This document discusses the behavior and key characteristics of quadratic functions and their graphs (parabolas). It begins by defining quadratic functions, expressions, and equations. It then explains that a quadratic function has two variables, x and y, where x is the independent variable and y is the dependent variable. The document goes on to discuss several important aspects of parabolas including the vertex (point of minimum or maximum), axis of symmetry, zeros (where the graph crosses the x-axis), and point of origin. It then analyzes the graphs of various quadratic functions of the form y=ax^2, observing that when a is positive the graph opens upward and the opening gets narrower as a increases, and when a

1. Behavior of the
Graph of a
Quadratic Function

2. Let us differentiate the following:
Quadratic
Function
Quadratic
Equation
Quadratic
Expression
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄 = 𝟎
𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
or
𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙+c
Clearly we can see that
𝒇 𝒙 = 𝒚
𝒐𝒓
𝒚 = 𝒇(𝒙)

3. A quadratic function has two variables the x and the y
x – is the independent variable
y – is the dependent variable or we can say that the
value of y depends on the value of x
The values of x are called the domain of the quadratic
function while the values of y are called the range of
the quadratic function

4. ax2 + bx + c = 0
Recall:
Quadratic term
Linear term
Constant term

5. The Graph of a
Quadratic
Function is called
a PARABOLA

6. •vertex: The point at which a parabola changes direction,
corresponding to the minimum or maximum value of the
quadratic function.
•axis of symmetry: A vertical line drawn through the vertex of a
parabola around which the parabola is symmetric.
•zeros: In a given function, the values of x at which y=0, also called
roots. It is where the graph of the function intersects the x-axis
•point of origin: the intersection of the x-axis and y-
axis
•line of symmetry: it is a line drawn intersecting the vertex and is
parallel to the y-axis
Key Terms
•x –intercept is where the graph of the function intersects the x-axis
•y –intercept is where the graph of the function intersects the y-axis

7. Let us observe the graph of the Quadratic Function of the
form 𝒚 = 𝒂𝒙𝟐
1) 𝑦 = 𝑥2
2) 𝑦 = 2𝑥2
3) 𝑦 =
1
2
𝑥2
4) 𝑦 =
1
3
𝑥2
5) 𝑦 = −𝑥2
6) 𝑦 = −2𝑥2

8. Analyze the graphs.
a. What happens to the graph as the value of a becomes larger?
b. What happens when 0 < 𝑎 < 1?
c. What happens when 𝑎 < 0? 𝑎 > 0?
d. Summarize your observations.

9. Let us observe the graph of the Quadratic Function of the
form 𝒚 = 𝒂𝒙𝟐
1) 𝑦 = 𝑥2
2) 𝑦 = 2𝑥2
3) 𝑦 =
1
2
𝑥2
4) 𝑦 =
1
3
𝑥2
5) 𝑦 = −𝑥2
6) 𝑦 = −2𝑥2
Opens upward
Opens upward, opening becomes wider the previous
Opens upward, opening becomes narrower
Opens upward, opening becomes wider
Opens downward,
Opens downward, opening becomes narrower
When a is positive it opens upward, as the value of a increases its opening becomes
narrower.
When a is negative it opens downward, as the value of a decreases its opening
becomes narrower

11. Let us observe the graph of the Quadratic Function of
the form 𝒚 = 𝒂(𝒙 − 𝒉)𝟐
+ k
1) 𝑦 = 𝑥2
2) 𝑦 = (𝑥 + 1)2
3) 𝑦 = (𝑥 + 1)2
+ 1
4) 𝑦 = 2(𝑥 + 2)2
+ 2
5) 𝑦 = (𝑥 − 1)2
f) 𝑦 = (𝑥 − 1)2
-1
f) 𝑦 = 2(𝑥 − 2)2
-2
f) 𝑦 = −2(𝑥 − 2)2
- 2

12. Analyze the graphs
a. What have you noticed about the graphs of the quadratic function of
the form𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘?
b. How would you compare the graph of 𝑦 = 𝑎(𝑥 − ℎ)2
+ 𝑘 and that
of 𝑦 = 𝑎𝑥2
c. Discuss your ideas and observations.
d. What Conclusions can you give based on your observations?

13. 𝒚 = 𝒂(𝒙 − 𝒉)𝟐
+ 𝐤
a indicates a
reflection across
the x-axis and/or
a vertical stretch
or compression
h indicates a
horizontal
translation
k indicates a
vertical translation

14. Write a REFLECTION about your experiences and what
you have learned in our Mathematics Class.
Special Task

15. Mylane S. Epino
Master Teacher I