Lesson 27
TAKS Grade 9 Objective 5
(A.9)(C)
Vertical Translations
of Parabolas
A parabola is the U-shaped graph of a quadratic function, a function in
the form y ax2 bx c. The simplest quadratic function, y x2, is
known as the parent function. Its graph is shown to the right. The vertex is
the minimum point on the parabola if it opens upward (or the maximum
point if the parabola opens downward). The line that divides a parabola into
two symmetrical halves is the axis of symmetry.
New Vocabulary
•
•
•
•
parabola
quadratic function
vertex
axis of symmetry
Graphing y x2 c
4
Consider the equation y x2 + c. Its graph is that of y x2 if c 0.
If c 0, then the value of c determines how far up or down along the
y-axis the graph of y x2 is translated. When a graph is translated,
its shape does not change.
y
y = x2
2
–4
–2 O
2
4x
–2
When c is positive, the graph moves up. When c is negative, the graph
moves down.
–4
EXAMPLE 1
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
Draw the graph of the function y x2 c when c 5.
Because c is positive, this graph is a translation of the graph of y x2 up 5 units.
Start by making a chart of points:
Now, plot the points:
x
y x2
y x2 5
(x, y)
2
4
9
(2, 9)
1
1
6
(1, 6)
0
0
5
(0, 5)
4
1
1
6
(1, 6)
2
2
4
9
(2, 9)
8
y
6
–4
y=x 2 5
–2 O
4x
2
Quick Check 1
1a. Graph y x2 4.
y
4
1b. Give the equation of the graph below.
8
2
–4
–2 O
y
6
2
4
4x
–2
–4
TAKS Review and Preparation Workbook
2
–4
–2 O
LESSON 27
■
2
4x
Vertical Translations of Parabolas
79
TAKS Objective 5 (A.9)(C)
LESSON 27
Changing the Value of c
Changing the value of c in the equation y x2 c translates the graph up
or down along the y-axis.
Example: Look at the graph of y x2 3. If 3 is changed to 1, this
becomes the graph of y x2 (1).
y x2 3
8
y x2 (1)
y
4
4
–4
2 y=
–2 O
y
2
6
–4
Both y x2 c and
f(x) x2 c describe
the same function.
The form f(x) x2 c
is called function
notation. It can be
read “f of x equals
x squared plus c.”
2
x2
–2 O
3
2
–2 y =
4x
x2
1
–4
4x
Changing c from 3 to 1 is a change of 1 3 4. Therefore, the graph
is translated down by 4 units. Always subtract the original value of c from
the new value of c to find the vertical change.
EXAMPLE 2
When graphed, what function would appear to be shifted 4 units up from the graph of
f(x) x2 6?
The graph of f(x) x2 6 looks like this:
Shifted up 4 units, it looks like this:
–4
–2 O
y
4
2
2
4x
–2
y
–4
–2 O
–4
–2
–6
–4
2
4x
The new graph is the graph of f(x) x2 2.
You can determine this by adding 4 to the original value of c:
6 4 2
Quick Check 2
2a. How would the graph of y x2 5 be changed if the function were
changed to y x2 10?
2b. Consider the graph of y x2 (1). Describe the shift in the vertex
of the parabola if the 1 in the function is changed to 7.
80
LESSON 27
■
Vertical Translations of Parabolas
When the graph of a
parabola is translated
vertically by changing
the value of c, the
graph moves as a
whole. Each point is
translated the same
distance as every
other point.
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
2
Name__________________________Class____________Date________
1 The graph of which function appears to
be shifted 3 units down from the graph of
y x2 3?
A y x2 3
4 In the graph of the function y x2 (3),
which describes the shift in the vertex of the
parabola if the 3 in the function is changed
to 8?
B y x2
F 8 units up
C y x2 3
G 11 units up
D y x2 6
H 8 units down
J 11 units down
2 How would the graph of y 5 be
affected if the 5 were changed to a 5?
x2
5 Which graph shows a function f(x) x2 c
in which c 2?
8
F The graph would shift 10 units left.
6
G The graph would shift 10 units right.
H The graph would shift 10 units up.
4
A
J The graph would shift 10 units down.
2
–4
–2 O
4
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
3 What will the equation of the graph below be
if it is shifted 5 units up?
4
y
–2 O
2
4x
2
4x
2
4x
2
4x
y
2
B
–4
–2 O
–2
2
–4
y
2
–4
4x
–2
4
y
–4
2
A f(x) x2 5
B f(x) x2 1
C
–4
–2 O
–2
C f(x) x2 1
–4
D f(x) x2 5
8
y
6
4
D
2
–4
TAKS Review and Preparation Workbook
LESSON 27
–2 O
■
Vertical Translations of Parabolas
81