5.1 modeling data with quadratic functions 1.quadratic functions and their graphs

5.1 Modeling Data with Quadratic Functions

1. Quadratic Functions and Their Graphs

1) Quadratic Formulas and Their Graphs

A quadratic function is a function that produces a parabola.

1) Quadratic Formulas and Their Graphs

A quadratic function is a function that produces a parabola.

1) Quadratic Formulas and Their Graphs

A quadratic function is a function that produces a parabola.

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3

1) Quadratic Formulas and Their Graphs

The equation of a quadratic function can be written in standard form. 

cbxaxxf 2)(

Quadratic term

Linear term

Constant term

Quadratic Function:f(x) = ax2 + bx + c

‘a’ cannot = 0

1) Quadratic Formulas and Their Graphs

Since the largest exponent of function is 2, we say that a quadratic equation has a degree of 2. 

Equations of second degree are called quadratic.

QUADRATICS - - what are they?

Important Detailso c is y-intercepto a determines shape and position

if a > 0, then opens up

if a < 0, then opens downo Vertex: x-coordinate is at –b/2a

FORM _______________________Y = ax² + bx + c Quadratic

term

Linear

term

Constant

Parts of a parabola

This is the y-intercept, cIt is where the parabola crosses the y-axis

This is the vertex, V

This is the calledthe axis of symmetry, a.o.s.Here a.o.s. is the line x = 2

These are the rootsRoots are also called:-zeros-solutions- x-intercepts

STEPS FOR GRAPHINGY = ax² + bx +c

1 HAPPY or SAD ?2 VERTEX = ( -b / 2a , f(-b / 2a) )3 T- Chart4 Axis of Symmetry

GRAPHING - - Standard Form (y = ax² + bx + c)

y = x² + 6x + 8 1) It is happy because a>0

2) FIND VERTEX (-b/2a) a =1 b=6 c=8 So x = -6 / 2(1) = -3 Then y = (-3)² + 6(-3) + 8 = -1

So V = (-3 , -1)

3) T-CHART

X Y = x² + 6x + 8

-2 y = (-2)² + 6(-2) + 8 = 0

0 y = (0)² + 6(0) + 8 = 8

Why -2 and 0?

Pick x values where the graph will cross an axiw

The graph will be symmetrical. Once you have half the graph, the other two points come from the mirror of the first set of points.

GRAPHING - - Standard Form (y = ax² + bx + c)

y = -x² + 4x - 5 1) It is sad because a<0

2) FIND VERTEX (-b/2a) a =-1 b=4 c=-5 So x = - 4 / 2(-1) = 2 Then y = -(2)² + 4(2) – 5 = -1

So V = (2 , -1)

3) T-CHART

X Y = -x² + 4x - 5

1 y = -(1)² + 4(1) - 5 = -2

0 y = -(0)² + 4(0) – 5 = -5

Here we only have one point where the graph will cross an axis. Choose one other point (preferably between the vertex and the intersection point) to graph.

1) Quadratic Formulas and Their Graphs

Example 1:

Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term.

)3()() xxxfa 22 )5()() xxxxfb

1) Quadratic Formulas and Their Graphs

Example 1:

Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term.

xx

xxxfa

3

)3()()2

22 )5()() xxxxfb

This IS a quadratic function.

QUADRATIC TERM: x2

LINEAR TERM: 3x

CONSTANT TERM: none

1) Quadratic Formulas and Their Graphs

Example 1:

Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term.

xx

xxxfa

3

)3()()2

x

xxx

xxxxfb

5

5

)5()()22

22

This IS a quadratic function.

QUADRATIC TERM: x2

LINEAR TERM: 3x

CONSTANT TERM: none

This is NOT a quadratic function.

QUADRATIC TERM: none

LINEAR TERM: 5x

CONSTANT TERM: none

EX 3

Find the vertex, axis of symmetry and the corresponding points to P and Q.

y = x2 – 6x + 11

Ex 1

Is the function linear or quadratic?

f(x) = (2x – 1)2

EX 2

Is the function linear or quadratic?

f(x) = x2 – (x + 1)(x – 1)

EX 4

Find a quadratic function to model the given points:

(-2, -17) (1, 10) (5, -10)

Ex 5

y = 2x2 + x – c contains the point (1, 2). Find c.

1) Quadratic Formulas and Their Graphs

We can graph parabolas using a table of values.

1) Quadratic Formulas and Their Graphs

We can graph parabolas using a table of values.

Recall…graphing linear functions…

1) Quadratic Formulas and Their Graphs

Example 2:

Graph the parent function f(x) = x2 using a table of values.

1) Quadratic Formulas and Their Graphs

Example 2:

Graph the parent function f(x) = x2 using a table of values.

x y

-2

-1

0

1

2

1) Quadratic Formulas and Their Graphs

Example 2:

Graph the parent function f(x) = x2 using a table of values.

x y

-2 (-2)2 = 4

-1 (-1)2 = 1

0 (0)2 = 0

1 (1)2 = 1

2 (2)2 = 4

1) Quadratic Formulas and Their Graphs

Example 2:

Graph the parent function f(x) = x2 using a table of values.

x y

-2 4

-1 1

0 0

1 1

2 4 

1) Quadratic Formulas and Their Graphs

The axis of symmetry is a line that divides the parabola in half.

1) Quadratic Formulas and Their Graphs

The axis of symmetry is a line that divides the parabola in half.

The vertex is a maximum or minimum of the parabola.

1) Quadratic Formulas and Their Graphs

The axis of symmetry here is

x = 0

The vertex here is a minimum at

(0, 0)

1) Quadratic Formulas and Their Graphs

Points on the parabola have corresponding points that are equidistant from the axis of symmetry.

A B

A’ B’

1) Quadratic Formulas and Their Graphs

Example 3:

Identify the vertex and axis of symmetry for the parabola. Identify points corresponding to P and Q.

-2

-1

0

1

2

3

4

-2 -1 0 1 2 3 4

P

Q 4321-1

-2

-2 -1

1

2

3

1) Quadratic Formulas and Their Graphs

Example 3:

Identify the vertex and axis of symmetry for each parabola. Identify points corresponding to P and Q.

P Vertex: (1, -1)

Axis of symmetry: x = 1

P’ (3, 3)

Q’ (0, 0)

-2

-1

0

1

2

3

4

-2 -1 0 1 2 3 4

P

QQ’

P’

4321-1

-2

-2 -1

1

2

3