

2223 215 0 5y x x 22 ( 5)y x

2( 5) 2y x Vertex is h= -5 k = -2.What is the vertex?

Example: Convert the quadratic f(x) = 3(x + 2)2 – 8 which is in vertex form to the form f(x) = ax2 + b + c

f(x) = 3(x + 2)2 – 8Given formula.

Expand the quantity squared.

f(x) = 3(x2 + 4x + 4) - 8

Multiply by the 3. f(x) = 3x2 + 12x + 12 - 8

Simplfy. f(x) = 3x2 + 12x + 4

Part 2: Polynomial Functions• Properties of Basic Polynomial Function

naxy

Basic Polynomial Function

NOTE: Linear and Quadratic functions we considered earlier belong to the family of polynomial functions!

Basic Polynomial Function

Basic Polynomial Function

Basic Polynomial Function

Example: Draw the graph of the following functions:

7

4

5)

3)

xyb

xya