The maximum or minimum point of the fraction x^2 +y^2+
Answers
Step-by-step explanation:
Given:
y
=
2
x
2
−
18
x
+
13
is a quadratic equation in standard form:
y
=
a
x
2
+
b
x
+
c
,
where:
a
=
2
,
b
=
−
18
, and
c
=
13
To graph a quadratic function, you need to have at least the vertex and x-intercepts. The y-intercept is helpful, also.
Axis of Symmetry: vertical line
(
x
,
±
∞
)
that divides the parabola into two equal halves. The variable for the line is
x
=
−
b
2
a
.
x
=
−
(
−
18
)
2
⋅
2
x
=
18
4
x
=
9
2
←
axis of symmetry and
x
-value for the vertex
Vertex: the maximum or minimum point of the parabola. If
a
>
0
, the vertex is the minimum point and the parabola will open upward. If
a
<
0
, the vertex is the maximum point and the parabola will open downward.
We have the
x
-value of the vertex. To determine the
y
-value, substitute
9
2
for
x
in the equation and solve for
y
.
y
=
2
(
9
2
)
2
−
18
(
9
2
)
+
13
Simplify.
y
=
2
(
81
4
)
−
162
2
+
13
All terms must have a common denominator of
4
. Multiply fractions without a denominator of
4
by a multiplier equal to
1
that will produce an equivalent fraction with a denominator of
4
. For example,
3
3
=
1
Recall that any whole number,
n
, is understood to have a denominator of
1
. So
13
=
13
1
y
=
162
4
−
162
2
×
2
2
+
13
1
×
4
4
y
=
162
4
−
324
4
+
52
4
y
=
(
162
−
324
+
52
)
4
y
=
−
110
4
y
=
−
55
2
Vertex:
(
9
2
,
−
55
2
)
←
minimum point of the parabola
Approximate vertex:
(
4.5
,
−
27.5
)
Substitute
0
for
y
and use the quadratic formula to find the roots and the x-intercepts.
0
=
2
x
2
−
18
x
+
13
x
=
−
b
±
√
b
2
−
4
a
c
2
a
Plug in the known values.
x
=
−
(
−
18
)
±
√
(
−
18
)
2
−
4
⋅
2
⋅
13
2
⋅
2
Simplify.
x
=
18
±
√
324
−
104
4
x
=
18
±
√
220
4
Prime factorize
220
.
x
=
18
±
√
(
2
×
2
)
×
5
×
11
4
x
=
18
±
2
√
55
4
Simplify.
x
=
9
±
√
55
2
Roots: values for
x
x
=
9
+
√
55
2
,
9
−
√
55
2
Approximate values for
x
.
x
=
8.21
,
0.792
X-intercepts: values of
x
when
y
=
0
x
-intercepts:
(
9
+
√
55
2
,
0
)
and
(
9
−
√
55
2
,
0
)
Approximate
x
-intercepts:
(
8.21
,
0
)
and
(
0.792
,
0
)
Y-Intercept: value of
y
when
x
=
0
y
=
2
(
0
)
2
−
18
(
0
)
+
13
y
=
13
Y-intercept:
(
0
,
13
)
Plot the vertex and x-intercepts and sketch a parabola through the points. Do not connect the dots.
graph{y=2x^2-18x+13 [-13.95, 18.07, -40.31, -24.29]}