10.1
Graph y 5 ax 2 1 c
Goal
Your Notes
p Graph simple quadratic functions.
VOCABULARY
Quadratic function
Parabola
Parent quadratic function
Vertex
Axis of Symmetry
PARENT QUADRATIC FUNCTION
The most basic quadratic function in the family of
quadratic functions, called the
, is y 5 x 2. The graph is shown below.
The line that passes through
the vertex and divides the
parabola into two symmetric
parts is called the
. The axis of
symmetry for the graph of
.
y 5 x 2 is the y-axis,
y
5
3
y 5 x2
23
x50
1
21
21
1
3 x
(0, 0)
The lowest or highest point on the parabola is
the
. The vertex of the graph of y 5 x 2
is (
,
).
250
Lesson 10.1 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
10.1
Graph y 5 ax 2 1 c
Goal
Your Notes
p Graph simple quadratic functions.
VOCABULARY
Quadratic function A non-linear function that can
be written in the standard form y 5 ax 2 1 bx 1 c
where a Þ 0
Parabola A U-shaped graph of a quadratic function
Parent quadratic function y 5 x 2
Vertex The lowest or highest point on a parabola
Axis of Symmetry The line that passes through the
vertex and divides the parabola into two symmetric
parts
PARENT QUADRATIC FUNCTION
The most basic quadratic function in the family of
quadratic functions, called the parent quadratic
function , is y 5 x 2. The graph is shown below.
The line that passes through
the vertex and divides the
parabola into two symmetric
parts is called the axis of
symmetry . The axis of
symmetry for the graph of
y 5 x 2 is the y-axis, x 5 0 .
y
5
3
y 5 x2
23
x50
1
21
21
1
3 x
(0, 0)
The lowest or highest point on the parabola is
the vertex . The vertex of the graph of y 5 x 2
is ( 0 , 0 ).
250
Lesson 10.1 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Graph y 5 ax 2 where ⏐a⏐< 1
Example 1
1
Graph y 5 } x 2. Compare the graph with the graph of
y5
2
x 2.
Solution
Step 1 Make a table of values
y
1
7
for y 5 }
x 2.
2
5
x
24
22
0
2
4
3
y
1
23
x
3
1
21
the points from the table.
Step 2
through the points.
Step 3 Draw a
1
x 2 and y 5 x 2. Both
Step 4 Compare the graphs of y 5 }
2
graphs have the same vertex, (
,
), and axis
. However, the graph of
of symmetry,
1
y5}
x 2 is
than the graph of y 5 x 2. This
2
1 2
is because the graph of y 5 }
x is a vertical
2
1 by a factor of
2 of the graph of
y 5 x 2.
Checkpoint Graph the function. Compare the graph
with the graph of y 5 x 2.
1. y 5 25x 2
y
3
23
21
23
1
3
x
29
215
221
Copyright © Holt McDougal. All rights reserved.
Lesson 10.1 • Algebra 1 Notetaking Guide
251
Your Notes
Graph y 5 ax 2 where ⏐a⏐< 1
Example 1
1
Graph y 5 } x 2. Compare the graph with the graph of
y5
2
x 2.
Solution
Step 1 Make a table of values
y
1
7
for y 5 }
x 2.
2
y 5 x2
5
x
24
22
0
8
2
0
y
2
4
3
2
8
1
y 5 2 x2
1
23
x
3
1
21
Step 2 Plot the points from the table.
Step 3 Draw a smooth curve through the points.
1
x 2 and y 5 x 2. Both
Step 4 Compare the graphs of y 5 }
2
graphs have the same vertex, ( 0 , 0 ), and axis
of symmetry, x 5 0 . However, the graph of
1
y5}
x 2 is wider than the graph of y 5 x 2. This
2
1 2
is because the graph of y 5 }
x is a vertical
2
shrink
1 by a factor of
1
2
}
2 of the graph of
y 5 x 2.
Checkpoint Graph the function. Compare the graph
with the graph of y 5 x 2.
1. y 5 25x 2
y 5 25x 2 is narrower and
opens down because it is
a vertical stretch (by a
factor of 5) and a reflection
in the x-axis of the graph
y 5 x 2.
y
3
23
21
23
29
y 5 x2
1
3
x
y 5 25x 2
215
221
Copyright © Holt McDougal. All rights reserved.
Lesson 10.1 • Algebra 1 Notetaking Guide
251
Your Notes
Graph y 5 x 2 1 c
Example 2
Graph y 5 x 2 2 2. Compare the graph with the graph of
y 5 x 2.
y
3
Step 1 Make a table of values for
y 5 x 2 2 2.
x
22
0
21
1
2
1
23
y
1
21
21
3 x
23
the points from the table.
Step 2
through the points.
Step 3 Draw a
Step 4 Compare the graphs of y 5 x 2 2 2 and y 5 x 2.
Both graphs open
and have the same axis of
symmetry,
. However, the vertex of the
,
), is different than
graph of y 5 x 2 2 2, (
2
the vertex of the graph of y 5 x , (
,
),
2
because the graph of y 5 x 2 2 is a
(of
units
) of the graph
2
of y 5 x .
Example 3
Graph y 5 ax 2 1 c
Graph y 5 23x 2 1 3. Compare the graph with the
graph of y 5 x 2.
y
Step 1 Make a table of values for
y 5 23x 2 1 3.
x
22
21
0
1
2
Step 3 Draw a
23
21
22
1
3 x
26
y
Step 2
2
210
the points from the table.
through the points.
Step 4 Compare the graphs. Both graphs have the
same axis of symmetry. However, the graph of
y 5 23x 2 1 3 is
and has a
2
vertex than the graph of y 5 x because the graph
and
of y 5 23x 2 1 3 is a
a
of the graph of y 5 x 2.
252
Lesson 10.1 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Graph y 5 x 2 1 c
Example 2
Graph y 5 x 2 2 2. Compare the graph with the graph of
y 5 x 2.
y
3
Step 1 Make a table of values for
y 5 x 2 2 2.
x
22
21
0
1
2
y
2
21
22
21
2
y 5 x2
1
23
1
21
21
23
3 x
y 5 x2 2 2
Step 2 Plot the points from the table.
Step 3 Draw a smooth curve through the points.
Step 4 Compare the graphs of y 5 x 2 2 2 and y 5 x 2.
Both graphs open up and have the same axis of
symmetry, x 5 0 . However, the vertex of the
graph of y 5 x 2 2 2, ( 0 , 22 ), is different than
the vertex of the graph of y 5 x 2, ( 0 , 0 ),
because the graph of y 5 x 2 2 2 is a vertical
translation (of 2 units down ) of the graph
of y 5 x 2.
Example 3
Graph y 5 ax 2 1 c
Graph y 5 23x 2 1 3. Compare the graph with the
graph of y 5 x 2.
y
Step 1 Make a table of values for
y 5 23x 2 1 3.
x
22
21
0
1
2
y
29
0
3
0
29
y 5 x2
2
23
21
22
1
3 x
26
y 5 23x 2 1 3
210
Step 2 Plot the points from the table.
Step 3 Draw a smooth curve through the points.
Step 4 Compare the graphs. Both graphs have the
same axis of symmetry. However, the graph of
y 5 23x 2 1 3 is narrower and has a higher
vertex than the graph of y 5 x 2 because the graph
of y 5 23x 2 1 3 is a vertical stretch and
a vertical translation of the graph of y 5 x 2.
252
Lesson 10.1 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Graph the function. Compare the graph
with the graph of y 5 x 2.
1
x2 2 6
2. y 5 }
4
y
1
23
x
1
21
21
3
23
25
Compared with the graph of
y 5 x 2, the graph of y 5 ax 2 is:
• a vertical
if a > 1,
• a vertical
if 0 < a < 1.
y 5 ax 2, a > 0
y
x
a.1
a51
0,a,1
Compared with the graph of
y 5 x 2, the graph of y 5 ax 2 is:
• a vertical
and a
in the x-axis if
a < 21,
• a vertical
y 5 ax 2, a < 0
y
a , 21
a 5 21
21 , a , 0
x
and a
in the x-axis if
21 < a < 0.
Compared with the graph of y 5 x 2,
the graph of y 5 x 2 1 c is:
Homework
y 5 x2 1 c
y
• an
vertical
translation if c > 0,
vertical
•a
translation if c < 0.
Copyright © Holt McDougal. All rights reserved.
c.0
c50
c,0
Lesson 10.1 • Algebra 1 Notetaking Guide
x
253
Your Notes
Checkpoint Graph the function. Compare the graph
with the graph of y 5 x 2.
1
x2 2 6
2. y 5 }
4
y
y 5 x2
1
1 2
y5}
x 2 6 is a vertical
4
23
shrink, and downward
vertical translation of the
graph y 5 x 2.
Compared with the graph of
y 5 x 2, the graph of y 5 ax 2 is:
x
1
21
21
3
23 1
y 5 4x2 2 6
25
y 5 ax 2, a > 0
y
• a vertical stretch if a > 1,
• a vertical
shrink
if 0 < a < 1.
x
a.1
a51
0,a,1
Compared with the graph of
y 5 x 2, the graph of y 5 ax 2 is:
• a vertical stretch and a
reflection in the x-axis if
a < 21,
y 5 ax 2, a < 0
y
a , 21
a 5 21
21 , a , 0
x
• a vertical shrink and a
reflection in the x-axis if
21 < a < 0.
Compared with the graph of y 5 x 2,
the graph of y 5 x 2 1 c is:
Homework
y 5 x2 1 c
y
• an
upward
vertical
translation if c > 0,
• a downward vertical
translation if c < 0.
Copyright © Holt McDougal. All rights reserved.
c.0
c50
c,0
Lesson 10.1 • Algebra 1 Notetaking Guide
x
253