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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
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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