Write the equation of a parabola in standard or the vertex form. Determine the vertex, and the axis of symmetry
Y=x^2+8x+17
Answers
Answer:
Vertex :
(
−
3
,
−
4
)
Axis of Symmetry is at :
x
=
(
−
3
)
x-intercepts:
(
−
1
,
0
)
and
(
−
5
,
0
)
y-intercept:
(
0
,
5
)
Explanation:
Given:
y
=
f
(
x
)
=
x
2
+
6
x
+
5
The Vertex Form of a quadratic function is given by:
f
(
x
)
=
a
(
x
−
h
)
2
+
k
, where
(
h
,
k
)
is the Vertex of the parabola.
x
=
h
is the axis of symmetry.
Use completing the square method to convert
f
(
x
)
into Vertex Form.
y
=
f
(
x
)
=
x
2
+
6
x
+
5
Standard Form
⇒
a
x
2
+
b
x
+
c
=
0
Consider the quadratic
x
2
+
6
x
+
5
=
0
a
=
1
;
b
=
6
and
c
=
5
Step 1 - Move the constant value to the right-hand side.
Subtract 5 from both sides.
x
2
+
6
x
+
5
−
5
=
0
−
5
x
2
+
6
x
+
5
−
5
=
0
−
5
x
2
+
6
x
=
−
5
Step 2 - Add a value to both sides.
What value to add?
Add the square of
b
2
Hence,
x
2
+
6
x
+
[
(
6
2
)
2
]
=
−
5
+
[
(
6
2
)
2
]
x
2
+
6
x
+
9
=
−
5
+
9
x
2
+
6
x
+
9
=
4
Step 3 - Write as Perfect Square.
(
x
+
3
)
2
=
4
Subtract
4
from both sides to get the vertex form.
(
x
+
3
)
2
−
4
=
4
−
4
f
(
x
)
=
(
x
+
3
)
2
−
4
Now, we have the vertex form.
f
(
x
)
=
a
(
x
−
h
)
2
+
k
, where
(
h
,
k
)
is the Vertex of the parabola.
Hence, Vertex is at
(
−
3
,
−
4
)
Axis of Symmetry is at
x
=
h
Note that
h
=
−
3
⇒
x
=
−
3
Step 4 - Write the x, y intercepts.
Consider
(
x
+
3
)
2
=
4
To find the solutions, take square root on both sides.
√
(
x
+
3
)
2
=
±
√
4
⇒
x
+
3
=
±
2
There are two solutions.
x
+
3
=
2
⇒
x
=
2
−
3
=
−
1
Hence,
x
=
−
1
is one solution.
Next,
x
+
3
=
−
2
x
=
−
2
−
3
=
−
5
Hence,
x
=
−
5
is the other solution.
Hence, we have two x-intercepts:
(
−
1
,
0
)
and
(
−
5
,
0
)
To find the y-intercept:
Let
x
=
0
We have,
f
(
x
)
=
(
x
+
3
)
2
−
4
f
(
0
)
=
(
0
+
3
)
2
−
4
⇒
3
2
−
4
=
9
−
4
=
5
Hence, y-intercept is at
y
=
5
⇒
(
0
,
5
)
Step-by-step explanation:
draw the graph:
Answer:
Vertex :
(
−
3
,
−
4
)
Axis of Symmetry is at :
x
=
(
−
3
)
x-intercepts:
(
−
1
,
0
)
and
(
−
5
,
0
)
y-intercept:
(
0
,
5
)
Explanation:
Given:
y
=
f
(
x
)
=
x
2
+
6
x
+
5
The Vertex Form of a quadratic function is given by:
f
(
x
)
=
a
(
x
−
h
)
2
+
k
, where
(
h
,
k
)
is the Vertex of the parabola.
x
=
h
is the axis of symmetry.
Use completing the square method to convert
f
(
x
)
into Vertex Form.
y
=
f
(
x
)
=
x
2
+
6
x
+
5
Standard Form
⇒
a
x
2
+
b
x
+
c
=
0
Consider the quadratic
x
2
+
6
x
+
5
=
0
a
=
1
;
b
=
6
and
c
=
5
Step 1 - Move the constant value to the right-hand side.
Subtract 5 from both sides.
x
2
+
6
x
+
5
−
5
=
0
−
5
x
2
+
6
x
+
5
−
5
=
0
−
5
x
2
+
6
x
=
−
5
Step 2 - Add a value to both sides.
What value to add?
Add the square of
b
2
Hence,
x
2
+
6
x
+
[
(
6
2
)
2
]
=
−
5
+
[
(
6
2
)
2
]
x
2
+
6
x
+
9
=
−
5
+
9
x
2
+
6
x
+
9
=
4
Step 3 - Write as Perfect Square.
(
x
+
3
)
2
=
4
Subtract
4
from both sides to get the vertex form.
(
x
+
3
)
2
−
4
=
4
−
4
f
(
x
)
=
(
x
+
3
)
2
−
4
Now, we have the vertex form.
f
(
x
)
=
a
(
x
−
h
)
2
+
k
, where
(
h
,
k
)
is the Vertex of the parabola.
Hence, Vertex is at
(
−
3
,
−
4
)
Axis of Symmetry is at
x
=
h
Note that
h
=
−
3
⇒
x
=
−
3
Step 4 - Write the x, y intercepts.
Consider
(
x
+
3
)
2
=
4
To find the solutions, take square root on both sides.
√
(
x
+
3
)
2
=
±
√
4
⇒
x
+
3
=
±
2
There are two solutions.
x
+
3
=
2
⇒
x
=
2
−
3
=
−
1
Hence,
x
=
−
1
is one solution.
Next,
x
+
3
=
−
2
x
=
−
2
−
3
=
−
5
Hence,
x
=
−
5
is the other solution.
Hence, we have two x-intercepts:
(
−
1
,
0
)
and
(
−
5
,
0
)
To find the y-intercept:
Let
x
=
0
We have,
f
(
x
)
=
(
x
+
3
)
2
−
4
f
(
0
)
=
(
0
+
3
)
2
−
4
⇒
3
2
−
4
=
9
−
4
=
5
Hence, y-intercept is at
y
=
5
⇒
(
0
,
5
)
Analyze the image of the graph below:
enter image source here
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