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Math Problems
Algebra 1
Write a quadratic function from its x-intercepts and another point
A curve is defined by the parametric equations
x
(
t
)
=
6
t
3
+
6
t
+
8
x(t)=6 t^{3}+6 t+8
x
(
t
)
=
6
t
3
+
6
t
+
8
and
y
(
t
)
=
−
4
cos
(
−
2
t
)
y(t)=-4 \cos (-2 t)
y
(
t
)
=
−
4
cos
(
−
2
t
)
. Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Answer:
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A curve is defined by the parametric equations
x
(
t
)
=
−
6
t
3
−
t
−
4
x(t)=-6 t^{3}-t-4
x
(
t
)
=
−
6
t
3
−
t
−
4
and
y
(
t
)
=
7
t
3
−
5
t
2
y(t)=7 t^{3}-5 t^{2}
y
(
t
)
=
7
t
3
−
5
t
2
. Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Answer:
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A curve is defined by the parametric equations
x
(
t
)
=
10
cos
(
9
t
)
x(t)=10 \cos (9 t)
x
(
t
)
=
10
cos
(
9
t
)
and
y
(
t
)
=
4
e
10
t
y(t)=4 e^{10 t}
y
(
t
)
=
4
e
10
t
. Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Answer:
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A curve is defined by the parametric equations
x
(
t
)
=
4
t
3
x(t)=4 t^{3}
x
(
t
)
=
4
t
3
and
y
(
t
)
=
7
sin
(
−
8
t
)
y(t)=7 \sin (-8 t)
y
(
t
)
=
7
sin
(
−
8
t
)
. Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Answer:
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If the equation
y
=
2
(
1.5
)
x
y=2(1.5)^{x}
y
=
2
(
1.5
)
x
is graphed in the
x
y
xy
x
y
-plane, what are the coordinates of its y-intercept?
\newline
Choose
1
1
1
answer:
\newline
(A)
(
0
,
0
)
(0,0)
(
0
,
0
)
\newline
(B)
(
0
,
1.5
)
(0,1.5)
(
0
,
1.5
)
\newline
(C)
(
0
,
2
)
(0,2)
(
0
,
2
)
\newline
(D)
(
0
,
3
)
(0,3)
(
0
,
3
)
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(
3
g
−
4
)
(
2
g
−
8
)
=
a
g
2
+
b
g
+
c
(3 g-4)(2 g-8)=a g^{2}+b g+c
(
3
g
−
4
)
(
2
g
−
8
)
=
a
g
2
+
b
g
+
c
\newline
In the given equation,
a
,
b
a, b
a
,
b
, and
c
c
c
are constants. What is the value of
a
a
a
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
32
-32
−
32
\newline
(B)
−
16
-16
−
16
\newline
(C)
5
5
5
\newline
(D)
6
6
6
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What is the area of the region between the graphs of
f
(
x
)
=
4
x
,
g
(
x
)
=
4
f(x)=\frac{4}{x}, g(x)=4
f
(
x
)
=
x
4
,
g
(
x
)
=
4
, and
x
=
4
x=4
x
=
4
?
\newline
Choose
1
1
1
answer:
\newline
(A)
24
−
4
ln
(
3
)
24-4 \ln (3)
24
−
4
ln
(
3
)
\newline
(B)
24
−
4
ln
(
4
)
24-4 \ln (4)
24
−
4
ln
(
4
)
\newline
(C)
12
−
4
ln
(
3
)
12-4 \ln (3)
12
−
4
ln
(
3
)
\newline
(D)
12
−
4
ln
(
4
)
12-4 \ln (4)
12
−
4
ln
(
4
)
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Let
h
(
x
)
=
2
x
h(x)=2^{x}
h
(
x
)
=
2
x
.
\newline
Can we use the intermediate value theorem to say the equation
h
(
x
)
=
4
h(x)=4
h
(
x
)
=
4
has a solution where
3
≤
x
≤
5
3 \leq x \leq 5
3
≤
x
≤
5
?
\newline
Choose
1
1
1
answer:
\newline
(A) No, since the function is not continuous on that interval.
\newline
(B) No, since
4
4
4
is not between
h
(
3
)
h(3)
h
(
3
)
and
h
(
5
)
h(5)
h
(
5
)
.
\newline
(C) Yes, both conditions for using the intermediate value theorem have been met.
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The graph of a sinusoidal function intersects its midline at
(
0
,
2
)
(0,2)
(
0
,
2
)
and then has a minimum point at
(
3
,
−
6
)
(3,-6)
(
3
,
−
6
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
□
f(x)=\square
f
(
x
)
=
□
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The graph of a sinusoidal function has a minimum point at
(
0
,
−
10
)
(0,-10)
(
0
,
−
10
)
and then has a maximum point at
(
2
,
−
4
)
(2,-4)
(
2
,
−
4
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
□
f(x)=\square
f
(
x
)
=
□
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A
=
2
π
r
2
+
2
π
r
h
A=2 \pi r^{2}+2 \pi r h
A
=
2
π
r
2
+
2
π
r
h
\newline
The surface area,
A
A
A
, of a cylinder of radius,
r
r
r
, and height,
h
h
h
, can be found with the given equation. Which of the following correctly shows the cylinder's height in terms of its radius and surface area?
\newline
Choose
1
1
1
answer:
\newline
(A)
h
=
r
−
A
2
π
r
h=r-\frac{A}{2 \pi r}
h
=
r
−
2
π
r
A
\newline
(B)
h
=
A
2
π
r
−
r
h=\frac{A}{2 \pi r}-r
h
=
2
π
r
A
−
r
\newline
(C)
h
=
r
−
A
r
h=r-\frac{A}{r}
h
=
r
−
r
A
\newline
(D)
h
=
A
r
−
r
h=\frac{A}{r}-r
h
=
r
A
−
r
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A function
p
p
p
is defined as
p
(
x
)
=
(
x
−
a
)
(
x
−
15
)
(
x
−
20
)
p(x)=(x-a)(x-15)(x-20)
p
(
x
)
=
(
x
−
a
)
(
x
−
15
)
(
x
−
20
)
where
a
a
a
is a constant. Given that
p
(
7
)
=
15
p(7)=15
p
(
7
)
=
15
, what is the value of
a
a
a
?
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In the
x
y
x y
x
y
-plane, the graph of a quadratic function has
x
x
x
-intercepts at
(
−
3
,
0
)
(-3,0)
(
−
3
,
0
)
and
(
7
,
0
)
(7,0)
(
7
,
0
)
. If the parabola has an axis of symmetry of
x
=
h
x=h
x
=
h
, then what is the value of
h
h
h
?
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In the
x
y
x y
x
y
-plane, the graph of the equation
y
=
3
(
x
+
6
)
2
+
5
y=3(x+6)^{2}+5
y
=
3
(
x
+
6
)
2
+
5
has a
y
y
y
-intercept of
(
0
,
b
)
(0, b)
(
0
,
b
)
. What is the value of
b
b
b
?
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The equation
y
=
30
(
1
2
)
x
y=30\left(\frac{1}{2}\right)^{x}
y
=
30
(
2
1
)
x
is graphed in the
x
y
x y
x
y
-plane. Which of the following statements about the graph is true?
\newline
Choose
1
1
1
answer:
\newline
(A) As
x
x
x
increases,
y
y
y
increases at an increasing rate.
\newline
(B) As
x
x
x
increases,
y
y
y
increases at a decreasing rate.
\newline
(C) As
x
x
x
increases,
y
y
y
decreases at an increasing rate.
\newline
(D) As
x
x
x
increases,
y
y
y
decreases at a decreasing rate.
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The equation
5
x
+
10
y
=
30
5 x+10 y=30
5
x
+
10
y
=
30
is graphed in the
x
y
x y
x
y
-plane. Which of the statements below is true of its graph?
\newline
Choose
1
1
1
answer:
\newline
(A) The graph's
x
x
x
-intercept is
3
3
3
and its
y
y
y
-intercept is
6
6
6
.
\newline
(B) The graph has a slope of
1
2
\frac{1}{2}
2
1
.
\newline
(C) The graph has a slope of
−
2
1
-\frac{2}{1}
−
1
2
.
\newline
D The graph's
x
x
x
-intercept is
6
6
6
and its
y
y
y
-intercept is
3
3
3
.
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A line is graphed in the
x
y
x y
x
y
-plane shown. Which of the following is an equation of the line, if the line passes through the points
(
0
,
0
)
(0, 0)
(
0
,
0
)
and
(
2
,
−
3
)
(2, -3)
(
2
,
−
3
)
?
?
?
\newline
Choose
1
1
1
answer:
\newline
(A)
y
=
3
2
x
y=\frac{3}{2} x
y
=
2
3
x
\newline
(B)
y
=
2
3
x
y=\frac{2}{3} x
y
=
3
2
x
\newline
(C)
y
=
−
3
2
x
y=-\frac{3}{2} x
y
=
−
2
3
x
\newline
(D)
y
=
−
2
3
x
y=-\frac{2}{3} x
y
=
−
3
2
x
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The equation
2
x
−
y
=
8
2 x-y=8
2
x
−
y
=
8
is graphed in the
x
y
x y
x
y
-plane. Which of the following is a true statement about the graph?
\newline
Choose
1
1
1
answer:
\newline
(A) The graph's
x
x
x
-intercept is
4
4
4
and its
y
y
y
-intercept is
−
8
-8
−
8
.
\newline
B The graph's
x
x
x
-intercept is
4
4
4
and its
y
y
y
-intercept is
8
8
8
.
\newline
(C) The graph has a slope of
−
2
-2
−
2
.
\newline
D The graph has a slope of
1
2
\frac{1}{2}
2
1
.
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The equation
6
−
2
y
=
x
6-2 y=x
6
−
2
y
=
x
is graphed in the
x
y
x y
x
y
-plane. Which of the following is a true statement about the graph?
\newline
Choose
1
1
1
answer:
\newline
(A) The graph's
x
x
x
-intercept is
6
6
6
and its
y
y
y
-intercept is
−
2
-2
−
2
.
\newline
B The graph's
x
x
x
-intercept is
6
6
6
and its
y
y
y
-intercept is
2
2
2
.
\newline
(C) The graph has a slope of
−
2
-2
−
2
.
\newline
D) The graph has a slope of
−
1
2
-\frac{1}{2}
−
2
1
.
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The equation
6
y
−
3
x
=
5
6 y-3 x=5
6
y
−
3
x
=
5
is graphed in the
x
y
x y
x
y
-plane. Which of the following is a true statement about the graph?
\newline
Choose
1
1
1
answer:
\newline
(A) The graph's
x
x
x
-intercept is
5
6
\frac{5}{6}
6
5
and its
y
y
y
-intercept is
−
5
3
-\frac{5}{3}
−
3
5
.
\newline
(B) The graph's
x
x
x
-intercept is
−
5
3
-\frac{5}{3}
−
3
5
and its
y
y
y
-intercept is
5
6
\frac{5}{6}
6
5
.
\newline
(C) The graph has a slope of
2
2
2
.
\newline
D The graph has a slope of
−
1
2
-\frac{1}{2}
−
2
1
.
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The line represented by the equation
3.4
y
=
0
3.4 y=0
3.4
y
=
0
is graphed in the
x
y
x y
x
y
-plane. Which of the following statements correctly describes the graph of the line?
\newline
Choose
1
1
1
answer:
\newline
(A) The line has a negative
y
y
y
intercept.
\newline
(B) The line goes through the point
(
0
,
3.4
)
(0,3.4)
(
0
,
3.4
)
.
\newline
(C) The line has a positive slope.
\newline
(D) The line is perpendicular to the
y
y
y
-axis.
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The equation
y
=
3
2
(
x
−
8
)
y=\frac{3}{2}(x-8)
y
=
2
3
(
x
−
8
)
is graphed in the
x
y
x y
x
y
-plane. Which of the following equations will have a graph that is parallel to the graph of the given equation and have an
x
x
x
intercept on the negative
x
x
x
-axis?
\newline
Choose
1
1
1
answer:
\newline
(A)
y
=
3
2
(
x
+
8
)
y=\frac{3}{2}(x+8)
y
=
2
3
(
x
+
8
)
\newline
(B)
y
=
3
2
x
−
8
y=\frac{3}{2} x-8
y
=
2
3
x
−
8
\newline
(C)
y
=
−
2
3
(
x
+
8
)
y=-\frac{2}{3}(x+8)
y
=
−
3
2
(
x
+
8
)
\newline
(D)
y
=
−
2
3
x
−
8
y=-\frac{2}{3} x-8
y
=
−
3
2
x
−
8
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The equations
2
y
=
−
4
x
+
8
2 y=-4 x+8
2
y
=
−
4
x
+
8
and
2
y
=
−
4
x
+
16
2 y=-4 x+16
2
y
=
−
4
x
+
16
are graphed in the
x
y
x y
x
y
-plane. Which of the following must be true of the graphs of the two equations?
\newline
Choose
1
1
1
answer:
\newline
(A) The graphs of the two equations are parallel lines.
\newline
(B) The graphs of the two equations are perpendicular lines.
\newline
(C) The graphs have the same
y
y
y
-intercept.
\newline
(D) The
y
y
y
-intercepts of the two graphs are reflected in the
x
x
x
-axis.
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Which of the following is an equation of the line in the
x
y
x y
x
y
plane with
x
x
x
-intercept
−
2
7
-\frac{2}{7}
−
7
2
and
y
y
y
intercept
8
7
\frac{8}{7}
7
8
?
\newline
Choose
1
1
1
answer:
\newline
(A)
7
y
−
28
x
=
8
7 y-28 x=8
7
y
−
28
x
=
8
\newline
(B)
7
y
+
28
x
=
8
7 y+28 x=8
7
y
+
28
x
=
8
\newline
(C)
−
7
y
+
28
x
=
8
-7 y+28 x=8
−
7
y
+
28
x
=
8
\newline
(D)
−
7
y
−
28
x
=
8
-7 y-28 x=8
−
7
y
−
28
x
=
8
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Which of the following is an equation of the line in the
x
y
x y
x
y
plane with
x
x
x
-intercept
5
3
\frac{5}{3}
3
5
and
y
y
y
intercept
10
7
\frac{10}{7}
7
10
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
7
y
+
6
x
=
10
-7 y+6 x=10
−
7
y
+
6
x
=
10
\newline
(B)
7
y
+
6
x
=
−
10
7 y+6 x=-10
7
y
+
6
x
=
−
10
\newline
(C)
7
y
+
6
x
=
10
7 y+6 x=10
7
y
+
6
x
=
10
\newline
(D)
7
y
−
6
x
=
10
7 y-6 x=10
7
y
−
6
x
=
10
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(
x
+
3
)
2
+
(
y
−
4
)
2
=
9
(x+3)^{2}+(y-4)^{2}=9
(
x
+
3
)
2
+
(
y
−
4
)
2
=
9
\newline
A circle in the
x
y
x y
x
y
-plane has the equation shown. If the
x
x
x
coordinate of a point on the circle is
−
3
-3
−
3
, what is a possible corresponding
y
y
y
-coordinate?
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