Quadratic Functions & Models

MATH 109 - Precalculus S. Rook

Overview

• Section 2.1 in the textbook: – Properties of quadratic functions – Finding the vertex of quadratic functions – Maximum & minimum of quadratic funcitons – Applications 2

Properties of Quadratic Functions

Quadratic Function

• • •

Quadratic Function:

a function that can be represented as f(x) =

a

x

2 +

b

x +

c

, where

a

,

b

, and

c

are constants and

a

≠ 0 Key features of a quadratic function: – Vertex – Axis of symmetry Has a U-shape called a

parabola

– If

a

> 0, opens up (smile) If

a

< 0, opens down (frown) 4

Axis of Symmetry

• •

Axis of Symmetry (x = k):

property that any point

m

a vertical line with the units in the horizontal direction from x = k will have a companion point units in the horizontal direction from x = k

–m

For a quadratic function, the axis of symmetry passes through a special point called the

vertex

– Occurs in the middle of the parabola – Points on one side of a parabola can be reflected across the axis of symmetry to obtain a companion point on the other side 5

Standard Form of a Quadratic Function • • • •

Standard Form of a Quadratic Function:

function of the form f(x) =

a

(x –

h

) 2 + k,

a

a quadratic ≠ 0 – vertex of (

h

, k) – axis of symmetry of x =

h

Obtained from f(x) =

a

x

2

square

+

b

x +

c

by

completing the

Powerful form of a quadratic function because BOTH the vertex and axis of symmetry are easily accessible Can easily sketch a quadratic function in standard form using transformations (1.7) 6

Properties of Quadratic Functions (Example)

Ex 1:

i) Sketch the graph of the quadratic function by converting it to standard form, ii) identify the vertex, iii) identify the axis of symmetry, iv) identify the x-intercepts a) f(x) = -x 2 + 2x + 5 b) g(x) = 4x 2 – 4x + 21 7

Writing the Equation of a Quadratic Function in Standard Form (Example)

Ex 2:

Write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point a) Vertex: (3, 4); point: (1, 2) b) Vertex: (-2, 5); point: (0, 9) 8

Finding the Vertex of Quadratic Functions

Vertex & Vertex Formula

• • Depending how the parabola opens, the

vertex

represents the minimum or maximum point of the quadratic function – There are occasions when we need only the vertex Given f(x) =

a

x

2 +

b

x +

c

, the vertex is (h, k) where

h

 

b

2

a

and

k



f



f b

2

a

– This is a much quicker way to find the vertex when we

DO NOT need to sketch the parabola

10

Vertex & Vertex Formula (Example)

Ex 3:

Use the vertex formula to find the vertex of the quadratic function: f(x) = -2x 2 – 4x + 1 11

Minimum & Maximum of Quadratic Functions

• Minimum & Maximum of Quadratic Functions Depending on the value of

a

in f(x) =

a

x

2 +

b

x +

c

, the

vertex

is

EITHER

the minimum (lowest) or the maximum (highest) point of the quadratic function – If

a

> 0, the vertex is the minimum point If

a

< 0, the vertex is the maximum point – The vertex determines the

range

of a quadratic function 13

Minimum & Maximum of Quadratic Functions (Example)

Ex 4:

i) Indicate whether the quadratic function has a minimum or maximum, ii) find the minimum or maximum value, iii) state the domain and range a) f(x) = 4x 2 + 16x – 3 b) g(x) = -x 2 + 2x + 5 14

Applications

Applications

• Applications for quadratic functions are fairly straightforward – Most problems ask for either the maximum or minimum

value

of some quadratic function • Referring to the

y-coordinate

of the

vertex

16

Application (Example)

Ex 5:

A manufacturer of lighting fixtures has daily C(x) = 800 – 10x + 0.25x 2 where C(x) is the total cost (in dollars) and x is the number of units produced. How many fixtures should be produced to yield a minimum cost AND what is that minimum cost?

17

Summary

• • • After studying these slides, you should be able to: – Determine which way a quadratic function will open given its equation – Write a quadratic function in standard form and then sketch it, state the vertex, and axis of symmetry – – Use the vertex formula Find the minimum or maximum value of a quadratic function and state its domain – Solve application problems Additional Practice – See the list of suggested problems for 2.1

Next lesson – Polynomial Functions of Higher Degree (Section 2.2) 18