3.1

Quadratic Functions and Models

Quadratic Functions

A quadratic function is of the form

f(x) = ax2 + bx + c,

where a, b, and c are real numbers, a ≠ 0.

The graph of a quadratic function is a parabola. The domain of a quadratic function is all real

numbers. These functions have a linear rate of change.

Vertex The maximum or minimum point of a parabola

Axis of symmetry The vertical line passing through the vertex

Leading coefficient In a quadratic function is this “a” (the coefficient of x2). When positive the graph opens up. When negative the graph opens down. Larger values of |a| result in a narrower parabola, smaller

values of |a| result in a wider parabola.

Vertex Form of a Quadratic Function Vertex Form

The parabolic graph of f(x) = a(x – h)2 + k has vertex (h,k). Graph opens up when a > 0, down when a < 0.

Examples Page 184 Identify f as being linear, quadratic,

or neither. If it is quadratic, identify the leading coefficient and evaluate f(-2). #2 f(x) = 1 – 2x + 3x2 #4 f(x) = (x2 + 1)2

#6 f(x) = 1/5 x2

Page 185 Identify the vertex and the leading coefficient. Then write the equation as f(x) = ax2 + bx + c #18 f(x) = 5(x + 2)2 – 5 #20 f(x) = ½(x + 3)2 – 5

Finding the vertex

Vertex Formula The vertex for the graph of f(x) = ax2 + bx + c with

a ≠ 0 is the point

,2 2

b bf

a a

Examples

Page 185 Use the vertex formula to determine the vertex of the graph of f. #26 f(x) = 2x2 – 2x + 1

#30 f(x) = -3x2 + x – 2

Completing the Square

y = x2 + 6x – 8

Examples

Page 185 Write the given equation in the form f(x) = (x – h)2 + k. #40 f(x) = x2 + 10x + 7

#50 f(x) = 6 + 5x – 10x2

Quadratic Regression on the Calculator Enter data into List 1 and List 2 

Choose Stat Calc 5: quadreg enter enter

To have the data go directly into y = Before you press the second enter Choose vars y-vars function y1

Examples

Page 187 #98 a) Make a scatterplot of the data.

b) Find the values for a, h and k. Graph f(x) together with the data in the same viewing rectangle.

c) Approximate the undetermined value(s) in the table.

U.S. population in millions

Year 1800 1820 1840 1860 1870 1880

Population 5 10 17 31 ? 50

Year 1900 1920 1940 1960 1980 2000

Population 76 106 132 178 226 ?

Problem Solving

Page 186 #82 Match the physical situation with the graph of the

quadratic function that models it best.

Example

Page 187 #102 The cables that support a suspension bridge,

such as the Golden Gate Bridge, can be modeled by parabolas.

Suppose that a 300-foot long suspension bridge has towers at its ends that are 120 feet tall, as illustrated in the accompanying figure.

If the cable comes within 20 feet of the road in the center of the bridge, find the quadratic function that models the height of the cable above the road a distance of x feet from the center of the bridge.

120 ft

300 ft20 ft

3.2

Quadratic Equations and Problem Solving

Examples

Page 201 #2

#10

2 9 10 8x x

28 63 46x x

Quadratic formula

The solutions to the quadratic equation

ax2 + bx + c = 0, where a ≠ 0, are given by

x = b b ac

a

2 4

2

#16

#18

23( 5) 6 0x

23 1 10

4 2 2x x

The Discriminant

The discriminant is used to determine the number of real solutions to ax2 + bx + c =0. If b2 – 4ac > 0, there are two real solutions. If b2 – 4ac = 0, there is one real solution. If b2 – 4ac < 0, there are no real solutions.

Examples Page 202

a. Write the equation in standard form

b. Calculate the discriminant and determine the number of real solutions

c. Solve the equation.

#46

#58

#60

28 2 14x

24 6x x

(5 3) 1x x

Solve graphically

Page 202 #4222 4 1.595x x

Problem Solving

Page 203 #100 From 1984 to 1994 the cumulative number of

AIDS cases can be modeled by the equation

Where x represents years after 1984. Estimate the year when 200,000 AIDS cases had been diagnosed.

2( ) 3034 14,018 6400,C x x x

Page 204 #108 A rectangular pen for a pet is under construction

using 100 feet of fence.a. Determine the dimension that result in an area of 576

square feet.

b. Find the dimensions that give the maximum area.

3.3

Quadratic Inequalities

Solving Quadratic Inequalities Write in Standard Form Solve Use the boundary numbers to test points Use the table or graph to write your solution

Examples

Page 213 Solve each equation and inequality. Write the solution set for each inequality in interval notation. #12

a.

b.

c. #14

a.

b.

c.

2 8 12 0x x 2 8 12 0x x 2 8 12 0x x

2 17 0n 2 17 0n

2 17 0n

#16a.

b.

c.

#18a.

b.

c.

#22a.

b.

c.

27 4 0x x 27 4 0x x 27 4 0x x

2 2 1 0x x 2 2 1 0x x 2 2 1 0x x

22 4 3 0x x 22 4 3 0x x 22 4 3 0x x

3.4

Transformations of Graphs

Shifting and Stretching

Graph y1 = x2

y2 = x2 + 3 y3 = x2 – 3

What pattern do you see?

Vertical Shifts

Vertical Shifts g(x) = f(x) + a, shift graph up a units g(x) = f(x) – a, shift graph down a units

Graph y1 = x2

y2 = (x + 3)2

y3 = (x – 3)2

What pattern do you see?

Horizontal Shifts

Horizontal Shifts g(x) = f(x + a), shift graph left a units g(x) = f(x – a), shift graph right a units

Graph y1 = x2

y2 = 3x2

y3 = 6x2

Stretching

Vertical and Horizontal stretches: For a >0, the graph g(x) = af(x) stretches the

graph vertically by a factor of a. For a >1, the graph g(x) = f(ax) compresses the

graph horizontally by a factor of a. h(x) = f(x/a) compresses the graph horizontally by

a factor of a.

Graph y1 = x2

y2 = -x2

Negative Coefficients

When you multiply by a negative it reflects (flips) the graph over the x-axis.

Predict what will happen y = -x2 + 3

y = -2(x + 5)2 - 3

f(x) = x2, af(x + b) + c

Examples

Page 229 Use the accompanying graph of y = f(x) to sketch a graph of each equation. #12

a. y = f(x + 1)

b. y = -f(x)

c. Y = 2f(x) #14

a. y = f(x – 1) - 2

b. y = -f(x) + 1

c. y = f(1/2x)

Other Parent Graphs

y = x

y = |x|

y = x3

y = √x

Examples

Page 230 Use transformations for graphs to sketch a graph of f. #50

#52

#68

( ) 1f x x

( ) | 4 |f x x

3( ) ( ) 1f x x