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Math Problems
Algebra 2
Domain and range of quadratic functions: equations
Find the area enclosed by the graphs of
f
(
x
)
=
2
−
x
f(x)=2-x
f
(
x
)
=
2
−
x
and
g
(
x
)
=
x
2
g(x)=x^{2}
g
(
x
)
=
x
2
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Find the area enclosed by the graphs of
f
(
x
)
=
2
−
x
f(x)=2-x
f
(
x
)
=
2
−
x
and
g
(
x
)
=
x
2
g(x)=x^{2}
g
(
x
)
=
x
2
.
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Which equation best matches the graph shown below?
\newline
Answer
\newline
y
=
(
x
−
5
)
2
−
2
y=(x-5)^{2}-2
y
=
(
x
−
5
)
2
−
2
\newline
y
=
−
(
x
+
5
)
2
−
2
y=-(x+5)^{2}-2
y
=
−
(
x
+
5
)
2
−
2
\newline
y
=
−
(
x
−
5
)
2
−
2
y=-(x-5)^{2}-2
y
=
−
(
x
−
5
)
2
−
2
\newline
y
=
(
x
+
5
)
2
−
2
y=(x+5)^{2}-2
y
=
(
x
+
5
)
2
−
2
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Which equations define
y
y
y
as a nonlinear function of
x
x
x
? Select all that apply.
\newline
Multi-select Choices:
\newline
(A)
y
=
4
x
4
+
5
y = 4x^4 + 5
y
=
4
x
4
+
5
\newline
(B)
y
=
x
4
y = \frac{x}{4}
y
=
4
x
\newline
(C)
y
=
x
+
2
2
y = x + 2^2
y
=
x
+
2
2
\newline
(D)
4
y
=
3
x
4y = 3x
4
y
=
3
x
\newline
(E)
y
=
5
x
2
y = 5x^2
y
=
5
x
2
\newline
(F)
y
=
3
−
5
x
y = 3 - 5x
y
=
3
−
5
x
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Which equations define
y
y
y
as a nonlinear function of
x
x
x
? Select all that apply.
\newline
Multi-select Choices:
\newline
(A)
y
=
3
x
2
y = \frac{3x}{2}
y
=
2
3
x
\newline
(B)
y
−
4
=
3
x
2
y - 4 = 3x^2
y
−
4
=
3
x
2
\newline
(C)
y
=
x
5
+
6
y = x^5 + 6
y
=
x
5
+
6
\newline
(D)
2
y
=
12
2y = 12
2
y
=
12
\newline
(E)
y
=
7
x
−
x
y = 7x - x
y
=
7
x
−
x
\newline
(F)
y
=
7
x
×
x
y = 7x \times x
y
=
7
x
×
x
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Find the exact area of the surface obtained by
\newline
x
=
e
t
−
t
,
y
=
4
e
t
2
,
0
≤
t
≤
1
x=e^{t}-t, \quad y=4 e^{\frac{t}{2}}, \quad 0 \leq t \leq 1
x
=
e
t
−
t
,
y
=
4
e
2
t
,
0
≤
t
≤
1
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8
8
8
−
95
-95
−
95
Use the graph on the right to evaluate each of the following.
\newline
a.
lim
x
→
4
f
(
x
)
\lim _{x \rightarrow 4} f(x)
lim
x
→
4
f
(
x
)
\newline
b.
lim
x
→
0
−
f
(
x
)
\lim _{x \rightarrow 0^{-}} f(x)
lim
x
→
0
−
f
(
x
)
\newline
c.
lim
x
→
−
∞
f
(
x
)
\lim _{x \rightarrow-\infty} f(x)
lim
x
→
−
∞
f
(
x
)
\newline
d.
lim
x
→
0
f
(
x
)
\lim _{x \rightarrow 0} f(x)
lim
x
→
0
f
(
x
)
\newline
e.
lim
x
→
−
3
f
(
x
)
\lim _{x \rightarrow-3} f(x)
lim
x
→
−
3
f
(
x
)
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Excercises: Determine the antiderivatives
\newline
1
1
1
.
f
(
x
)
=
(
x
+
1
)
100
f(x)=(x+1)^{100}
f
(
x
)
=
(
x
+
1
)
100
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find the area of the figure bonded by
y
=
x
2
−
2
x
+
5
y=x^2-2x+5
y
=
x
2
−
2
x
+
5
,
y
=
5
x
−
5
y=5x-5
y
=
5
x
−
5
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st \#
7
7
7
on Solving Quadratics (Sq Rt...
\newline
37
37
37
Solve the system:
\newline
{
y
=
7
x
+
1
y
=
−
5
x
+
13
\left\{\begin{array}{l} y=7 x+1 \\ y=-5 x+13 \end{array}\right.
{
y
=
7
x
+
1
y
=
−
5
x
+
13
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Example
\newline
Linear Equat
\newline
Example
\newline
y
=
2
x
+
3
y=2 x+3
y
=
2
x
+
3
\newline
Gradient
\newline
y
y
y
-intercept
\newline
Table of Values
\newline
\begin{tabular}{|l|l|l|l|l|}
\newline
\hline
x
x
x
&
−
1
-1
−
1
&
0
0
0
&
1
1
1
&
2
2
2
\\
\newline
\hline
y
y
y
& & & & \\
\newline
\hline
\newline
\end{tabular}
\newline
Question
1
1
1
\newline
y
=
2
x
−
4
y=2 x-4
y
=
2
x
−
4
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The function
k
k
k
is given by
k
(
x
)
=
3
x
2
−
19
x
−
14
k(x)=3 x^{2}-19 x-14
k
(
x
)
=
3
x
2
−
19
x
−
14
. Find all values of
x
x
x
for which
k
(
x
)
>
0
k(x)>0
k
(
x
)
>
0
.
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Find the set values for
p
p
p
such that
p
x
2
+
2
x
+
4
p
<
0
px^2 + 2x + 4p < 0
p
x
2
+
2
x
+
4
p
<
0
for all
c
c
c
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The set
S
=
{
x
∣
−
2
<
x
<
8
}
S=\{x \mid-2<x<8\}
S
=
{
x
∣
−
2
<
x
<
8
}
is described by which interval notation?
\newline
(
−
2
,
8
)
(-2,8)
(
−
2
,
8
)
\newline
[
−
2
,
8
]
[-2,8]
[
−
2
,
8
]
\newline
[
−
2
,
8
)
[-2,8)
[
−
2
,
8
)
\newline
(
−
2
,
8
]
(-2,8]
(
−
2
,
8
]
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What is the range of this quadratic function?
\newline
y
=
x
2
−
10
x
+
21
y = x^2 - 10x + 21
y
=
x
2
−
10
x
+
21
\newline
Choices:
\newline
(A)
y
∣
y
≥
5
{y | y \geq 5}
y
∣
y
≥
5
\newline
(B)
y
∣
y
≤
5
{y | y \leq 5}
y
∣
y
≤
5
\newline
(C)
y
∣
y
≥
−
4
{y | y \geq -4}
y
∣
y
≥
−
4
\newline
(D)all real numbers
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5
5
5
Find the values of the constant
c
c
c
for which the line
4
y
=
2
x
+
c
4 y=2 x+c
4
y
=
2
x
+
c
is a tangent to th curve
y
=
4
x
+
8
x
y=4 x+\frac{8}{x}
y
=
4
x
+
x
8
.
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Select all the expressions that are equivalent to
3
6
×
9
6
3^6 \times 9^6
3
6
×
9
6
.
\newline
Multi-select Choices:
\newline
(A)
1
2
7
−
6
\frac{1}{27^{-6}}
2
7
−
6
1
\newline
(B)
2
7
12
27^{12}
2
7
12
\newline
(C)
1
2
7
6
\frac{1}{27^6}
2
7
6
1
\newline
(D)
2
7
36
27^{36}
2
7
36
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The graph of
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
between the lines
x
≡
a
x \equiv a
x
≡
a
and
x
≡
b
x \equiv b
x
≡
b
is rotated around the line
y
≡
K
y \equiv K
y
≡
K
to form a solid, as shown in the figure.
\newline
Which of the following integrals represents the volume of this solid?
\newline
25
25
25
\newline
∫
a
b
π
(
f
(
x
)
−
K
)
2
d
x
\int_{a}^{b} \pi(f(x)-K)^{2} d x
∫
a
b
π
(
f
(
x
)
−
K
)
2
d
x
\newline
Top:
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
\newline
Line:
y
=
K
y=K
y
=
K
\newline
each slice is a circle
\newline
∫
a
b
π
[
(
f
(
x
)
)
2
−
K
2
]
d
x
\int_{a}^{b} \pi\left[(f(x))^{2}-K^{2}\right] d x
∫
a
b
π
[
(
f
(
x
)
)
2
−
K
2
]
d
x
\newline
∫
a
b
2
π
x
∣
f
(
x
)
−
K
]
d
x
\left.\int_{a}^{b} 2 \pi x \mid f(x)-K\right] d x
∫
a
b
2
π
x
∣
f
(
x
)
−
K
]
d
x
\newline
Tutoring
\newline
∫
a
b
π
(
f
(
x
)
)
2
d
x
\int_{a}^{b} \pi(f(x))^{2} d x
∫
a
b
π
(
f
(
x
)
)
2
d
x
Get tutor help
Qacstion
\newline
Determine each feature of the graph of the given function.
\newline
f
(
x
)
=
5
x
+
5
2
x
+
2
f(x)=\frac{5x+5}{2x+2}
f
(
x
)
=
2
x
+
2
5
x
+
5
\newline
Answer sthempti cut of
2
2
2
\newline
Horizontal Asymptote:
\newline
y
=
y=
y
=
\newline
Vertical Asymptote:
\newline
x
=
x=
x
=
\newline
\left[\begin{array}{cc}\(\newline\text{x-Intercept: } & (\square,0)\square (\newline\)\text{y-Intercept: } & (0,\square)\text{ Noytares }
\newline
\end{array}\right]\)
\newline
Hole:
\newline
□
,
□
\square,\square
□
,
□
Ssb bolk
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Qacstion
\newline
Determine each feature of the graph of the given function.
\newline
f
(
x
)
=
5
x
+
5
2
x
+
2
f(x)=\frac{5 x+5}{2 x+2}
f
(
x
)
=
2
x
+
2
5
x
+
5
\newline
Answer sthempti out of
2
2
2
\newline
Horizontal Asymptote:
y
=
y=
y
=
\newline
Vertical Asymptote:
x
=
x=
x
=
\newline
x
x
x
-Intercept:
□
,
0
)
\square, 0)
□
,
0
)
Nox-inerest
\newline
y
y
y
-Intercept:
(
0
,
□
)
(0, \square)
(
0
,
□
)
Noy-thecest
\newline
Hole:
(
□
,
□
)
(\square, \square)
(
□
,
□
)
Noble
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a.)
y
=
x
−
6
y=x-6
y
=
x
−
6
Graph each linear equation usens the table point plotting method
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Rewrite the following equation in slope-intercept form.
7
x
+
2
y
=
−
11
7x + 2y = -11
7
x
+
2
y
=
−
11
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Which of the following is the graph of
−
x
2
-x^{2}
−
x
2
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How many distinct real solutions does the given equation have?
1
,
000
=
20
z
2
1{,}000=20z^2
1
,
000
=
20
z
2
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Multiply complex numbers
\newline
−
12
⋅
(
8
−
2
i
)
=
□
-12 \cdot(8-2 i)=\square
−
12
⋅
(
8
−
2
i
)
=
□
\newline
Your answer should be a complex number in the form
a
+
b
i
a+b i
a
+
bi
where
a
a
a
and
b
b
b
are real numbers.
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Which of the following is a correct interpretation of the expression
\newline
−
6
+
9
-6+9
−
6
+
9
?
\newline
Choose
1
1
1
answer:
\newline
(A) The number that is
9
9
9
to the left of
−
6
-6
−
6
on the number line
\newline
(B) The number that is
9
9
9
to the right of
−
6
-6
−
6
on the number line
\newline
(C) The number that is
6
6
6
to the left of
−
9
-9
−
9
on the number line
\newline
(D) The number that is
6
6
6
to the right of
−
9
-9
−
9
on the number line
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Let
g
g
g
be a continuous function on the closed interval
[
−
1
,
3
]
[-1,3]
[
−
1
,
3
]
, where
g
(
−
1
)
=
−
2
g(-1)=-2
g
(
−
1
)
=
−
2
and
g
(
3
)
=
−
5
g(3)=-5
g
(
3
)
=
−
5
.
\newline
Which of the following is guaranteed by the Intermediate Value Theorem?
\newline
Choose
1
1
1
answer:
\newline
(A)
g
(
c
)
=
−
3
g(c)=-3
g
(
c
)
=
−
3
for at least one
c
c
c
between
−
1
-1
−
1
and
3
3
3
\newline
(B)
g
(
c
)
=
−
3
g(c)=-3
g
(
c
)
=
−
3
for at least one
c
c
c
between
−
5
-5
−
5
and
−
2
-2
−
2
\newline
(C)
g
(
c
)
=
0
g(c)=0
g
(
c
)
=
0
for at least one
c
c
c
between
−
5
-5
−
5
and
−
2
-2
−
2
\newline
(D)
g
(
c
)
=
0
g(c)=0
g
(
c
)
=
0
for at least one
c
c
c
between
−
1
-1
−
1
and
3
3
3
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What is the amplitude of
y
=
3
sin
(
2
x
−
1
)
+
4
y=3 \sin (2 x-1)+4
y
=
3
sin
(
2
x
−
1
)
+
4
?
\newline
units
Get tutor help
What is the amplitude of
y
=
−
3
cos
(
π
x
+
2
)
−
6
?
y=-3 \cos (\pi x+2)-6 ?
y
=
−
3
cos
(
π
x
+
2
)
−
6
?
\newline
units
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What is the amplitude of
y
=
5
sin
(
4
x
−
2
)
−
3
y=5 \sin (4 x-2)-3
y
=
5
sin
(
4
x
−
2
)
−
3
?
\newline
units
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What is the amplitude of
\newline
g
(
x
)
=
−
2
sin
(
π
2
x
−
3
)
+
5
?
g(x)=-2 \sin \left(\frac{\pi}{2} x-3\right)+5 ?
g
(
x
)
=
−
2
sin
(
2
π
x
−
3
)
+
5
?
\newline
units
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What is the amplitude of
h
(
x
)
=
−
4
sin
(
2
x
−
7
)
+
3
h(x)=-4 \sin (2 x-7)+3
h
(
x
)
=
−
4
sin
(
2
x
−
7
)
+
3
?
\newline
units
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The functions
f
(
x
)
=
8
(
2
5
)
x
f(x)=8\left(\frac{2}{5}\right)^{x}
f
(
x
)
=
8
(
5
2
)
x
and
g
(
x
)
=
8
(
b
)
x
g(x)=8(b)^{x}
g
(
x
)
=
8
(
b
)
x
are graphed in the
x
y
x y
x
y
-plane. For what value of
b
b
b
would the graphs of functions
f
f
f
and
g
g
g
be symmetric with respect to the
y
y
y
-axis?
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The equation
s
=
(
t
+
3
)
2
(
t
+
2
)
(
t
+
1
)
(
t
)
(
t
−
1
)
s=(t+3)^{2}(t+2)(t+1)(t)(t-1)
s
=
(
t
+
3
)
2
(
t
+
2
)
(
t
+
1
)
(
t
)
(
t
−
1
)
is graphed on the
s
t
s t
s
t
-plane. What is the product of the unique
t
t
t
-intercepts of the graph?
\newline
□
\square
□
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What is the range of this quadratic function?
\newline
y
=
x
2
+
8
x
+
16
y = x^2 + 8x + 16
y
=
x
2
+
8
x
+
16
\newline
Choices:
\newline
(A)
{
y
∣
y
≥
0
}
\left\{y\mid y \geq 0\right\}
{
y
∣
y
≥
0
}
\newline
(B)
{
y
∣
y
≤
0
}
\left\{y\mid y \leq 0\right\}
{
y
∣
y
≤
0
}
\newline
(C)
{
y
∣
y
≥
−
4
}
\left\{y\mid y \geq -4\right\}
{
y
∣
y
≥
−
4
}
\newline
(D) all real numbers
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