2. OBJECTIVES:
At the end of the lesson, the students are expected to:
1. Define the terms used and analyze the basic concepts of
quadratic functions.
2. Illustrate the graph of quadratic functions.
3. Solve equations and word problems involving
quadratic functions.
3. DEFINITION OF TERMS
Quadratic Function is one of the form f(x)=ax 2+bx+c, where a, b
and c are real numbers and a not equal to zero (a≠0).
Parabola is the graph of quadratic function.
Quadratic Equation is the result when the quadratic function is set
to zero.
The Factored Form is the equation f(x)=a(x-x1)(x-x2) where x1 and
x2 are the roots of the quadratic function.
4. DEFINITION OF TERMS
The Vertex Form is the f(x)=a(x-h)² + k where h and k are the x
and y coordinates of the vertex.
Minimum of a Quadratic Function happens if a>0 and the graph of
f(x)=ax 2+bx+c opens up and the vertex (-b/2a,f(-b/2a)) is the
lowest point on the graph, where f(-b/2a) is the minimum value of
the function.
Maximum of a Quadratic Function is when a<0, then the graph of
f(x)=ax 2+bx+c opens down and the vertex (-b/2a,f(-b/2a)) is the
highest point on the graph, where f(-b/2a) is the maximum value of
the function.
5. PRE-TEST
Solve the following:
1. Find the zeros of the function: f(x)= 3x2 + 3x + 2.
2. Find either a maximum or minimum value of f(x) = -2x2 + 8x -
5.
3. Find two numbers whose sum is 10 and whose product is a
maximum.
4. A clock manufacturer can produce a particular clock at a cost of
P 600 per clock. It is estimated that if the selling price of the
clock is “x” pesos, then the number of clocks sold per week is
2,000-x. Determine what the selling price should be in order for
the manufacturer’s weekly profit to be maximum.
6. GRAPH OF QUADRATIC FUNCTIONS
Suppose we’re given an equation
y = x². How do we illustrate the graph?
Here are a few steps in graphing the
points:
1. Determine the value of x and y.
2. Plot the points in the graph.
3. Trace the points to form a parabola.
7.
8. QUADRATIC EQUATION
The Quadratic Formula 𝒙 =
−𝒃± 𝒃 𝟐−𝟒𝒂𝒄
𝟐𝒂
can be used to
solve any quadratic equation in the form ax 2+bx+c. The
final answer should be reduced and have the radical in
lowest term. One needs to be careful in reducing the final
answer because this step can often be the source of an
incorrect answer.
9. QUADRATIC EQUATION
Example:
Solve x² + 6x + 2 = 0 by the quadratic formula.
Solution:
Values are, a = 1, b = 6, and c = 2.
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
𝑥 =
−(6)± 62−4(1)(2)
2(1)
𝑥 =
−6± 36−8
2
𝑥 =
−6± 28
2
Therefore, x1 = -3+√7 and x2 = -3-√7.
11. FACTORED FORM OF
QUADRATIC FUNCTIONS
A quadratic equation in the variable x in the form of f(x)=ax 2+bx+c,
where a, b and c are real numbers and a≠0 and are factorable using
integers , can be solved by factoring and applying the property ab=0.
The factored form of quadratic function is the f(x)=a(x-x1)(x-x2) where
x1 and x2 are the roots of the quadratic function
Example:
Solve x² + 4x = 21.
Solution:
x² + 4x = 21
(x+7)(x-3) =0
Therefore, x1 = -7 and x2 = 3.
12. VERTEX FORM OF QUADRATIC
FUNCTION
Vertex Form: f(x)=a(x-h)² + k where h and k are the x and y coordinates
of the vertex.
h= -b/(2a) and k = f ( -b/(2a))
The coordinates of the vertex of f(x)=ax 2+bx+c are
[ -b/(2a), f ( -b/(2a)) ]
13. VERTEX FORM OF QUADRATIC
FUNCTION
Example: Find the vertex form of f(x) = 2x² - 8x + 3.
Solution:
h= -b/(2a) = -(-8) / (2·2) = 2
k = f ( -b/(2a)) = 2(2)² - 8(2) + 3 = -5
The vertex is (2,-5). Substituting into the vertex form
f(x)=a(x-h)² + k yields the equation f(x) =2(x-2)² - 5.
14. MAXIMUM AND MINIMUM
OF QUADRATIC FUNCTION
Theorem 1:
The quadratic function defined by f(x)=ax 2+bx+c, where (a≠0), has an
extreme value at the point x = -b/(2a).
If a > 0 , the extreme value is a minimum value, and if a < 0, the extreme
value is a maximum value.
15. Examples:
1. Use Theorem 1 to find either the maximum or minimum value of
f(x) = -3x²/2 + 6x – 10.
Solution:
From the equation ax² + bx + c = 0, let a = -3/2, b = 6, and c = 6.
x = - b / (2a)
= -6 / [2 ( -3/2) ]
x = 2
F(-b/2a) = -3(2)²/2 + 6(2) - 10 = -4 a<0
Therefore, the extreme value is a minimum value that equals to -4.
MAXIMUM AND MINIMUM
OF QUADRATIC FUNCTION
16. 2. Use Theorem 1 to find either the maximum or minimum value of
f(x) = 4x² + 8x + 7
Solution:
From the equation ax² + bx + c = 0, let a = 4, b = 8, and c = 7.
x = - b / (2a)
= -8/ [2 ( 4) ]
x = -1
F(-b/2a) = 4(-1)² + 8(-1) + 7 = 3 a>0
Therefore, the extreme value is a maximum value that equals to 3.
MAXIMUM AND MINIMUM
OF QUADRATIC FUNCTION
17. Example:
Find the range of f(x) = -2x² - 6x – 1. Determine the values of x for which
f(x) = 3.
Solution:
h = -b/2a = -6/ (2)(-2) = -3/2
k = f(-3/2) = -2(-3/2)-6(-3/2)-1 = 7/2
The vertex is (-3/2,7/2). Because the parabola opens down, 7/2 is the
maximum value of f. Therefore, the range of f is {y|y ≤ 7/2}
f(x) = 3
-2x² - 6x – 1 = 3
-2(x+1)(x+2) = 0
Therefore, the values of x are -1 and -2.
RANGE OF A QUADRATIC FUNCTION
18. Problem:
The height h(t), in feet, of a snowboarder t seconds after beginning a
certain jump can be approximated by h(t) = -16t² + 22.9t + 9. If the
snowboarder lands at a point that is 3 feet below the base of the jump,
determine the time the snowboarder is in the air for this jump.
Solution:
h(t) = -3, where the snowboarder lands below the base of the jump
h(t) = -16t² + 22.9t + 9
-3 = -16t² + 22.9t + 9
0 = -16t² + 22.9t + 12
t =
−22.9± 22.92−4(−16)(12)
2(−16)
t = -0.4 or 1.8
Since negative time is not possible, the time for this jump is 1.8 sec.
APPLICATION OF QUADRATIC FUNCTIONS
19. ACTIVITY
Group 1
Factor: 3x² + 10x – 8.
Group 2
Solve for x: 2x² - x = 1.
Group 3
Find f(-3) for f(x) = 2x²
- 5x -7.
Group 4
Find two numbers such
that their sum is 8 and their
product is 6.
20. SUMMARY
To complete the factored or vertex form to standard form, one should
multiply, expand or distribute the factors.
To convert the standard form to factored form, the quadratic formula 𝒙 =
−𝒃± 𝒃 𝟐−𝟒𝒂𝒄
𝟐𝒂
is used to determine the roots.
To convert the standard form to vertex form, the process called completing
the square can be used.
21. POST-TEST
Solve the following:
1. Find the zeros of the function: f(x) = -2x2 + 8x - 5..
2. Find either a maximum or minimum value of the function : f(x)= 3x2 +
3x + 2.
3. Find two numbers whose difference is 14 and whose product is a
minimum.
4. A rectangular field is to be fenced off along a river bank, and no fence
is required along the river. The material for the fence costs P320 per
linear foot for the two ends and P480 per linear foot for the side
parallel to the river; P144,000 worth of fence is to be used. Find the
dimensions of the field of largest possible area that can be enclosed
by the P144,000 worth of fence. What is the largest area?
