1
Quadratic functions
A. Quadratic functions
B. Quadratic equations
C. Quadratic inequalities
2
A. Quadratic functions
3
A. Quadratic functions
Example
Remember exercise 4 (linear functions):
For a local pizza parlor the weekly demand
function is 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
Fixed cost: 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  6  2) = 1890
5
A. Quadratic functions
Example
Group excursion
♦
♦
♦
♦
Minimum 20 participants
Fixed cost: 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?
total revenue = y = 122 + (20 + x)  (80  x  2)
= 2x2 + 40x + 1722
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A. Quadratic functions
revenue function y = 2x2 + 40x + 1722
is a quadratic function
Definition
A function f is a quadratic function if and only if
its equation 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 p. 141)
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A. Quadratic functions
Example
Equation: y  2 x 2  40 x  1722
Table:
x
0
1
2
…
y
1722
1760
1794
…
Graph: PARABOLA
2000
y
500
x
10
40
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B. Quadratic equations
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B. Quadratic equations
Example
Group excursion Revenue equal to 1872?
 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
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B. Quadratic equations
Definition
A quadratic equation is an equation that can be
written in the form
ax² + bx + c = 0
where a, b and c are constants and a  0.
(Section 0.8 p. 36)
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B. Quadratic equations
Solving a quadratic equation - strategy 1:
based on factoring, …
Exercises
1. Solve x²+x12=0
(Section 0.8 – example 1 p. 36)
2. Solve (3x  4)(x+1)= 2
(Section 0.8 – example 2 p. 37)
3. Solve 4x  4x³=0
(Section 0.8 – example 3 p. 37)
y  1 y  5 7(2 y  1)


4. Solve
y  3 y  2 y²  y  6
5. Solve x²=3
(Section 0.8 – ex. 4 p. 37)
(Section 0.8 – example 5 p. 38)
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B. Quadratic equations
Solving a quadratic equation - strategy 2:
formula based on the discriminant
Discriminant = d = b²  4ac
b d
b d
• if d > 0: two solutions: x1 
, x2 
2a
2a
b
• if d = 0: one solution: x 
2a
• if d < 0: no solutions
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B. Quadratic equations
Exercises
1. Solve 4x² - 17x + 15 = 0
(Section 0.8 – example 6 p. 36)
2. Solve 2x² + 40x  150 = 0
3. Solve 2 + 6
2 y + 9y² = 0
(example slide 9)
(Section 0.8 – example 7 p. 37)
4. Solve z² + z + 1 = 0
(Section 0.8 – example 8 p. 37)
1 9
 3 8 0
6
x x
(Section 0.8 – example 9 p. 37)
5. Solve
Supplementary exercises: exercise 1
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A. Quadratic functions
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A. Quadratic functions
Graph
Quadratic functions: graph is a PARABOLA
What does the sign of a mean?
♦ if a>0, the parabola opens upward
♦ if a<0, the parabola opens downward
Example (group excursion)
2000
y
y = 2x² + 40x + 1722
a = 2 < 0
(Section 3.3 p. 142-144)
500
x
10
40
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A. Quadratic functions
Graph
Quadratic functions: graph is a PARABOLA
Graphical interpretation of y=ax²+bx+c=0?
Sign of the discriminant determines the number of
intersections with the horizontal axis
Zero’s, also called x-intercepts, solutions of the
quadratic equation y=ax²+bx+c=0, correspond to
intersections with the horizontal x-axis
Example
Group excursion: y=-2x²+40x+1722
d=124²>0 x=41; (x=-21)
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A. Quadratic functions
Graph
sign of the
coefficient
of x2
determines
the
orientation
of the
opening
sign of the discriminant
determines the number of
intersections with horizontal axis
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A. Quadratic functions
Graph
Quadratic functions: graph is a PARABOLA
What is the Y-intercept?
2000
y
Example
Group excursion:
y=-2x²+40x+1722
500
The y-intercept is 1722.
The y-intercept is c.
x
10
40
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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)
axis of symmetry is line x=-b/(2a)
x = -b/(2a)
•
•
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A. Quadratic functions
Example
Group excursion: Maximum revenue?
y  2 x 2  40 x  1722
vertex is “highest” point
x-coordinate of vertex equals -b/(2a)
40
x
 10
2   2 
maximum revenue = y-coo of vertex = y(10) = 1922
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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)
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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?
(Section 3.3 – example 1 p. 143)
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 initial height of the ball?
What is the maximum height?
When is the ball back at a height of 8 feet?
(Section 3.2 – Apply it 14 p. 144)
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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
2000
r
Exercise 7
1500
1000
500
x
0
-500 0
-1000
200
line
400
600
800
parabola
1000
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C. Quadratic inequalities
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C. Quadratic inequalities
Definition
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 .
Example
Solve the inequality (2x  5) x  3x  14
2
i.e. find all x for which (2x  5) x  3x  14
2
standard form  x  5 x  14  0
2
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C. Quadratic inequalities
Example (2x  5) x  3x 2  14
1. Write in standard form:  x  5 x  14  0
2
2. Study the equality:
 x  5x  14  0
2
x = -2; x = 7
3. Determine type of graph:
4. Solve inequality:
conclusion: x-2 or x7
interval notation: ]-,-2][7,[
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C. Quadratic inequalities
inequalities that can be reduced to the form


ax 2  bx  c 0


determine the common
points with the x-axis by
solving the EQUATION and
…
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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
zeroes
vertex
symmetry axis
minimum/maximum
• Quadratic inequalities: solutions
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Quadratic functions
Extra exercises: Handbook –
Problems 0.8: Ex 31, 37, 45, 55, 57, 79
Problems 3.3: Ex 11, 13, 23, 29, 37, 41