This document provides information about quadratic functions including: 1. Quadratic functions have graphs that form a U-shaped curve called a parabola with a single turning point called the vertex. 2. The axis of symmetry of the parabola passes through the vertex and divides the graph into two symmetrical parts. 3. Quadratic functions can be transformed between the standard form f(x)=ax^2 + bx + c and the vertex form f(x)=a(x-h)^2 + k using completing the square. This allows finding the vertex (h,k) of the parabola. 4. Examples show how to transform quadratic functions between the standard and vertex

1. Quadratic Functions
Learning Targets
• Give an example of real life situations
using quadratic functions.
• Differentiate quadratic functions from
linear functions.

2. Match the items in column A with the item in column B that
is related to it.
Column A Column B
A newborn baby children
A bar code in a shop date of birth
A real number square of that number
A mother price

3. A newborn baby is related to his or her date of birth. This
relationship is a function since to each baby have its own date of
birth. In other words, no baby can have two dates of birth.
-3
-2
1
0
3
9
4
1
0
9
x x2
A bar code in a store is related to a price. This also shows a
function since to each bar code, there corresponds a price. In
fact, no bar code can have two or more different prices.
A real number is related to a square of that number. This is
also a function since to each real number there corresponds a
square of the number. In other words, no real number can have
two or more different squares.
On the other hand, a mother can have two or more
children. Hence, this relation is NOT a function.

4. DOMAIN RANGE
Correspondence
FUNCTION
A function is a correspondence between a first set called domain, and
a second set, called range, such that each member of that domain
corresponds to exactly one member of the range.
QUADRATIC FUNCTION
A quadratic function is polynomial of degree 2 and defined by the
equation 𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄, where a, b and c are real numbers and a≠ 0.

5. The following are examples of quadratic
function:
1. 𝒇 𝒙 = 𝟑𝒙𝟐
− 𝟔𝒙 + 𝟓
2. 𝒇 𝒙 = −𝟑𝒙𝟐
+ 𝟑𝒙 + 𝟏
3. 𝒇 𝒙 = 𝟖𝒙𝟐
− 𝟑
4. 𝒇 𝒙 = 𝟐𝒙𝟐
5. 𝒇 𝒙 = 𝒙𝟐
− 𝟐𝒙

6. Quadratic Term Linear Term Constant Term
TAKE NOTE!
The most general form of quadratic function is 𝒇 𝒙 = 𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄 or 𝒚 =
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄.
The first term in the quadratic function 𝒂𝒙𝟐 is called quadratic term, 𝒃𝒙 is the
linear term and 𝒄 is the constant term.
𝒇 𝒙 = −𝟑𝒙𝟐 + 𝟑𝒙 + 𝟏
6x2
𝒇 𝒙 = 𝟔𝒙𝟐 − 𝟐𝒙 + 𝟕
𝒇 𝒙 = (𝟓𝒙 + 𝟑)𝟐−(𝟐𝒙 + 𝟓)𝟐
-3x2
21x2
-2x
3x
10x
7
1
-16

7. 𝒇 𝒙 = (5𝑥 + 3)𝟐
−(2𝑥 + 5)2
𝒇 𝒙 = (25𝑥𝟐+𝟑𝟎𝒙 + 𝟗) − (4𝑥2+𝟐𝟎𝒙 + 𝟐𝟓)
𝒇 𝒙 = 25𝑥𝟐
+ 𝟑𝟎𝒙 + 𝟗 − 4𝑥2
− 𝟐𝟎𝒙 − 𝟐𝟓
𝒇 𝒙 = 21𝑥𝟐 + 𝟏𝟎𝒙 − 𝟏𝟔
SECOND DIFFERENCE TEST
A relation f is a quadratic function if equal differences
in the independent variable x produce nonzero equal
second difference in the function value f(x).

8. 𝒇 𝒙 = 𝒙𝟐
+ 𝟐𝒙 + 𝟏
X -2 -1 0 1 2
y 1 0 1 4 9

9. 𝒇 𝒙 = 𝒙𝟐
+ 𝟐𝒙 + 𝟏
X -2 -1 0 1 2
y 1 0 1 4 9
1
1
1
1
𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒙
𝟏𝒔𝒕 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒚 5
3
1
-1
𝟐𝒏𝒅 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒚 2
2
2

10. Given the table of values, determine if it is
a quadratic or not.
x -3 -2 -1 0 1 2 3
f(x)=y 9 4 1 0 1 3 9
1
1
1
1
𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒙 1 1
6
1
-1
-5 -3 2
4
1
2
2
2
𝑵𝑶𝑻 𝑸𝑼𝑨𝑫𝑹𝑨𝑻𝑰𝑪

11. Concept Review page 49

13. Learning Targets
 transform the quadratic function
defined by 𝑦=ax²+bx+c into the form
y=a(x-h)2 + k.
 Determine the values of (h,k0) using
completing square or the formula:
𝒉 =
−𝒃
𝟐𝒂
and 𝒌 =
𝟒𝒂𝒄−𝒃𝟐
𝟒𝒂

14. QUADRATIC FUNCTION IN THE FORM
y=ax2+bx+c into y = a(x-h)2 + k or VICE VERSA
Using the principle of completing the square, we can rewrite the quadratic
function is 𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 into is 𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 where (h, k) is the vertex.
𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 Standard form of the quadratic function
𝒇 𝒙 = 𝒂(𝒙𝟐 +
𝒃
𝒂
𝒙) + 𝒄 Factor a form first 2 terms
𝒇 𝒙 = 𝒂 𝒙𝟐
+
𝒃
𝒂
𝒙 +
𝒃𝟐
𝟒𝒂𝟐 + 𝒄 −
𝒃𝟐
𝟒𝒂𝟐
𝒇 𝒙 = 𝒂(𝒙 +
𝒃
𝟐𝒂
)𝟐 +
𝟒𝒂−𝒃𝟐
𝟒𝒂
Letting 𝒉 =
−𝒃
𝟐𝒂
and 𝒌 =
𝟒𝒂𝒄−𝒃𝟐
𝟒𝒂
f(x) = a(x-h)2 + k

15. Example 1: Transform 𝒚 = 𝒙𝟐
+ 𝟐𝒙 + 𝟑 into
𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐
+ 𝒌.
SOLUTION:
𝒂 = 𝟏
𝒌 =
𝟒𝒂𝒄 − 𝒃𝟐
𝟒𝒂
𝒉 =
−𝒃
𝟐𝒂
𝒃 = 𝟐 𝒄 = 𝟑
𝒉 =
−𝒃
𝟐𝒂
𝒉 =
−(𝟐)
𝟐(𝟏)
𝒉 =
−𝟐
𝟐
𝒉 = −𝟏
𝒌 =
𝟒𝒂𝒄 − 𝒃𝟐
𝟒𝒂
𝒌 =
𝟏𝟐 − 𝟒
𝟒
𝒌 =
𝟒(𝟏)(𝟑) − (𝟐)𝟐
𝟒(𝟏)
𝒌 =
𝟖
𝟒 𝒌 = 𝟐
𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐
+ 𝒌
𝒇 𝒙 = 𝟏(𝒙 − (−𝟏))𝟐
+ 𝟐
𝒇 𝒙 = (𝒙 + 𝟏)𝟐
+ 𝟐

16. Example 2: Transform 𝒚 = 𝟑𝒙𝟐
+ 𝟔𝒙 + 𝟒
into 𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐
+ 𝒌.
SOLUTION:
𝒂 = 𝟑
𝒌 =
𝟒𝒂𝒄 − 𝒃𝟐
𝟒𝒂
𝒉 =
−𝒃
𝟐𝒂
𝒃 = 𝟔 𝒄 = 𝟒
𝒉 =
−𝒃
𝟐𝒂
𝒉 =
−(𝟔)
𝟐(𝟑)
𝒉 =
−𝟔
𝟔
𝒉 = −𝟏
𝒌 =
𝟒𝒂𝒄 − 𝒃𝟐
𝟒𝒂
𝒌 =
𝟒𝟖 − 𝟑𝟔
𝟏𝟐
𝒌 =
𝟒(𝟑)(𝟒) − (𝟔)𝟐
𝟒(𝟑)
𝒌 =
𝟏𝟐
𝟏𝟐 𝒌 = 𝟏
𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐
+ 𝒌
𝒇 𝒙 = 𝟑(𝒙 − (−𝟏))𝟐
+ 𝟏
𝒇 𝒙 = 𝟑(𝒙 + 𝟏)𝟐
+ 𝟏
𝒉 =
−𝒃
𝟐𝒂

17. Example 3: Transform 𝒇 𝒙 = 𝟐(𝒙 + 𝟑)𝟐
− 𝟒
into 𝒇 𝒙 = 𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄.
SOLUTION:
𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐
+ 𝒌
𝒇 𝒙 = 𝟐(𝒙 + 𝟑)𝟐
− 𝟒
𝒇 𝒙 = 𝟐(𝒙𝟐
+ 𝟔𝒙 + 𝟗)
−𝟒
𝒇 𝒙 = (𝟐𝒙𝟐
+ 𝟏𝟐𝒙 + 𝟏𝟖)
𝒇 𝒙 = 𝟐𝒙𝟐
+ 𝟏𝟐𝒙 + 𝟏𝟒
−𝟒

18. Example 4: Transform 𝒇 𝒙 = −𝟐(𝒙 − 𝟒)𝟐
+ 𝟓
into 𝒇 𝒙 = 𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄.
SOLUTION:
𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐
+ 𝒌
𝒇 𝒙 = −𝟐(𝒙 − 𝟒)𝟐
+ 𝟓
𝒇 𝒙 = −𝟐(𝒙𝟐
− 𝟖𝒙 + 𝟏𝟔)
+𝟓
𝒇 𝒙 = (−𝟐𝒙𝟐
+ 𝟏𝟔𝒙 − 𝟑𝟐)
𝒇 𝒙 = -𝟐𝒙𝟐
+ 𝟏𝟔𝒙 − 𝟐𝟕
+𝟓

19. Concept Review Page 53
1-5 only

20. GRAPHS OF QUADRATIC FUNCTION
All quadratic functions have graphs
similar to this graph. This U-shaped is
called a parabola. 𝑷𝒂𝒓𝒂𝒃𝒐𝒍𝒂
𝒗𝒆𝒓𝒕𝒆𝒙
( 0, -1 )
The “turning points” of the graph is
called the vertex of the parabola.
The parabola is symmetric with respect
to a line that passes through the vertex.
This line is called the axis of symmetry. It
divides the parabola into two parts so
that one part is a reflection of the other
part.
𝒂𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚

21. Vlog it!
1. Choose 3 things/objects in your house that will represent
graph of quadratic function.
2. Show it and explain how it will represent a graph of quadratic
function or concept quadratic function by making a short
vlog.
3. You will be graded using the criteria below:
Presentation 5
Content 6
Explanation 9
20