Introduction of Quadratic Functions and its graph

Lesson on quadratic functions

1. 1
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-1
Quadratic
Functions
Chapter 8

2. 2
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-2
§8.5
Graphing Quadratic
Functions

3. 3
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-3
Quadratic Functions
Quadratic Function
A quadratic function is a function that can be written
in the form
f(x) = ax
2
+ bx + c
For real numbers a, b, and c, with a ≠ 0.

4. 4
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-4
The graph of every quadratic function is a
parabola.
Definitions
The vertexis the lowest point on a parabola
that opens upward, or the highest point on
a parabola that opens downward.

5. 5
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-5
Graphs of quadratic equations have symmetry
about a line through the vertex. This line is called
the axis of symmetry.
The sign of a, the numerical coefficient of the
squared term, determines whether the parabola
will open upward or downward.
Definitions

6. 6
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-6
Vertex of a Parabola
Vertex of a Parabola
The parabola represented by the function f(x) = ax
2
+
bx + c will have vertex
Since we often find the y-coordinate of the vertex by
substituting the x-coordinate of the vertex into f(x), the
vertex may also be designated as




 

a
bac
a
b
4
4
,
2
2 












a
b
f
a
b
2
,
2

7. 7
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-7
Axis of Symmetry of a Parabola
Axis of Symmetry
For a quadratic function of the form f(x) = ax
2
+ bx + c,
the equation of the axis of symmetry of the parabola isa
b
x
2


8. 8
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-8
x-Intercepts of a Parabola
x-Intercepts of a Parabola
To find the x-intercepts (if there are any) of a quadratic
function, solve the equation ax
2
+ bx + c = 0 for x.
This equation may be solved by factoring, by using the
quadratic formula, or by completing the square.

9. 9
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-9
Graph Quadratic Functions
Example Consider the quadratic function
y = –x
2
+ 8x –12.
a.Determine whether the parabola opens upward
or downward.
b.Find the y-intercept.
c.Find the vertex.
d.Find the equation of the axis of symmetry.
e.Find the x-intercepts, if any.
f.Draw the graph.
continued

10. 10
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-10
Graph Quadratic Functions
a.Since a is -1, which is less than 0, the parabola
opens downward.
b. To find the y-intercept, set x = 0 and solve for y.
continued1212)0(8)0(
2
y
The y-intercept is (0, 12)

11. 11
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-11
Graph Quadratic Functions
c. First, find the x-coordinate, then find the y-
coordinate of the vertex. From the function, a = -1,
b = 8, and c = -12.
continued4
)1(2
8
2



a
b
x
Since the x-coordinate of the vertex is not a fraction,
we will substitute x = 4 into the original function to
determine the y-coordinate of the vertex.4123216
12)4(8)4(
128
2
2



y
xxy
The vertex is (4, 4).

12. 12
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-12
Graph Quadratic Functions
continued
d. Since the axis of symmetry is a vertical line
through the vertex, the equation is found using the
same formula used to find the x-coordinate of the
vertex (see part c). Thus, the equation of the axis of
symmetry is x = 4.

13. 13
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-13
Graph Quadratic Functions
continued
e. To find the x-intercepts, set y = 0. 2 x 6
02or 06
0)2)(6(
0128
2




x
xx
xx
xx
Thus, the x-intercepts are (2, 0) and (6, 0). These
values could also be found by the quadratic formula
(or by completing the square).

14. 14
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-14
Graph Quadratic Functions
f. Draw the graph.

15. 15
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-15
Solve Maximum and Minimum Problems
A parabola that opens upward has a minimum value
at its vertex, and a parabola that opens downward has
amaximum value at its vertex.

16. 16
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-16
Understand Translations of Parabolas
-4
x
y
4
4
21
2
h x x 
2
f x x 
2
2g x x
Start with the basic graph of f(x) = ax
2
and translate, or shift, the
position of the graph to obtain the graph of the function you are
seeking.
Notice that the value of ain the graph f(x) = ax
2
determines the
width of the parabola. As |a| gets larger, the parabola gets
narrower, and as |a| gets smaller, the parabola gets wider.

17. 17
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-17
-4
x
y
4
4
2
( 2)h x x 
2
f x x 
2
( 2)g x x
Start with the basic graph of f(x) = ax
2
and translate, or shift, the
position of the graph to obtain the graph of the function you are
seeking.
If his a positive real number, the graph of g(x) = a(x–h)
2
will be
shifted hunits to the right of the graph g(x) = ax
2
. If his a negative
real number, the graph of g(x) = a(x–h)
2
will be shifted |h| units to
the left.
Understand Translations of Parabolas

18. 18
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-18
-4
x
y
4
4
2
1h x x 
2
f x x 
2
1g x x
Start with the basic graph of f(x) = ax
2
and translate, or shift, the
position of the graph to obtain the graph of the function you are
seeking.
In general, the graph of g(x) = ax
2
+ k is the graph of f(x) = ax
2
shifted kunits up if kis a positive real number and |k|units down if
kis a negative real number.
Understand Translations of Parabolas

19. 19
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-19
Understand Translations of Parabolas
Parabola Shifts
For any function f(x) = ax
2
, the graph of g(x) = a(x-h)
2
+ k will
have the same shape as the graph of f(x). The graph of g(x)
will be the graph of f(x) shifted as follows:
•If h is a positive real number, the graph will be shifted h
units to the right.
•If h is a negative real number, the graph will be shifted |h|
units to the left.
•If k is a positive real number, the graph will be shifted k
units up.
•If k is a negative real number, the graph will be shifted |k|
units down.

20. 20
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-20
Understand Translations of Parabolas
Axis of Symmetry and Vertex of a Parabola
The graph of any function of the form
f(x) = a(x –h)
2
+ k
will be a parabola with axis of symmetry x = h and vertex at
(h, k).

21. 21
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-21
Write Functions in the Form f(x) = a(x –h)
2
+ k
If we wish to graph parabolas using translations, we
need to change the form of a function from f(x) = ax
2
+ bx+ c to f(x) = a(x –h)2 + k. To do this we complete
the square as we discussed in Section 8.1.
Example Given f(x) = x2 –6x + 10,
a)Write f(x) in the form of f(x) = a(x –h)2 + k.
b)Graph f(x).

22. 22
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-22
Write Functions in the Form f(x) = a(x –h)
2
+ k
a)We use the x2 and -6x terms to obtain a perfect
square trinomial.
Now we take half the coefficient of the x-term and
square it.
We then add this value, 9, within the parentheses.10)6()(
2
 xxxf 9)6(
2
1
2







 109)96()(
2
 xxxf
continued

23. 23
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-23
Write Functions in the Form f(x) = a(x –h)
2
+ k
By doing this we have created a perfect square
trinomial within the parentheses, plus a constant
outside the parentheses. We express the perfect
square trinomial as the square of a binomial.
The function is now in the form we are seeking.1)3()(
2
xxf
continued

24. 24
Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-24
Write Functions in the Form f(x) = a(x –h)
2
+ k
b) Graph f(x).