Family of Quadratic Functions

Lesson 5.5a

General Form• Quadratic functions have the standard

formy = ax2 + bx + c

a, b, and c are constants a ≠ 0 (why?)

• Quadratic functions graph as a parabola

Zeros of the Quadratic

• Zeros are where the function crosses the x-axis Where y = 0

• Consider possible numbers of zeros

None (or two complex)None (or two complex) OneOne TwoTwo

Axis of Symmetry

• Parabolas are symmetric about a vertical axis

• For y = ax2 + bx + c the axisof symmetry is at

• Given y = 3x2 + 8x What is the axis of symmetry?

2

bx

a

Vertex of the Parabola

• The vertex is the “point” of theparabola The minimum value Can also be a maximum

• What is the x-value of thevertex?

• How can we find the y-value?

2

bx

a

( )2

by f x f

a

Vertex of the Parabola

• Given f(x) = x2 + 2x – 8

• What is the x-value of the vertex?

• What is the y-value of the vertex?

• The vertex is at (-1, -9)

21

2 2 1

bx

a

( 1) 1 2 9 9f

Vertex of the Parabola

• Given f(x) = x2 + 2x – 8 Graph shows vertex at (-1, -9)

• Note calculator’s ability to find vertex (minimum or maximum)

Shifting and Stretching

• Start with f(x) = x2

• Determine the results of transformations ___ f(x + a) = x2 + 2ax + a2

___ f(x) + a = x2 + a ___ a * f(x) = ax2

___ f(a*x) = a2x2

a) horizontal shift

b) vertical stretch or

squeeze

c) horizontal stretch or

squeeze

d) vertical shift

e) none of these

Other Quadratic Forms

• Standard formy = ax2 + bx + c

• Vertex formy = a (x – h)2 + k Then (h,k) is the vertex

• Given f(x) = x2 + 2x – 8 Change to vertex form Hint, use completing the square

Experiment with Quadratic Function

Spreadsheet

Experiment with Quadratic Function

Spreadsheet

Vertex Form

• Changing to vertex form

2

2

2

2 8

2 8

y x x

y x x

y x

Add something in to make a perfect square trinomial

Add something in to make a perfect square trinomial

Subtract the same amount to keep it even.

Subtract the same amount to keep it even.

Now create a binomial squared

Now create a binomial squared This gives us the

ordered pair (h,k)

This gives us the ordered pair (h,k)

Assignment

• Lesson 5.5a

• Page 231

• Exercises 1 – 25 odd