1 Quadratic functions A. Quadratic functions B. Quadratic equations C. Quadratic inequalities

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Quadratic functions

A. Quadratic functionsB. Quadratic equations

C. Quadratic inequalities

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A. Quadratic functions

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A. Quadratic functions

Example

Remember exercise 4 (linear functions): For a local pizza parlor the weekly demand functionis given by p=26-q/40. Express the revenue as a function of the demand q.

Solution:revenue= price x quantity = 26q –q²/40

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A. Quadratic functions

Example

Group excursion • Minimum 20 participants• Price of the guide: 122 EUR• For 20 participants: 80 EUR per person• For every supplementary participant: for everybody

(also the first 20) a price reduction of 2 EUR per supplementary participant

Revenue of the travel agency when there are 6 supplementary participants?

total revenue = 122 + (20 + 6) (80 2 6) = 1890

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A. Quadratic functions

Example

• Minimum 20 participants• Price of the guide: 122 EUR• For 20 participants: 80 EUR per person• For every supplementary participant: for everybody

(also the first 20) a price reduction of 2 EUR per supplementary participant

Revenue y of the travel agency when there are x supplementary participants?

QUADRATIC FUNCTION!

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A. Quadratic functions

Definition

A function f is a quadratic function if and only if f(x) can be written in the form f(x) = y=ax² + bx + c where a, b and c are constants and a 0.

(Section 3.3 p141)

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A. Quadratic functions

Example

Equation:

Graph:

x y

0 1722

1 1760

2 1794

… …

10 40

500

2000

x

y

22 40 1722y x x

Table: PARABOLA

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B. Quadratic equations

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B. Quadratic equations

Example

2x² + 40x + 1722 = 1872 2x² + 40x + 1722 1872 = 0

2x² + 40x 150 = 0

We have to solve the equation 2x² + 40x 150 =0

Quadratic equation

Revenue equal to 1872?Group excursion

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B. Quadratic equations

Definition

(Section 0.8 p36)

A quadratic equation is an equation that canbe written in the form f(x) = y=ax² + bx + cwhere a, b and c are constants and a 0.

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B. Quadratic equations

Exercises(Section 0.8 – example 1 p36)1. Solve x²+x-12=0

2. Solve (3x-4)(x+1)=-2

3. Solve 4x-4x³=0

4. Solve

5. Solve x²=3

(Section 0.8 – example 2 p37)

(Section 0.8 – example 3 p37)

(Section 0.8 – example 4 p37)

(Section 0.8 – example 5 p38)

Solving a quadratic equation - strategy 1: based on factoring

6²

)12(7

2

5

3

1

yy

y

y

y

y

y

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B. Quadratic equations

Solution

Solving a quadratic equation - strategy 2:

if discriminant d > 0: two solutions if discriminant d = 0: one solution

if discriminant d < 0: no solutions

Discriminant: d = b² 4ac

a

dbx

21

a

dbx

22

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B. Quadratic equations

Exercises(Section 0.8 – example 6 p36)1. Solve 4x² - 17x + 15 = 0

2. Solve 2 + 6 y + 9y² = 0

3. Solve z² + z + 1 = 0

4. Solve

(Section 0.8 – example 7 p37)

(Section 0.8 – example 8 p37)

(Section 0.8 – example 9 p37)0891

36

xx

2

Supplementary exercisesExercise 1

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A. Quadratic functions

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A. Quadratic functions

Graph

Quadratic functions: graph is a PARABOLA

If a>0, the parabola opens upward.If a<0, the parabola opens downward

What does the sign of a mean ?

Group excursion: y=-2x²+40x+1722Example a<0

(Section 3.3 p142-144)

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A. Quadratic functions

Graph

Quadratic functions: graph is a PARABOLA

Sign of the discriminant determines the number of intersections with the horizontal axis

Graphical interpretation of y=ax²+bx+c=0 ?

Zero’s, also called x-intercepts, solutions of the quadratic equation y=ax²+bx+c=0 correspond to intersections with the horizontal x-axis

Group excursion: y=-2x²+40x+1722Example

d=124²>0 x=41; (x=-21)

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A. Quadratic functions

Graph sign of the discriminant determines the number of intersections with the horizontal axis

sign of the coefficient

of x 2

determines the

orientation of the

opening

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A. Quadratic functions

Graph

Quadratic functions: graph is a PARABOLA

The y-intercept is c.

What is the Y-intercept ?

10 40

500

2000

x

y

Group excursion: y=-2x²+40x+1722

Example

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A. Quadratic functions

Graph

Each parabola is symmetric about a vertical line. Which line ?

Both parabola’s at the right show a point labeled vertex, where the symmetry axis cuts the parabola. If a>0, the vertex is the “lowest” point on the parabola. If a<0, the vertex refers to the “highest” point.

•

•

x-coordinate of vertex equals -b/(2a)

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A. Quadratic functions

Example Group excursion: Maximum revenue?

In this case you can find it e.g. using the table:

So: 10 supplementary participants (30 participants in total)

This can also be determined algebraically, based on a general study of quadratic functions!

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A. Quadratic functions

Graph

x-coordinate of vertex equals -b/(2a)

Group excursion:Example 22 40 1722y x x

40

102 2

x

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A. Quadratic functions

Exercises

1. Graph the quadratic function y = -x² - 4x + 12. Sign a? Sign d? Zeros? Y-intercept? Vertex? 2. A man standing on a pitcher’s mound throws a ball straight up with an initial velocity of 32 feet per second. The height of the ball in feet t seconds after it was thrown is described by the function h(t)= - 16t²+32t+8, for t ≥ 0. What is the maximum height? When does the ball hit the ground?

(Section 3.3 – example 1 p143)

(Section 3.2 – Apply it 14 p144)

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A. Quadratic functions

Supplementary exercises

• Exercise 2 (f1 and f5),

• Exercise 3, 7, 5• rest of exercise 2• Exercise 4, 6, 8 and 9

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A. Quadratic functions

Supplementary exercises

-1000

-500

0

500

1000

1500

2000

0 200 400 600 800 1000

x

rExercise 7

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C. Quadratic inequalities

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Definition

C. Quadratic inequalities

A quadratic inequality is one that can be written in the form ax² + bx + c ‘unequal’ 0, where a, b and c are constants and a 0 and where ‘unequal’ stands for <, , > or .

ExampleSolve the inequality 143)52( 2 xxx

i.e. Find all x for which 143)52( 2 xxx

standard form 01452 xx

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C. Quadratic inequalities

ExampleStudy the equality first:

01452 xx

Solve now inequality

01452 xx

x=-2; x=7

conclusion: x-2 or x7 interval notation: ]-,-2][7,[

Next, determine type of graph

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C. Quadratic inequalities

02

cbxax

inequalities that can be reduced to the form

... and determine the common points with the x-axis by solving the EQUATION

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C. Quadratic inequalities

Supplementary exercises

• Exercise 10 (a)• Exercises 11 (a), (c)• Exercises 10 (b), (c), (d)• Exercises 11 (b), (d)

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Quadratic functions

Summary• Quadratic equations : discriminant d, solutions• Quadratic functions : Graph: Parabola, sign of a, sign of d, zeros vertex, symmetry axis,

minimum/maximum• Quadratic inequalities : solutions

Extra: Handbook –Problems 0.8: Ex 31, 37, 45, 55, 57, 79Problems 3.3: Ex 11, 13, 23, 29, 37, 41