1. Algebra
1
QUADRATIC FUNCTIONS
• Quadratic Expressions, Rectangles and Squares
• Absolute Value, Square Roots and Quadratic Equations
• The Graph Translation Theorem
• Graphing 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐
• Completing the Square
• Fitting a Quadratic Model to Data
• The Quadratic Formula
• Analyzing Solutions to Quadratic Equations
• Solving Quadratic Equations and Inequalities

2. Algebra
2
QUADRATIC FUNCTIONS
Quadratic – quadratus (Latin) , ‘to make
square’
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
f 𝑥 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
Standard form of a quadratic: 𝑎𝑥2 + 𝑏𝑥 + 𝑐

3. Algebra
3
QUADRATIC FUNCTIONS
Quadratic expressions from Rectangles and
Squares
 Suppose a rectangular swimming pool 50 m by 20 m is
to be built with a walkway around it. If the walkway is w
meters wide, write the total area of the pool and walkway
in standard form.
 Write the area of the square with sides of length 𝑥 + y in
standard form

4. Algebra
4
QUADRATIC FUNCTIONS
Binomial Square Theorem
For all real numbers x and y,
𝑥 + 𝑦 2
= 𝑥2
+ 2xy + 𝑦2
𝑥 − 𝑦 2 = 𝑥2 − 2xy + 𝑦2
Note: When discussing this, ask students whether any real-number values of
the variable give a negative value to the expression. [ The square of any
real number is nonnegative].

5. Algebra
5
QUADRATIC FUNCTIONS
Challenge
Have students give quadratic expressions for the areas
described below.
1. The largest possible circle inside a square whose side
is x.
2. The largest possible square inside a circle whose radius
is x.

6. Algebra
6
QUADRATIC FUNCTIONS
Absolute Value, Square Roots and Quadratic Equations
In Geometry
• 𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑛 𝑓𝑟𝑜𝑚 0 𝑜𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒
In Algebra
• 𝑥 =
𝑥, 𝑓𝑜𝑟 𝑥 ≥ 0
−𝑥, 𝑓𝑜𝑟 𝑥 < 0

7. Algebra
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QUADRATIC FUNCTIONS
Activity
1. Evaluate each of the following.
42, −4 2, 9.32, −9.3 2
2. Find a value of x that is a solution to 𝑥2 = 𝑥.
3. Find a value of x that is not a solution to 𝑥2 = 𝑥.

8. Algebra
8
QUADRATIC FUNCTIONS
Absolute Value – Square Root Theorem
For all real numbers x, 𝑥2 = 𝑥
Example 1
Solve 𝑥2 = 40
Example 2
A square and a circle have the same area. The square
has side 10. What is the radius of the circle?

9. Algebra
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QUADRATIC FUNCTIONS
Challenge
The Existence of Irrational Numbers
Prove that 2 cannot be written as a simple fraction.

10. Algebra
10
QUADRATIC FUNCTIONS
Graphs and Translations
 Consider the graphs of 𝑦1 = 𝑥2 and 𝑦2 = 𝑥 − 8 2
 What transformation maps the graph of the first function
onto the graph of the second?
Graph – Translation Theorem
In a relation described by a sentence in x and y, the following two processes
yield the same graph:
1. replacing 𝑥 by 𝑥 − ℎ and 𝑦 by 𝑦 − 𝑘
2. applying the translation 𝑇ℎ,𝑘 to the graph of the original relation.

11. QUADRATIC FUNCTIONS
6/1/2014
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12. Algebra
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QUADRATIC FUNCTIONS
Example 1
Find an equation for the image of the graph of 𝑦 = 𝑥 under the
translation 𝑇5,−3.
Corollary
The image of the parabola 𝑦 = 𝑎𝑥2 under the translation 𝑇ℎ,𝑘 is the
parabola with the equation
𝑦 − 𝑘 = 𝑎 𝑥 − ℎ 2
Example 2
a. Sketch the graph of 𝑦 − 7 = 3 𝑥 − 6 2
b. Give the coordinates of the vertex of the parabola
c. Tell whether the parabola opens up or down
d. Give the equation for the axis of symmetry.

13. Algebra
13
QUADRATIC FUNCTIONS
Graphing 𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
Suppose ℎ = −16𝑡2
+ 44𝑡 + 5
a. Find ℎ when 𝑡 = 0, 1, 2 𝑎𝑛𝑑 3
b. Explain what each pair 𝑡, ℎ tells you about the
height of the ball.
c. Graph the pairs 𝑡, ℎ over the domain of the function.
Note: Two natural questions about the thrown ball are related to
questions about this parabola.
1. How high does the ball get? The largest possible value of h.
2. When does the ball hit the ground?

14. Algebra
14
QUADRATIC FUNCTIONS
Newton’s Formula
ℎ = −
1
2
𝑔𝑡2 + 𝑣0 𝑡 + ℎ0
• 𝑔 is a constant measuring the acceleration due to gravity
• 𝑣0 is the initial upward velocity
• ℎ0 is the initial height
• the equation represents the height ℎ of the ball off the ground at time
𝑡.

15. Algebra
15
QUADRATIC FUNCTIONS
Completing the Square
𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑜𝑟𝑚
𝑦 − 𝑘 = 𝑎 𝑥 − ℎ 2
𝑣𝑒𝑟𝑡𝑒𝑥 𝑓𝑜𝑟𝑚
 Completing the square geometrically and algebraically
Theorem
To complete the square on 𝑥2
+ 𝑏𝑥, 𝑎𝑑𝑑
1
2
𝑏
2
.
𝑥2 + 𝑏𝑥 +
1
2
𝑏
2
= 𝑥 +
1
2
𝑏
2
= 𝑥 +
𝑏
2
2

16. Algebra
16
QUADRATIC FUNCTIONS
Practice
• Going back to ℎ = −16𝑡2 + 60𝑡 + 5, find the maximum
height of the ball.
• Rewrite the equation 𝑦 = 𝑥2 + 10𝑥 + 8 in vertex form.
Locate the vertex of the parabola.
• Suppose 𝑓 𝑥 = 3𝑥2 + 12𝑥 + 16
a. What is the domain of 𝑓?
b. What is the vertex of the graph?
c. What is the range of 𝑓?

17. Algebra
17
QUADRATIC FUNCTIONS
Fitting a Model to Data

18. Algebra
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QUADRATIC FUNCTIONS
Practice
The number of handshakes ℎ needed for everyone in a
group of 𝑛 people, 𝑛 ≥ 2, to shake the hands of every
other person is a quadratic function of 𝑛. Find three
points of the function relating ℎ and 𝑛. Use these points
to find a formula for this function.

19. Algebra
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QUADRATIC FUNCTIONS

20. Algebra
20
QUADRATIC FUNCTIONS
The Angry Blue Bird Problem
What if Blue Bird’s flight path is described by the function
ℎ 𝑥 = −0.005𝑥2 + 2𝑥 + 3.5
Where is Blue Bird when she’s 8 feet high?

21. Algebra
21
QUADRATIC FUNCTIONS
The Quadratic Formula
If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑎𝑛𝑑 𝑎 ≠ 0 , 𝑡ℎ𝑒𝑛
𝑥 =
−𝑏± 𝑏2−4𝑎𝑐
2𝑎
Challenge
How was it derived?

22. Algebra
22
QUADRATIC FUNCTIONS
Practice
 Solve 3𝑥2 + 11𝑥 − 4 = 0
 The 3-4-5 right triangle has sides which are consecutive
integers. Are there any other right triangles with this
property?
 Challenge: Find a number such that 1 less than the
number divided by the reciprocal of the number is equal
to 1.

23. Algebra
23
QUADRATIC FUNCTIONS
How Many Real Solutions Does a Quadratic Equation
Have?
Discriminant Theorem
Suppose 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 are real numbers with 𝑎 ≠ 0.
Then the equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 has
i. two real solutions if 𝑏2 − 4𝑎𝑐 > 0.
ii. one real solution if 𝑏2 − 4𝑎𝑐 = 0.
iii. two complex conjugate solutions if 𝑏2 − 4𝑎𝑐 < 0.

24. Algebra
24
QUADRATIC FUNCTIONS
Practice
Determine the nature of the roots of the following
equations. Then solve.
a. 4𝑥2
− 12𝑥 + 9 = 0
b. 2𝑥2 + 3𝑥 + 4 = 0
c. 2𝑥2
− 3𝑥 − 9 = 0

25. Algebra
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QUADRATIC FUNCTIONS
Solving Quadratic Equations
 Extracting Square Roots
 Factoring
 Completing the Square
 Quadratic Formula

26. Algebra
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QUADRATIC FUNCTIONS
Factoring
Steps:
• Transform the quadratic equation into standard form if necessary.
• Factor the quadratic expression.
• Apply the zero product property by setting each factor of the quadratic
expression equal to 0.
Zero Product Property
– If the product of two real numbers is zero, then either of the two
is equal to zero or both numbers are equal to zero.
• Solve each resulting equation.
• Check the values of the variable obtained by substituting each in the original
equation.

27. Algebra
27
QUADRATIC FUNCTIONS
Practice
Solve the following equations.
1. 𝑥2 + 7𝑥 = 0
2. 𝑥2 + 9𝑥 = −8
3. 4𝑥2 − 81 = 0
4. 4𝑥2 + 9 = 12𝑥
5. 𝑥 − 1 2 − 5 𝑥 − 1 + 6 = 0

28. Algebra
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QUADRATIC FUNCTIONS

29. Algebra
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QUADRATIC FUNCTIONS

30. Algebra
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QUADRATIC FUNCTIONS
HOW TO SOLVE?
1. find the "=0" points
2. in between the "=0" points, are intervals that are either
greater than zero (>0), or
less than zero (<0)
3. then pick a test value to find out which it is
(>0 or <0)

31. Algebra
31
QUADRATIC FUNCTIONS

32. Algebra
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QUADRATIC FUNCTIONS
Here is the plot of 𝑦 = 𝑥2
− 𝑥 − 6
The equation equals zero at -2 and 3
The inequality "<0" is true
between -2 and 3.

33. Algebra
33
QUADRATIC FUNCTIONS
Practice
1. Find the solution set of 𝑥2 − 5𝑥 − 14 > 0.
2. Graph 𝑦 > 𝑥2 − 5𝑥 − 14
3. A stuntman will jump off a 20 m building. A high-speed
camera is ready to film him between 15 m and 10 m
above the ground. When should the camera film him?