Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 1 of 5)
Here are the two forms in which quadratic functions can be written:
Vertex Form:
y  a( x  h ) 2  k
General (Standard) Form:
y  ax 2  bx  c
In these equations, a, b, and c, h, and k represent constants, but a cannot equal zero.
 Why must we say a  0 ?
Comparison of Characteristics
Standard Form
y  ax 2  bx  c
Characteristic
( 2ab , ?)
Plug x = 2ab into the
equation to find y
b
x
2a
vertex
axis of
symmetry
y-intercept
point symmetric
to y-intercept
x-intercept(s)
Vertex Form
y  a( x  h ) 2  k
The two forms of
quadratic equations
provide information
about the function’s
graph in different ways.
(h, k)
x=h
(0, ?)
Plug x = 0 into the
equation to find y
The y-intercept (and other points) can be reflected
across the axis of symmetry to find other points on the
graph of the function.
These points can be read from the graph or table.
When in doubt, use the calculator’s CALC 2: zero
command (2nd, TRACE).
(0, c)
Find each characteristic for the functions described.
y  x 2  2x  3
Characteristic
However, some things
are the same,
regardless of which
form you use.
y  ( x  1) 2  4
vertex
axis of
symmetry
y-intercept
symmetric
point to
y-intercept
x-intercept(s)
Compare with the results from the handout: Investigating Characteristics of Quadratic
Functions.
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 2 of 5)
Sample Problems
Find the characteristic parts of each function. Use this information to produce the graph.
A)
y   x 2  6x  2
Characteristic
Value
x
y
x
y
Vertex
Axis of
Symmetry
y-intercept
Point
symmetric to
y-intercept
x-intercept(s)
B)
f ( x )  2( x  1) 2  3
Characteristic
Value
Vertex
Axis of
Symmetry
y-intercept
Point
symmetric to
y-intercept
x-intercept(s)
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 3 of 5)
Practice Problems
For problems #1-4 make a table of values, graph the function, find the vertex, determine if the vertex
is a maximum or minimum, write the equation of the line for the axis of symmetry, find the y-intercept
and symmetric point, and give the x-intercepts.
1)
f ( x )  x 2  4x  5
Characteristic
Value
x
y
x
y
Vertex
Axis of
Symmetry
y-intercept
Point
symmetric to
y-intercept
x-intercept(s)
2)
y  ( x  2) 2
Characteristic
Value
Vertex
Axis of
Symmetry
y-intercept
Point
symmetric to
y-intercept
x-intercept(s)
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 4 of 5)
3)
y   x 2  4 x  12
Characteristic
Value
x
y
x
y
Vertex
Axis of
Symmetry
y-intercept
Point
symmetric to
y-intercept
x-intercept(s)
4)
y  2( x  1) 2  1
Characteristic
Value
Vertex
Axis of
Symmetry
y-intercept
Point
symmetric to
y-intercept
x-intercept(s)
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 5 of 5)
5) True Value Fabricators produces
circular iron cast disks to be used as
endplates for pipes. The cost of the disks
is a quadratic function of the diameter.
The cost of some disks is given at right.
1 inch diameter …. $12.00
2 inch diameter …. $18.00
3 inch diameter …. $28.00
4 inch diameter …. $42.00
5 inch diameter …. $60.00
A)
In this situation, what are the independent and
dependent variables?
B)
C)
Sketch a scatterplot of the data. Label the axes.
Enter the data into the graphing calculator. Use
transformations of y  x 2 to determine a
representative function for the data set in
y  a( x  h ) 2  k form.
D)
What would be a reasonable domain and range for this
function?
E)
Find each of the characteristics and explain their meaning in the problem situation.
Characteristic
Value(s)
Meaning in Problem Situation
vertex
axis of
symmetry
y-intercept
x-intercept(s)
F)
What would be the cost of a disk with a diameter of 12 inches?
G)
If the cost of the disk is $522, what would be the diameter of the disk?
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 1 of 5) KEY
Here are the two forms in which quadratic functions can be written:
Vertex Form:
y  a( x  h ) 2  k
General (Standard) Form:
y  ax 2  bx  c
In these equations, a, b, and c, h, and k represent constants, but a cannot equal zero.
 Why must we say a  0 ?
Sample: If a were equal to zero, the function would no longer have an x2-term and would then
be linear instead of quadratic.
Comparison of Characteristics
Standard Form
Characteristic
y  ax 2  bx  c
( 2ab , ?)
Plug x = 2ab into the
equation to find y
b
x
2a
vertex
axis of
symmetry
y-intercept
point symmetric
to y-intercept
x-intercept(s)
Vertex Form
y  a( x  h ) 2  k
The two forms of
quadratic equations
provide information
about the function’s
graph in different ways.
(h, k)
x=h
(0, ?)
Plug x = 0 into the
equation to find y
The y-intercept (and other points) can be reflected
across the axis of symmetry to find other points on the
graph of the function.
These points can be read from the graph or table.
When in doubt, use the calculator’s CALC 2: zero
command (2nd, TRACE).
(0, c)
Find each characteristic for the functions described.
y  x 2  2x  3
Characteristic
vertex
( 2)
=1
x = 2(1)
(1, -4)
However, some things
are the same,
regardless of which
form you use.
y  ( x  1) 2  4
(1, -4)
axis of
symmetry
( 2)
, or x = 1
x = 2(1)
x=1
y-intercept
(0, -3)
(0, -3)
symmetric
point to
y-intercept
(2, -3)
(2, -3)
(-1, 0) (3, 0)
(-1, 0) (3, 0)
x-intercept(s)
Compare with the results from the handout: Investigating Characteristics of Quadratic
Functions.
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 2 of 5) KEY
Sample Problems
Find the characteristic parts of each function. Use this information to produce the graph.
A)
y   x 2  6x  2
Characteristic
Vertex
B)
Value
(3, 7) Max
Axis of
Symmetry
x=3
y-intercept
(0, -2)
Point
symmetric to
y-intercept
(6, -2)
x-intercept(s)
(0.35, 0)
(5.65, 0)
f ( x )  2( x  1) 2  3
Characteristic
Vertex
Value
(-1, 3) Min
Axis of
Symmetry
x = -1
y-intercept
(0, 5)
Point
symmetric to
y-intercept
(-2, 5)
x-intercept(s)
None
©2010, TESCCC
08/01/10
x
0
y
-2
1
3
2
6
3
7
4
6
5
3
6
-2
x
-4
y
21
-2
11
-2
5
-1
3
0
5
1
11
2
21
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 3 of 5) KEY
Practice Problems
For problems #1-4 make a table of values, graph the function, find the vertex, determine if the vertex
is a maximum or minimum, write the equation of the line for the axis of symmetry, find the y-intercept
and symmetric point, and give the x-intercepts.
1)
f ( x )  x 2  4x  5
Characteristic
Vertex
2)
Value
(-2, -9) Min
Axis of
Symmetry
x=-2
y-intercept
(0, -5)
Point
symmetric to
y-intercept
(-4, -5)
x-intercept(s)
(-5, 0)
(1, 0)
y  ( x  2) 2
Characteristic
Vertex
Value
(2, 0)
Min
Axis of
Symmetry
x=2
y-intercept
(0, 4)
Point
symmetric to
y-intercept
(4, 4)
x-intercept(s)
(2, 0)
©2010, TESCCC
08/01/10
x
-4
y
-5
-3
-8
-2
-9
-1
-8
0
-5
x
0
y
4
1
1
2
0
3
1
4
4
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 4 of 5) KEY
3)
y   x 2  4 x  12
Characteristic
Vertex
4)
Value
(-2, 16)
Max
Axis of
Symmetry
x = -2
y-intercept
(0, 12)
Point
symmetric to
y-intercept
(-4, 12)
x-intercept(s)
(-6, 0)
(2, 0)
y  2( x  1) 2  1
Characteristic
Vertex
y
12
-3
15
-2
16
-1
15
0
12
x
-3
y
9
-2
3
-1
1
0
3
1
9
Value
(-1, 1)
Min
Axis of
Symmetry
x = -1
y-intercept
(0, 3)
Point
symmetric to
y-intercept
(-2, 3)
x-intercept(s)
None
©2010, TESCCC
x
-4
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 02
Characteristics of Quadratic Functions (pp. 5 of 5) KEY
5) True Value Fabricators produces
circular iron cast disks to be used as
endplates for pipes. The cost of the disks
is a quadratic function of the diameter.
The cost of some disks is given at right.
A)
In this situation, what are the independent and
dependent variables?
Ind. – Diameter (in.)
Dep. – Cost ($)
B)
C)
Sketch a scatterplot of the data. Label the axes.
Enter the data into the graphing calculator. Use
transformations of y  x 2 to determine a
representative function for the data set in
y  a( x  h ) 2  k form.
y = 2x2 + 10
D)
What would be a reasonable domain and range for this
function?
1 inch diameter …. $12.00
2 inch diameter …. $18.00
3 inch diameter …. $28.00
4 inch diameter …. $42.00
5 inch diameter …. $60.00
Answers may vary. For the situation, we might say that the domain must be positive values of x
(or, x > 0), and that the cost will never fall below $10 (y  10).
E)
Find each of the characteristics and explain their meaning in the problem situation.
Characteristic
vertex
Value(s)
Meaning in Problem Situation
(0, 10)
This is a $10 basic fee charged on all
disks regardless of diameter.
axis of
symmetry
x=0
This is the line denoting the basic fee
charged on all disks regardless of
diameter.
y-intercept
(0, 10)
This is a $10 basic fee charged on all
disks regardless of diameter.
none
Customers will never be charged a fee of
$0 when buying disk.
x-intercept(s)
F)
What would be the cost of a disk with a diameter of 12 inches? $298
G)
If the cost of the disk is $522, what would be the diameter of the disk? 16 inches
©2010, TESCCC
08/01/10