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Math Problems
Algebra 1
Compare linear, exponential, and quadratic growth
y
+
y
2
y
−
2
3
\frac{y+y^{2}}{y^{-\frac{2}{3}}}
y
−
3
2
y
+
y
2
\newline
Which of the following expressions is equivalent to the given expression assuming
y
y
y
is nonzero?
\newline
Choose
1
1
1
answer:
\newline
(A)
y
−
3
2
+
y
−
3
y^{-\frac{3}{2}}+y^{-3}
y
−
2
3
+
y
−
3
\newline
(B)
y
3
2
+
y
3
y^{\frac{3}{2}}+y^{3}
y
2
3
+
y
3
\newline
(C)
y
1
3
+
y
4
3
y^{\frac{1}{3}}+y^{\frac{4}{3}}
y
3
1
+
y
3
4
\newline
(D)
y
5
3
+
y
8
3
y^{\frac{5}{3}}+y^{\frac{8}{3}}
y
3
5
+
y
3
8
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h
(
t
)
=
−
10
t
+
6
h
(
□
)
=
−
44
\begin{array}{l}h(t)=-10 t+6 \\ h(\square)=-44\end{array}
h
(
t
)
=
−
10
t
+
6
h
(
□
)
=
−
44
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h
(
x
)
=
8
x
−
10
h
(
□
)
=
62
\begin{array}{l}h(x)=8 x-10 \\ h(\square)=62\end{array}
h
(
x
)
=
8
x
−
10
h
(
□
)
=
62
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−
27
x
+
54
y
=
9
x
2
−
19
3
x
−
6
y
=
−
5
\begin{aligned} -27 x+54 y & =9 x^{2}-19 \\ 3 x-6 y & =-5 \end{aligned}
−
27
x
+
54
y
3
x
−
6
y
=
9
x
2
−
19
=
−
5
\newline
If
(
x
1
,
y
1
)
\left(x_{1}, y_{1}\right)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
\left(x_{2}, y_{2}\right)
(
x
2
,
y
2
)
are distinct solutions to the system of equations shown, what is the sum of the
y
y
y
-values
y
1
y_{1}
y
1
and
y
2
y_{2}
y
2
?
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y
=
3
(
2
x
2
−
x
−
1
)
y
=
4
x
+
2
\begin{array}{l} y=3\left(2 x^{2}-x-1\right) \\ y=4 x+2 \end{array}
y
=
3
(
2
x
2
−
x
−
1
)
y
=
4
x
+
2
\newline
If
(
x
1
,
y
1
)
\left(x_{1}, y_{1}\right)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
\left(x_{2}, y_{2}\right)
(
x
2
,
y
2
)
are distinct solutions to the system of equations shown, what is the value of
y
1
+
y
2
y_{1}+y_{2}
y
1
+
y
2
?
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Viet solved a quadratic equation. His work is shown below.
\newline
In which step did Viet make an error?
\newline
2
(
x
−
3
)
2
+
4
=
102
2(x-3)^{2}+4=102
2
(
x
−
3
)
2
+
4
=
102
\newline
2
(
x
−
3
)
2
=
106
Step 1
(
x
−
3
)
2
=
53
Step 2
x
−
3
=
±
53
Step 3
x
=
±
53
+
3
Step 4
\begin{aligned} 2(x-3)^{2} & =106 & & \text { Step 1 } \\ (x-3)^{2} & =53 & & \text { Step 2 } \\ x-3 & = \pm \sqrt{53} & & \text { Step 3 } \\ x & = \pm \sqrt{53}+3 & & \text { Step 4 } \end{aligned}
2
(
x
−
3
)
2
(
x
−
3
)
2
x
−
3
x
=
106
=
53
=
±
53
=
±
53
+
3
Step 1
Step 2
Step 3
Step 4
\newline
Step
3
3
3
\newline
Choose
1
1
1
answer:
\newline
(A) Step
1
1
1
\newline
(B) Step
2
2
2
\newline
(C) Step
3
3
3
\newline
(D) Step
4
4
4
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g
(
x
)
=
2
x
+
9
g
(
□
)
=
15
\begin{array}{l}g(x)=2 x+9 \\ g(\square)=15\end{array}
g
(
x
)
=
2
x
+
9
g
(
□
)
=
15
Get tutor help
If
f
(
x
)
=
−
2
x
3
+
3
x
2
−
11
x
+
1
f(x)=-2 x^{3}+3 x^{2}-11 x+1
f
(
x
)
=
−
2
x
3
+
3
x
2
−
11
x
+
1
, what is the value of
f
(
0
)
f(0)
f
(
0
)
?
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y
=
10
+
16
x
−
x
2
y
=
3
x
+
50
\begin{array}{l} y=10+16 x-x^{2} \\ y=3 x+50 \end{array}
y
=
10
+
16
x
−
x
2
y
=
3
x
+
50
\newline
If
(
x
1
,
y
1
)
\left(x_{1}, y_{1}\right)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
\left(x_{2}, y_{2}\right)
(
x
2
,
y
2
)
are distinct solutions to the system of equations shown, what is the sum of the
y
1
y_{1}
y
1
and
y
2
y_{2}
y
2
?
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If
y
=
(
4
x
−
1
)
(
x
+
1
)
(
x
+
2
)
+
5
y=(4 x-1)(x+1)(x+2)+5
y
=
(
4
x
−
1
)
(
x
+
1
)
(
x
+
2
)
+
5
, what is the value of
y
y
y
when
x
=
−
1
?
x=-1 \text { ? }
x
=
−
1
?
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−
2
−
x
=
y
2
−
4
y
+
10
11
−
11
x
+
3
y
=
62
\begin{aligned} -2-x & =\frac{y^{2}-4 y+10}{11} \\ -11 x+3 y & =62 \end{aligned}
−
2
−
x
−
11
x
+
3
y
=
11
y
2
−
4
y
+
10
=
62
\newline
If
(
x
1
,
y
1
)
\left(x_{1}, y_{1}\right)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
\left(x_{2}, y_{2}\right)
(
x
2
,
y
2
)
are two distinct solutions to the system of equations shown, what is the product of the
x
x
x
values of the two solutions
x
1
⋅
x
2
x_{1} \cdot x_{2}
x
1
⋅
x
2
?
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y
−
3
=
3
x
y-3=3 x
y
−
3
=
3
x
\newline
y
+
11
=
50
−
10
x
+
x
2
y+11=50-10 x+x^{2}
y
+
11
=
50
−
10
x
+
x
2
\newline
If
(
x
1
,
y
1
)
\left(x_{1}, y_{1}\right)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
\left(x_{2}, y_{2}\right)
(
x
2
,
y
2
)
are distinct solutions to the system of equations shown, what is the product of the
y
1
y_{1}
y
1
and
y
2
y_{2}
y
2
?
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y
=
x
2
+
7
x
−
7
y
=
x
+
9
\begin{array}{l} y=x^{2}+7 x-7 \\ y=x+9 \end{array}
y
=
x
2
+
7
x
−
7
y
=
x
+
9
\newline
If
(
x
,
y
)
(x, y)
(
x
,
y
)
is a solution to the system of equations and
x
<
0
x<0
x
<
0
, what is the value of
y
y
y
?
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(
ℓ
5
n
2
ℓ
7
n
)
(
ℓ
3
n
9
ℓ
n
3
)
\frac{\left(\frac{\ell^{5} n^{2}}{\ell^{7} n}\right)}{\left(\frac{\ell^{3} n^{9}}{\ell n^{3}}\right)}
(
ℓ
n
3
ℓ
3
n
9
)
(
ℓ
7
n
ℓ
5
n
2
)
\newline
Which expression is equivalent to the given quotient for all
ℓ
<
−
2
\ell<-2
ℓ
<
−
2
and
n
>
4
n>4
n
>
4
?
\newline
Choose
1
1
1
answer:
\newline
(A)
0
0
0
\newline
(B)
1
1
1
\newline
(C)
ℓ
4
n
5
\ell^{4} n^{5}
ℓ
4
n
5
\newline
(D)
1
ℓ
4
n
5
\frac{1}{\ell^{4} n^{5}}
ℓ
4
n
5
1
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(
x
2
y
2
z
2
)
5
(
x
3
y
3
z
3
)
7
\frac{\left(x^{2} y^{2} z^{2}\right)^{5}}{\left(x^{3} y^{3} z^{3}\right)^{7}}
(
x
3
y
3
z
3
)
7
(
x
2
y
2
z
2
)
5
\newline
Which of the following is equivalent to the given expression?
\newline
Choose
1
1
1
answer:
\newline
(A)
(
x
−
2
y
−
2
z
−
2
)
5
(
x
3
y
3
z
3
)
7
\left(x^{-2} y^{-2} z^{-2}\right)^{5}\left(x^{3} y^{3} z^{3}\right)^{7}
(
x
−
2
y
−
2
z
−
2
)
5
(
x
3
y
3
z
3
)
7
\newline
(B)
(
x
5
y
5
z
5
)
2
(
x
−
3
y
−
3
z
−
3
)
7
\left(x^{5} y^{5} z^{5}\right)^{2}\left(x^{-3} y^{-3} z^{-3}\right)^{7}
(
x
5
y
5
z
5
)
2
(
x
−
3
y
−
3
z
−
3
)
7
\newline
(C)
(
x
5
y
5
z
5
)
8
(
x
7
y
7
z
7
)
27
\frac{\left(x^{5} y^{5} z^{5}\right)^{8}}{\left(x^{7} y^{7} z^{7}\right)^{27}}
(
x
7
y
7
z
7
)
27
(
x
5
y
5
z
5
)
8
\newline
(D)
(
x
5
y
5
z
5
)
6
(
x
7
y
7
z
7
)
9
\frac{\left(x^{5} y^{5} z^{5}\right)^{6}}{\left(x^{7} y^{7} z^{7}\right)^{9}}
(
x
7
y
7
z
7
)
9
(
x
5
y
5
z
5
)
6
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8
−
1
2
⋅
8
7
6
8^{-\frac{1}{2}} \cdot 8^{\frac{7}{6}}
8
−
2
1
⋅
8
6
7
\newline
What is the value of the given expression?
Get tutor help
(
π
a
4
12
r
3
)
(
18
π
a
3
r
2
5
)
\left(\frac{\pi a^{4}}{12 r^{3}}\right)\left(\frac{18 \pi a^{3} r^{2}}{5}\right)
(
12
r
3
π
a
4
)
(
5
18
π
a
3
r
2
)
\newline
Which expression is equivalent to the given product for all
r
>
6
r>6
r
>
6
?
\newline
Choose
1
1
1
answer:
\newline
(A)
3
π
2
a
7
10
r
\frac{3 \pi^{2} a^{7}}{10 r}
10
r
3
π
2
a
7
\newline
(B)
5
a
216
r
5
\frac{5 a}{216 r^{5}}
216
r
5
5
a
\newline
(C)
3
π
2
a
7
10
\frac{3 \pi^{2} a^{7}}{10}
10
3
π
2
a
7
\newline
(D)
216
r
5
5
a
\frac{216 r^{5}}{5 a}
5
a
216
r
5
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72
=
2
x
2
72=2 x^{2}
72
=
2
x
2
\newline
What are the solutions to the given equation?
\newline
Choose
1
1
1
answer:
\newline
(A)
x
=
6
x=6
x
=
6
only
\newline
(B)
x
=
−
6
x=-6
x
=
−
6
and
x
=
6
x=6
x
=
6
\newline
(C)
x
=
−
2
+
6
2
x=-2+6 \sqrt{2}
x
=
−
2
+
6
2
\newline
(D)
x
=
−
2
−
6
2
x=-2-6 \sqrt{2}
x
=
−
2
−
6
2
and
x
=
−
2
+
6
2
x=-2+6 \sqrt{2}
x
=
−
2
+
6
2
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A florist orders exactly
1
3
\frac{1}{3}
3
1
gallons of nutrient-rich water for each bushel of flowers he buys. The florist buys bushels of flowers at
$
1.20
\$ 1.20
$1.20
per bushel and gallons of nutrient-rich water at
$
0.60
\$ 0.60
$0.60
per gallon. Which of the following equations gives the total cost,
C
(
b
)
C(b)
C
(
b
)
, in dollars, for
b
b
b
bushels of flowers and the nutrientrich water ordered for them?
\newline
Choose
1
1
1
answer:
\newline
(A)
C
(
b
)
=
(
1.2
+
0.2
)
⋅
b
C(b)=(1.2+0.2) \cdot b
C
(
b
)
=
(
1.2
+
0.2
)
⋅
b
\newline
(B)
C
(
b
)
=
3
⋅
0.6
⋅
b
C(b)=3 \cdot 0.6 \cdot b
C
(
b
)
=
3
⋅
0.6
⋅
b
\newline
(C)
C
(
b
)
=
(
1.2
+
0.6
)
⋅
b
C(b)=(1.2+0.6) \cdot b
C
(
b
)
=
(
1.2
+
0.6
)
⋅
b
\newline
(D)
C
(
b
)
=
7
⋅
0.6
⋅
b
C(b)=7 \cdot 0.6 \cdot b
C
(
b
)
=
7
⋅
0.6
⋅
b
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h
(
t
)
=
{
17
t
,
t
=
17
−
34
t
,
t
=
19
23
−
t
,
t
≠
17
,
19
h
(
17
)
=
□
\begin{array}{l}h(t)=\left\{\begin{array}{cl}\sqrt{17 t} & , \quad t=17 \\ -\frac{34}{t} & , \quad t=19 \\ 23-t & , \quad t \neq 17,19\end{array}\right. \\ h(17)=\square\end{array}
h
(
t
)
=
⎩
⎨
⎧
17
t
−
t
34
23
−
t
,
t
=
17
,
t
=
19
,
t
=
17
,
19
h
(
17
)
=
□
Get tutor help
The functions
s
(
x
)
s(x)
s
(
x
)
and
t
(
x
)
t(x)
t
(
x
)
are differentiable.
\newline
The function
z
(
x
)
z(x)
z
(
x
)
is defined as:
z
(
x
)
=
s
(
x
)
t
(
x
)
z(x)= \frac{s(x)}{t(x)}
z
(
x
)
=
t
(
x
)
s
(
x
)
\newline
If
s
(
4
)
=
7
s(4)= 7
s
(
4
)
=
7
,
s
′
(
4
)
=
2
s'(4)= 2
s
′
(
4
)
=
2
,
t
(
4
)
=
9
t(4)= 9
t
(
4
)
=
9
, and
t
′
(
4
)
=
−
3
t'(4)= -3
t
′
(
4
)
=
−
3
, what is
z
′
(
4
)
z'(4)
z
′
(
4
)
?
\newline
Simplify any fractions.
\newline
z
′
(
4
)
=
z'(4)=
z
′
(
4
)
=
_____
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The functions
a
(
x
)
a(x)
a
(
x
)
and
b
(
x
)
b(x)
b
(
x
)
are differentiable.
\newline
The function
c
(
x
)
c(x)
c
(
x
)
is defined as:
c
(
x
)
=
a
(
x
)
b
(
x
)
c(x)= \frac{a(x)}{b(x)}
c
(
x
)
=
b
(
x
)
a
(
x
)
\newline
If
a
(
2
)
=
5
a(2)= 5
a
(
2
)
=
5
,
a
′
(
2
)
=
3
a'(2)= 3
a
′
(
2
)
=
3
,
b
(
2
)
=
8
b(2)= 8
b
(
2
)
=
8
, and
b
′
(
2
)
=
−
1
b'(2)= -1
b
′
(
2
)
=
−
1
, what is
c
′
(
2
)
c'(2)
c
′
(
2
)
?
\newline
Simplify any fractions.
\newline
c
′
(
2
)
=
c'(2)=
c
′
(
2
)
=
_____
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Both of these functions grow as
x
x
x
gets larger and larger. Which function eventually exceeds the other?
\newline
Choices:
\newline
(
A
)
f
(
x
)
=
2
x
+
9
(A) \ f(x) = 2x + 9
(
A
)
f
(
x
)
=
2
x
+
9
\newline
(
B
)
g
(
x
)
=
2
x
(B)\ g(x) = 2^x
(
B
)
g
(
x
)
=
2
x
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