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Math Problems
Precalculus
Solve trigonometric equations
In a regular hexagonal pyramid
S
A
B
C
D
E
F
SABCDEF
S
A
BC
D
EF
, the base side
A
B
=
4
AB = 4
A
B
=
4
, and the side edge
S
A
=
7
SA = 7
S
A
=
7
. Point
M
M
M
lies on edge
B
C
BC
BC
, with
B
M
=
1
BM = 1
BM
=
1
, point
K
K
K
lies on edge
S
C
SC
SC
, with
S
K
=
4
SK = 4
S
K
=
4
. a) Prove that the
M
K
D
MKD
M
KD
plane is perpendicular to the plane of the base of the pyramid. b) Find the volume of the pyramid
A
B
=
4
AB = 4
A
B
=
4
0
0
0
.
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Find all solutions with
−
π
2
≤
θ
≤
π
2
- \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}
−
2
π
≤
θ
≤
2
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
csc
(
θ
)
=
1
\csc(\theta)=1
csc
(
θ
)
=
1
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Find all solutions with
0
≤
θ
≤
π
0\leq\theta\leq\pi
0
≤
θ
≤
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
–
7
sec
(
θ
)
–
7
=
0
– 7\sec(\theta)–7=0
–7
sec
(
θ
)
–7
=
0
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Find all solutions with
0
≤
θ
≤
π
0\leq\theta\leq\pi
0
≤
θ
≤
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
7
sec
(
θ
)
−
7
=
0
- 7\sec(\theta)-7=0
−
7
sec
(
θ
)
−
7
=
0
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Find all solutions with
−
π
2
≤
θ
≤
π
2
- \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}
−
2
π
≤
θ
≤
2
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
9
csc
(
θ
)
−
18
=
0
9\csc(\theta)-18=0
9
csc
(
θ
)
−
18
=
0
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Find all solutions with
−
π
2
≤
θ
≤
π
2
- \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}
−
2
π
≤
θ
≤
2
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
14
csc
(
θ
)
+
14
=
0
14\csc(\theta)+14=0
14
csc
(
θ
)
+
14
=
0
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Find all solutions with
−
π
2
<
θ
<
π
2
- \frac{\pi}{2} < \theta < \frac{\pi}{2}
−
2
π
<
θ
<
2
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
7
tan
(
θ
)
+
7
=
0
- 7\tan(\theta) + 7 = 0
−
7
tan
(
θ
)
+
7
=
0
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Find all solutions with
−
90
°
<
θ
<
90
°
- 90°<\theta<90°
−
90°
<
θ
<
90°
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
8
tan
(
θ
)
–
8
=
0
- 8\tan(\theta)–8=0
−
8
tan
(
θ
)
–8
=
0
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Find all solutions with
−
90
°
<
θ
<
90
°
- 90°<\theta<90°
−
90°
<
θ
<
90°
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
2
tan
(
θ
)
=
0
2\tan(\theta)=0
2
tan
(
θ
)
=
0
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Find all solutions with
−
9
0
∘
≤
θ
≤
9
0
∘
-90^\circ \leq \theta \leq 90^\circ
−
9
0
∘
≤
θ
≤
9
0
∘
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
2
sin
(
θ
)
+
9
2
=
−
13
sin
(
θ
)
+
10
- 2\sin(\theta)+ \frac{9}{2} = - 13\sin(\theta)+10
−
2
sin
(
θ
)
+
2
9
=
−
13
sin
(
θ
)
+
10
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Find all solutions with
−
9
0
∘
≤
θ
≤
9
0
∘
-90^\circ\leq\theta\leq90^\circ
−
9
0
∘
≤
θ
≤
9
0
∘
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
2
sin
(
θ
)
+
9
2
=
−
13
sin
(
θ
)
+
10
-2\sin(\theta)+\frac{9}{2}=-13\sin(\theta)+10
−
2
sin
(
θ
)
+
2
9
=
−
13
sin
(
θ
)
+
10
Get tutor help
Find all solutions with
−
90
°
≤
θ
≤
90
°
- 90°\leq\theta\leq90°
−
90°
≤
θ
≤
90°
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
sin
(
θ
)
=
−
3
sin
(
θ
)
- \sin(\theta)= - 3\sin(\theta)
−
sin
(
θ
)
=
−
3
sin
(
θ
)
Get tutor help
Find all solutions with
0
≤
θ
≤
π
0\leq\theta\leq\pi
0
≤
θ
≤
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
14
cos
(
θ
)
+
7
=
0
-14\cos(\theta)+7=0
−
14
cos
(
θ
)
+
7
=
0
Get tutor help
Find all solutions with
0
≤
θ
≤
π
0\leq\theta\leq\pi
0
≤
θ
≤
π
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
14
cos
(
θ
)
+
7
=
0
-14\cos(\theta)+7=0
−
14
cos
(
θ
)
+
7
=
0
Get tutor help
Find all solutions with
−
9
0
∘
≤
θ
≤
9
0
∘
-90^\circ\leq\theta\leq90^\circ
−
9
0
∘
≤
θ
≤
9
0
∘
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
−
7
sin
(
θ
)
−
7
2
=
0
-7\sin(\theta)-\frac{7}{2}=0
−
7
sin
(
θ
)
−
2
7
=
0
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Find all solutions with
0
°
≤
θ
≤
180
°
0°\leq\theta\leq180°
0°
≤
θ
≤
180°
. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.
–
15
cos
(
θ
)
–
15
2
=
0
– 15\cos(\theta)– \frac{15}{2} =0
–15
cos
(
θ
)
–
2
15
=
0
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Find the area,
A
A
A
, of the region between the curve
y
=
2
x
1
+
x
2
y=\frac{2x}{1+x^{2}}
y
=
1
+
x
2
2
x
and the interval
−
2
≤
x
≤
2
-2 \leq x \leq 2
−
2
≤
x
≤
2
of the
x
x
x
-axis. The area is
A
=
□
A=\square
A
=
□
. (Type an exact answer.)
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36
36
36
. Which statements are true about the line shown on the graph?
\newline
Select two answer choices.
\newline
(a) The point
(
−
4
,
0
)
(-4,0)
(
−
4
,
0
)
is on the line.
\newline
(B) The point
(
0
,
−
4
)
(0,-4)
(
0
,
−
4
)
is on the line.
\newline
(c) The point
(
36
,
28
)
(36,28)
(
36
,
28
)
is on the line.
\newline
() The graph represents the equation
2
x
−
3
y
=
−
12
2 x-3 y=-12
2
x
−
3
y
=
−
12
.
\newline
() The graph represents the equation
2
x
+
3
y
=
−
12
2 x+3 y=-12
2
x
+
3
y
=
−
12
.
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There are somureftobunters and some whiteeounters in a bag. At the start,
7
7
7
of the counters are red and the rest of the counters are whitc. Woody takes two counters from the bag. First he takes at random a counter from the bag. He does not put the counter back in the bag. Woody then takes at random another counter from the bag.
\newline
(a) Let the number of white counters in the bag be
x
x
x
.
\newline
Draw a tree diagram that represents the above scenario, showing all relevant information. counter Woody takes is red is
21
80
\frac{21}{80}
80
21
.
\newline
Find the number of white counters in the bag at the start.
\newline
2
2
2
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If
m
m
m
and
b
b
b
are constants, which equation represents a linear function where
x
x
x
is the input and
y
y
y
is the output?
\newline
Choices:
\newline
(A)
y
=
m
x
+
b
y = \frac{m}{x} + b
y
=
x
m
+
b
\newline
(B)
y
=
m
x
+
b
y = mx + b
y
=
m
x
+
b
\newline
(C)
y
=
m
x
2
+
b
y = mx^2 + b
y
=
m
x
2
+
b
\newline
Which statement about the function is true for any values of
m
m
m
and
b
b
b
?
\newline
Choices:
\newline
(A)The graph has a constant rate of change.
\newline
(B)The graph has a positive
y
y
y
-intercept.
\newline
(C)The graph has a positive slope.
\newline
(D)The graph is not a straight line.
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Which formulas represent linear relationships? Select all that apply.
\newline
Multi-select Choices:
\newline
(A)the area of a semicircle,
A
=
π
r
2
2
A = \frac{\pi r^2}{2}
A
=
2
π
r
2
\newline
(B)the volume of a square pyramid with a height of
1
1
1
,
V
=
1
3
s
2
V = \frac{1}{3}s^2
V
=
3
1
s
2
\newline
(C)the perimeter of a regular pentagon,
P
=
5
s
P = 5s
P
=
5
s
\newline
(D)the volume of a hemisphere,
V
=
2
3
π
r
3
V = \frac{2}{3} \pi r^3
V
=
3
2
π
r
3
Get tutor help
Which formulas represent linear relationships? Select all that apply.
\newline
Multi-select Choices:
\newline
(A)the area of a triangle whose height and base are the same,
A
=
1
2
h
2
A = \frac{1}{2}h^2
A
=
2
1
h
2
\newline
(B)the perimeter of a regular octagon,
P
=
8
s
P = 8s
P
=
8
s
\newline
(C)the surface area of a cube,
S
A
=
6
a
2
SA = 6a^2
S
A
=
6
a
2
\newline
(D)the perimeter of a semicircle,
P
=
π
r
+
2
r
P = \pi r + 2r
P
=
π
r
+
2
r
Get tutor help
Haley solved the equation
25
+
3
n
−
12
=
13
+
7
n
−
4
n
25 + 3n - 12 = 13 + 7n - 4n
25
+
3
n
−
12
=
13
+
7
n
−
4
n
. Here are her last two steps:
\newline
13
+
3
n
=
13
+
3
n
13 + 3n = 13 + 3n
13
+
3
n
=
13
+
3
n
\newline
13
=
13
13 = 13
13
=
13
\newline
Which statement is true about the equation?
\newline
(A)The solution is
n
=
13
n = 13
n
=
13
.
\newline
(B)There is no solution because
13
=
13
13 = 13
13
=
13
is a true equation.
\newline
(C)There are infinitely many solutions because
13
=
13
13 = 13
13
=
13
is a true equation.
\newline
(D)The solution is
(
13
,
13
)
(13, 13)
(
13
,
13
)
.
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Jill solved the equation
8
−
p
+
7
=
7
p
−
5
−
3
p
8 - p + 7 = 7p - 5 - 3p
8
−
p
+
7
=
7
p
−
5
−
3
p
. Here are her last two steps:
\newline
20
=
5
p
20 = 5p
20
=
5
p
\newline
4
=
p
4 = p
4
=
p
\newline
Which statement is true about the equation?
\newline
Choices:
\newline
(A)The solution is
p
=
4
p = 4
p
=
4
.
\newline
(B)There is no solution.
\newline
(C)There are infinitely many solutions.
\newline
(D)The solution is
(
20
,
4
)
(20, 4)
(
20
,
4
)
.
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Ann solved the equation
9
+
2
n
−
3
=
1
2
(
4
n
+
6
)
9 + 2n - 3 = \frac{1}{2}(4n + 6)
9
+
2
n
−
3
=
2
1
(
4
n
+
6
)
. Here are her last two steps:
\newline
6
+
2
n
=
2
n
+
3
6 + 2n = 2n + 3
6
+
2
n
=
2
n
+
3
\newline
6
=
3
6 = 3
6
=
3
\newline
Which statement is true about the equation?
\newline
(A)The solution is
n
=
6
3
n = \frac{6}{3}
n
=
3
6
.
\newline
(B)There is no solution because
6
=
3
6 = 3
6
=
3
is a false equation.
\newline
(C)There are infinitely many solutions because
6
=
3
6 = 3
6
=
3
is a true equation.
\newline
(D)The solution is
(
6
,
3
)
(6, 3)
(
6
,
3
)
.
Get tutor help
Which of these equations has no solutions?
\newline
Choices:
\newline
(A)
1
3
(
9
x
+
3
)
=
4
x
+
5
+
x
\frac{1}{3}(9x + 3) = 4x + 5 + x
3
1
(
9
x
+
3
)
=
4
x
+
5
+
x
\newline
(B)
−
3
(
x
+
5
)
=
−
x
−
2
x
−
15
-3(x + 5) = -x - 2x - 15
−
3
(
x
+
5
)
=
−
x
−
2
x
−
15
\newline
(C)
−
2
x
+
5
+
8
x
=
3
(
2
x
−
1
)
-2x + 5 + 8x = 3(2x - 1)
−
2
x
+
5
+
8
x
=
3
(
2
x
−
1
)
\newline
Which statement explains a way you can tell the equation has no solutions?
\newline
Choices:
\newline
(A) It is equivalent to an equation that has the same variable terms but different constant terms on either side of the equal sign.
\newline
(B) It is equivalent to an equation that has the same variable terms and the same constant terms on either side of the equal sign.
\newline
(C) It is equivalent to an equation that has different variable terms on either side of the equation.
Get tutor help
Which of these equations has no solutions?
\newline
Choices:
\newline
(A)
1
4
(
12
x
+
16
)
=
5
x
+
4
−
2
x
\frac{1}{4}(12x + 16) = 5x + 4 - 2x
4
1
(
12
x
+
16
)
=
5
x
+
4
−
2
x
\newline
(B)
3
x
+
1
−
5
x
=
4
x
−
11
3x + 1 - 5x = 4x - 11
3
x
+
1
−
5
x
=
4
x
−
11
\newline
(C)
4
(
2
x
−
1
)
=
8
x
−
1
4(2x - 1) = 8x - 1
4
(
2
x
−
1
)
=
8
x
−
1
\newline
Which statement explains a way you can tell the equation has no solutions?
\newline
Choices:
\newline
(A) It is equivalent to an equation that has the same variable terms but different constant terms on either side of the equal sign.
\newline
(B) It is equivalent to an equation that has the same variable terms and the same constant terms on either side of the equal sign.
\newline
(C) It is equivalent to an equation that has different variable terms on either side of the equation.
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Devin solved the equation
12
+
5
x
−
7
=
13
x
−
4
−
8
x
12 + 5x - 7 = 13x - 4 - 8x
12
+
5
x
−
7
=
13
x
−
4
−
8
x
. Here are his last two steps:
\newline
5
+
5
x
=
5
x
−
4
5 + 5x = 5x - 4
5
+
5
x
=
5
x
−
4
\newline
5
=
−
4
5 = -4
5
=
−
4
\newline
Which statement is true about the equation?
\newline
(A)The solution is
x
=
−
4
x = -4
x
=
−
4
.
\newline
(B)There is no solution because
5
=
−
4
5 = -4
5
=
−
4
is a false equation.
\newline
(C)There are infinitely many solutions because
5
=
−
4
5 = -4
5
=
−
4
is a false equation.
\newline
(D)The solution is
(
5
,
−
4
)
(5, -4)
(
5
,
−
4
)
.
Get tutor help
Which of these equations has infinitely many solutions?
\newline
Choices:
\newline
(A)
2
(
x
+
3
)
=
4
x
+
6
2(x + 3) = 4x + 6
2
(
x
+
3
)
=
4
x
+
6
\newline
(B)
3
x
−
5
+
2
x
=
5
(
x
−
1
5
)
3x - 5 + 2x = 5(x - \frac{1}{5})
3
x
−
5
+
2
x
=
5
(
x
−
5
1
)
\newline
(C)
−
2
x
+
3
+
4
x
=
1
2
(
4
x
+
6
)
-2x + 3 + 4x = \frac{1}{2}(4x + 6)
−
2
x
+
3
+
4
x
=
2
1
(
4
x
+
6
)
\newline
Which statement explains a way you can tell the equation has infinitely many solutions?
\newline
Choices:
\newline
(A) It is equivalent to an equation that has the same variable terms but different constant terms on each side of the equal sign.
\newline
(B) It is equivalent to an equation that has the same variable terms and the same constant terms on each side of the equal sign.
\newline
(C) It is equivalent to an equation that has different variable terms on each side of the equation.
Get tutor help
Solve this system of equations by graphing. Graph the equations, then find the solution.
\newline
y
=
−
x
−
5
y = -x - 5
y
=
−
x
−
5
\newline
y
=
−
1
2
x
−
3
y = -\frac{1}{2}x - 3
y
=
−
2
1
x
−
3
\newline
Click to select points on the graph.
\newline
\newline
The solution is (
_
,
_
\_, \_
_
,
_
).
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How many solutions does the system of equations below have?
\newline
y
=
8
7
x
+
10
y = \frac{8}{7}x + 10
y
=
7
8
x
+
10
\newline
y
=
8
7
x
+
10
y = \frac{8}{7}x + 10
y
=
7
8
x
+
10
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
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How many solutions does the system of equations below have?
\newline
y
=
6
x
+
10
y = 6x + 10
y
=
6
x
+
10
\newline
y
=
6
x
+
10
y = 6x + 10
y
=
6
x
+
10
\newline
Choices:
\newline
(A) no solution
\newline
(B) one solution
\newline
(C) infinitely many solutions
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How many solutions does the system of equations below have?
\newline
y
=
−
x
−
2
y = -x - 2
y
=
−
x
−
2
\newline
y
=
−
x
−
2
y = -x - 2
y
=
−
x
−
2
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
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How many solutions does the system of equations below have?
\newline
y
=
x
+
6
y = x + 6
y
=
x
+
6
\newline
y
=
x
+
9
4
y = x + \frac{9}{4}
y
=
x
+
4
9
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
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How many solutions does the system of equations below have?
\newline
y
=
9
x
−
2
9
y = 9x - \frac{2}{9}
y
=
9
x
−
9
2
\newline
y
=
9
x
−
2
9
y = 9x - \frac{2}{9}
y
=
9
x
−
9
2
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
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How many solutions does the system of equations below have?
\newline
y
=
−
5
x
+
4
y = -5x + 4
y
=
−
5
x
+
4
\newline
y
=
−
5
x
+
5
2
y = -5x + \frac{5}{2}
y
=
−
5
x
+
2
5
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
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Determine the values of
a
a
a
and
b
b
b
for which the system
{
2
x
+
y
+
a
z
=
−
1
3
x
−
2
y
+
z
=
b
5
x
−
8
y
+
9
z
=
3
\left\{\begin{array}{l}2 x+y+a z=-1 \\ 3 x-2 y+z=b \\ 5 x-8 y+9 z=3\end{array}\right.
⎩
⎨
⎧
2
x
+
y
+
a
z
=
−
1
3
x
−
2
y
+
z
=
b
5
x
−
8
y
+
9
z
=
3
\newline
(a) has no solution, (b) has only one solution, (c) has infinitely many solutions.
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Graph the curve whose parametric equations are given, and show its orientation. Find the rectangular equation of the curve.
\newline
x
=
t
+
14
,
y
=
t
;
t
≥
0
x=t+14, y=\sqrt{t} ; t \geq 0
x
=
t
+
14
,
y
=
t
;
t
≥
0
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Abby and Robert are each trying to solve the equation
x
2
−
10
x
+
26
=
0
x^{2}-10x+26=0
x
2
−
10
x
+
26
=
0
. They know that the solutions to
x
2
=
−
1
x^{2}=-1
x
2
=
−
1
are
i
i
i
and
−
i
-i
−
i
, but they are not sure how to use this information to solve for
x
x
x
in their equation. Solve the equation and explain to them where the
i
i
i
is needed.
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LT
15
15
15
Abby and Robert are each trying to solve the equation
x
2
−
10
x
+
26
=
0
x^{2}-\sqrt{10 x+26}=0
x
2
−
10
x
+
26
=
0
. They know that the solutions to
x
2
=
−
1
x^{2}=-1
x
2
=
−
1
are
i
i
i
and
−
i
-i
−
i
, but they are not sure how to use this information to solve for
x
x
x
in their equation. Solve the equation and explain to them where the
i
i
i
is needed.
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Consider
n
n
n
pairs of numbers. Suppose
x
ˉ
=
4
,
s
x
=
3
,
y
ˉ
=
2
\bar{x}=4, s_{x}=3, \bar{y}=2
x
ˉ
=
4
,
s
x
=
3
,
y
ˉ
=
2
, and
s
y
=
5
s_{y}=5
s
y
=
5
.
\newline
Of the following which could be the least squares line?
\newline
(A)
y
=
2
+
x
y=2+x
y
=
2
+
x
\newline
(B)
y
=
−
6
+
2
x
y=-6+2 x
y
=
−
6
+
2
x
\newline
(C)
y
=
−
10
+
3
x
y=-10+3 x
y
=
−
10
+
3
x
\newline
(D)
y
=
5
/
3
−
x
y=5 / 3-x
y
=
5/3
−
x
\newline
(E)
y
=
6
−
x
y=6-x
y
=
6
−
x
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What is the value of the discriminant of
x
2
+
5
x
=
−
6
x^{2}+5 x=-6
x
2
+
5
x
=
−
6
? How many real solutions does the equation have?
\newline
(A) The discriminant is
49
49
49
and the equation has two real solutions.
\newline
(B) The discriminant is
1
1
1
and the equation has two real solutions.
\newline
(C) The discriminant is
1
1
1
and the equation has one real solution.
\newline
(D) The discriminant is
0
0
0
and the equation has one real solution.
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NOTES \& LESSON:
\newline
Grammer
\newline
Celebrating Texas -...
\newline
Question
11
11
11
*
\newline
The graph of a linear function is shown on the grid.
\newline
Which equation is best represented by this graph?
\newline
A
y
+
2
=
7
5
(
x
+
7
)
y+2=\frac{7}{5}(x+7)
y
+
2
=
5
7
(
x
+
7
)
\newline
B
y
−
2
=
7
5
(
x
−
7
)
y-2=\frac{7}{5}(x-7)
y
−
2
=
5
7
(
x
−
7
)
\newline
c
y
+
2
=
5
7
(
x
+
7
)
y+2=\frac{5}{7}(x+7)
y
+
2
=
7
5
(
x
+
7
)
\newline
D
y
−
2
=
5
7
(
x
−
7
)
y-2=\frac{5}{7}(x-7)
y
−
2
=
7
5
(
x
−
7
)
\newline
A
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Find the intercepts of the equation.
\newline
y
=
x
−
6
y=x-6
y
=
x
−
6
\newline
What are the intercepts?
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7
7
7
.SP.
5
5
5
- Escape Room
\newline
consonta
\newline
/
1
1
1
FAlpQLScHP
2
2
2
LX_-k
4
4
4
Kz
6
6
6
_qVWICenKTsu
3
3
3
amrQKpV
5
5
5
BdWaPIPfiMin
\newline
Puzzle Three
\newline
Consider the spinner shown to answer the following questions. To break the cod puzzle three you will need to enter the numeric answer to each question, separa by a dash. Make sure to not add any labels, spaces, or commas. Be sure to conv all your answers to decimals.
\newline
For example, .
32
−
.
085
−
.
7
−
.
63
−
.
2
32-.085-.7-.63-.2
32
−
.085
−
.7
−
.63
−
.2
\newline
1
1
1
)
\qquad
P(e)
\newline
2
2
2
)
\qquad
P
(
D
)
\mathrm{P}(\mathrm{D})
P
(
D
)
\newline
3
3
3
)
\qquad
P(vowel)
\newline
4
4
4
)
\qquad
P(C)
\newline
5
5
5
)
\qquad
P(consonant)
\newline
6
6
6
)
\qquad
P(Capital letters)
\newline
ㄱ)
\qquad
Pllowercase letters)
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Find the intercepts of the equation.
\newline
x
=
−
6
y
x=-6 y
x
=
−
6
y
Get tutor help
Draw a graph of the line that contains the given point and has the given slope.
\newline
(
0
,
6
)
,
m
=
−
1
(0,6), m=-1
(
0
,
6
)
,
m
=
−
1
Get tutor help
Algebra
1
1
1
Exit Slip
4
4
4
/
24
24
24
/
24
24
24
Directions: Graph the quadratic equation using your knowledge of how the
a
a
a
,
k
k
k
, and
h
h
h
values will change the graph from the parent function.
y
=
−
2
⋅
(
x
−
4
)
2
+
1
y = -2 \cdot (x - 4)^2 + 1
y
=
−
2
⋅
(
x
−
4
)
2
+
1
The function represented by this graph has a (maximum/minimum) of The axis of symmetry is
4
−
4-
4
−
The vertex of this graph is located at
3
+
3+
3
+
2
+
2+
2
+
1
+
1+
1
+
1
1
1
2
2
2
k
k
k
0
0
0
k
k
k
1
1
1
k
k
k
2
2
2
k
k
k
3
3
3
y
=
−
2
⋅
(
x
−
4
)
2
+
1
y = -2 \cdot (x - 4)^2 + 1
y
=
−
2
⋅
(
x
−
4
)
2
+
1
k
k
k
5
5
5
4
−
4-
4
−
k
k
k
7
7
7
k
k
k
8
8
8
k
k
k
9
9
9
h
h
h
0
0
0
h
h
h
1
1
1
h
h
h
2
2
2
h
h
h
3
3
3
h
h
h
4
4
4
\newline
Learning Intention: Students will learn how to use functions of the form
h
h
h
5
5
5
to represent some quadratic relationships as a graph and a table.
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For each pair of functions
f
f
f
and
g
g
g
below, find
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
and
g
(
f
(
x
)
)
g(f(x))
g
(
f
(
x
))
. Then, determine whether
f
f
f
and
g
g
g
are inverses of each other.
\newline
Simplify your answers as much as possible.
\newline
(Assume that your expressions are defined for all
x
x
x
in the domain of the composition You do not have to indicate the domain.)
\newline
\begin{tabular}{|l|c|}
\newline
\hline (a)
f
(
x
)
=
3
x
,
x
≠
0
f(x)=\frac{3}{x}, x \neq 0
f
(
x
)
=
x
3
,
x
=
0
& (b)
f
(
x
)
=
−
x
+
4
f(x)=-x+4
f
(
x
)
=
−
x
+
4
\\
\newline
g
(
x
)
=
3
x
,
x
≠
0
g(x)=\frac{3}{x}, x \neq 0
g
(
x
)
=
x
3
,
x
=
0
&
g
g
g
0
0
0
\\
\newline
g
g
g
1
1
1
&
g
g
g
1
1
1
\\
\newline
g
g
g
3
3
3
&
g
g
g
3
3
3
\\
\newline
g
g
g
5
5
5
and
g
g
g
are inverses of each other &
g
g
g
7
7
7
and
g
g
g
are inverses of each other \\
\newline
f
f
f
and
g
g
g
are not inverses of each other &
g
g
g
7
7
7
and
g
g
g
are not inverses of each other \\
\newline
\hline
\newline
\end{tabular}
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9
9
9
. Find the center of mass of the region bounded by the following functions.
\newline
y
=
ln
x
,
x
=
1
,
x
=
2
,
y
=
0
y=\ln x, \quad x=1, \quad x=2, \quad y=0
y
=
ln
x
,
x
=
1
,
x
=
2
,
y
=
0
Get tutor help
1
2
3
...
4
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