3. THE ZEROS OF QUADRATIC FUNCTIONS
Any value of x that makes the function f(x) equal to zero is called a zero
of the given function and consequently, a root of its corresponding
equation.
The zeros of the quadratic function y = 𝑎𝑥2 + bx + c are the roots of the
equation 𝑎𝑥2
+ bx + c = 0.
1 2 3 4
5
3
2
-1
-2
-3
-4
-5 -4 -3 -2
-1
-5
-6
Quadratic Function y = 𝑥2
− x −
6
(-2 , 0) (3 , 0)
3 and -2 are the zeros of the
function since these are the
values of x when y is equals 0.
zeros of the
function
4. THE ZEROS OF QUADRATIC FUNCTIONS
FACTORING:
𝑥2
+ bx + c
(x+_)(x+_)
COMPLETING THE
SQUARE:
𝑥2 + bx = −c
(𝑎𝑥2 + bx + _) = −c + _
Quadratic formula: 𝑥 =
−𝑏± 𝑏2−4𝑎𝑐
2𝑎
Methods for finding the roots
of quadratic equations:
5. OBTAINING THE ZEROS OF A QUADRATIC
FUNCTION
Problem 1:
Find the zeros of the quadratic function y = 2𝑥2
+ 7𝑥 −
4
Solution:
y = 2𝑥2
+ 7𝑥 − 4
Thus,
2𝑥2 + 7𝑥 − 4 = 0
(2x-1)(x+4) = 0
2x-1 = 0 x+4 = 0
x =
1
2
x = -4
In order to find the
roots,
we will let y = 0
2x = 1
2 2
6. OBTAINING THE ZEROS OF A Q.FUNCTION
Thus, The zeros of f(x) are
1
2
and -4. These are the x-intercepts of the graph of
f(x) and are equivalent to the points (
1
2
, 0) and (-4 , 0). The graph of f(x)
is shown below:
(
1
2
, 0)
v(
−7
4
,
−81
8
)
(-4 , 0)
f(x) = 2𝑥2 + 7𝑥 − 4
v(
−𝑏
2𝑎
,
4𝑎𝑐 −𝑏2
4𝑎
)
v(
−7
4
,
−81
8
)
v(
−7
4
,
−32 −49
4(2)
)
7. OBTAINING THE ZEROS OF A QUADRATIC
FUNCTION
Problem 2:
Find the zeros of the quadratic function y = 4𝑥2
+ 3𝑥 − 1
Solution:
if we let y = 0 and solve the resulting equation 4𝑥2 + 3𝑥 − 1 = 0
by factoring, we get the following:
4𝑥2 + 3𝑥 − 1 = 0
(4x – 1)(x + 1) = 0
4x – 1 = 0 x + 1 = 0
x =
1
4
x = -1
4x = 1
4 4
8. OBTAINING THE ZEROS OF A QUADRATIC
FUNCTION
Thus, The zeros of f(x) are
1
4
and -1. These are the x-intercepts of the graph of
f(x) and are equivalent to the points (
1
4
, 0) and (-1 , 0). The graph of f(x) is
shown below:
(
1
4
, 0)
(-1 , 0)
f(x) = 4𝑥2 + 3𝑥 − 1
v(
−𝑏
2𝑎
,
4𝑎𝑐 −𝑏2
4𝑎
)
v(
−3
2(4)
,
4 4 −1 −(3)2
4(4)
)
v(
−3
8
,
−25
16
)
v(
−3
8
,
−25
16
)
9. OBTAINING THE ZEROS OF A QUADRATIC
FUNCTION
Problem 3:
find the zeros of a quadratic function f(x) = 𝑥2 + 𝑥 − 12.
Solution:
set y = 0.
thus, 0 = 𝑥2 + 𝑥 − 12 where a = 1, b = 1, c = -12.
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
𝑥 =
−(1) ± (1)2−4(1)(−12)
2(1)
𝑥 =
−1 ± 1 + 48
2
Substitute a, b and c:
Simplify:
𝑥 =
−1 ± 49
2
10. OBTAINING THE ZEROS OF A QUADRATIC
FUNCTION
𝑥 =
−1 ± 49
2
𝑥 =
−1 + 7
2
𝑥 =
−1 − 7
2
𝑥 =
6
2
𝑥 =
−8
2
𝑥 = 3 𝑥 = −4
Therefore, the zeros of f(x) = 𝑥2 + 𝑥 − 12 are 3 and -4.
11.
12. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
Problem 4:
write a quadratic function in the form f(x) = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
when its zeros are
1
2
and -3.
Solution:
the zeros of a function are also the roots of its
corresponding equation. Thus,
=
1
2
2x = 1
2x – 1 = 0
x = -3
x + 3 = 0
x
1
13. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
2x – 1 = 0 x + 3 = 0
(2x – 1)(x + 3) = 0
2𝑥2 + 5𝑥 − 3 = 0
So, the equation with the given root is:
Thus, the function of the given zeros is f(x) = 2𝑥2 + 5𝑥 − 3.
14. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
Problem 5:
Determine the equation of the parabola whose vertex is
located at (2 , -1) and whose y-intercept is equal to -3.
Solution:
since the vertex (h , K) is given, use the equation f(x) = 𝑎(𝑥 − ℎ)2 + 𝐾.
Substitute: v(2 , -1)
f(x) = 𝑎(𝑥 − 2)2 − 1
A y-intercept of -3 means that the graph crosses the y-axis at the
point (0 , -3). Substitute x = 0 and f(x) = -3 in the equation.
15. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
f(x) = 𝑎(𝑥 − 2)2 − 1
-3 = 𝑎(0 − 2)2
− 1
-3 = 𝑎(2)2 − 1
-3 = 4𝑎 − 1
-4a = 3 − 1
-4a = 2
−4 − 4
a = −
1
2
The equation of the given
parabola is f(x) = −
1
2
(𝑥 − 2)2 −
1.
Substitute:
f(x) = −
1
2
(𝑥 − 2)2 − 1
Substitute the value of a from the
equation:
16. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
Problem 6:
find the equation of the quadratic function determined
from the graph below:
0 1 2 3 4 5
-1
-2
-3
-4
2
1
-2 -1
(-1 , 0) (5 , 0)
(2 , -3)
17. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
Solution:
the vertex of the graph of the function is (2 , -3). The graph
passes through the point (5 , 0). By replacing x and y with 5 and 0,
respectively, and h and K with 2 and -3, respectively, we have:
Y = 𝑎(𝑥 − ℎ)2 − 𝐾
0 = 𝑎(5 − 2)2 − 3
0 = 𝑎(3)2 − 3
0 =9𝑎 − 3
-9a = -3
a =
1
3
−9 − 9
Thus, the quadratic function
is y =
1
3
(𝑥 − 2)2 − 3.
Substitute the values of
x and y:
18. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
Problem 7:
find a quadratic function whose zeros are -1 and
4.
Solution:
if the zeros are 𝑟1 = -1 and 𝑟2 = 4, then:
f(x) = (x - 𝑟1)(x - 𝑟2)
f(x) = [x – (-1)] (x - 4) Thus, the function is
f(x) = 𝑥2 − 3𝑥 − 4
f(x) = (x + 1) (x - 4)
19. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
But the equation f(x) = 𝑥2 − 3𝑥 − 4 is not
unique since there are other quadratic functions
whose zeros are -1 and 4 such as f(x) = 2𝑥2 − 6𝑥 −
8 and f(x) = 3𝑥2 − 9𝑥 − 12 and many more.
These equations are obtained by multiplying the
right hand side of the equation by a nonzero
constant.
Thus, the answer is f(x) = a(𝑥2 − 3𝑥 − 4), where a
is any nonzero constant.
20. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
Problem 8:
find the equation of a quadratic function whose zeros are
3 ± 2
3
.
Solution:
A quadratic equation with irrational roots cannot be
written as a product of linear factors with irrational coefficients.
In this case, we can use another method. Since the zeros are
3 ± 2
3
then:
=
3 ± 2
31
x
3x = 3 ± 2
3x - 3 = ± 2
21. FINDING THE EQUATION OF A QUADRATIC
FUNCTION
Square both sides of the equation and simplify:
3x - 3 = 2( )2 ( )2
Therefore we get:
3𝑥2 − 18𝑥 + 9 = 2
3𝑥2
− 18𝑥 + 7 = 0
Thus, the equation of quadratic function
is f(x) = 3𝑥2 − 18𝑥 + 9.
Editor's Notes
(GREETINGS)
So WE ARE ALREADY DONE WITH THE introduction of quadratic function, the forms of quadratic functions, the graphs and the properties of quadratic function.
So…
IN THIS DISCUSSION, we’re going to learn about the zeros of a quadratic function.
and finding the equation of a quadratic function. So at the end of this lessons, I hope you will learn the meaning and to understand quadratic function more.
So… what do we mean when we say zeros of a quadratic function?
To give you a better understanding, I’m going to show you a graph of a parabola
The graph of a quadratic function is a parabola.
--- don’t be confuse in the substitution of “y” and “f(x)” because they are the same. Y=f(x) or y is a function of x.
--- the y-intercept of the parabola is the function of x or f(x)
The horizontal line is x axis and the vertical line is y axis. Observe the parabola, we can see that it passes in the x axis at the points (3,0) and (-2, 0).
The x-coordinates of this points of intersection are called x-intercepts.
Similarly, 3, and -2 are the zeros of the function since these are the values of x when y = 0.
Recall that there are many algebraic methods for finding the roots of quadratic equations.
` We can find the roots by factoring, completing the square or by using the quadratic formula.
We are also going to use these methods in finding the zeros of a q.f.
Having recalled the different methods of solving q.f. we will now consider problems involving zeros of functions
if we let y = 0 and solve the resulting equation 2𝑥 2 +7𝑥−4 = 0 by factoring, we get the following:
Next problem. Lets solve by factoring
Another problem. Lets find the zeros of a quadratic function using quadratic formula
That’s it. And now we’re moving on to another lesson.
The preceding lessons have been a discussion of the geometric properties of quadratic functions. We have described the graphs of quadratic functions and determined their properties by simply manipulating their equations. This lesson will do the reverse of some processes we have learned previously. We will form the equation of a quadratic function given its zeros or set of points on its graph.
Let’s have an example.
Lets have another example.
This problem is harder because the roots are not given. Therefore we’re going to use another method.
Lets have another problem
What if the given problem is the graph of a parabola
We’re going to introduce another method of finding the equation of a quadratic function
In this situation we’re going to use: f(x) = a(x - 𝑟 1 )(x - 𝑟 2 ) since we already discuss about the sum and product of a quadratic equation
Also, we can use the sum and product of the zeros to fid the equation of the quadratic function.