I need help solving this I’m having trouble with it is from my trigonometry prep bookIf you can **** use Desmos to graph the function that is provided in the picture
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So we have to graph the function:
[tex]f(x)=\cot (x+\frac{\pi}{6})[/tex]First is important to note that the cotangent can be defined by the quotient between the cosine and the sine:
[tex]\cot (x+\frac{\pi}{6})=\frac{\cos(x+\frac{\pi}{6})}{\sin(x+\frac{\pi}{6})}[/tex]By looking at this new expression we can infer a few things about the graph. First of all, we have a sine in the denominator which means that the denominator can be equal to 0. Let's assume that the denominator is 0 at x=a. Then the graph has a vertical asymptote at x=a. What's more, the sine is a periodic funtion that is equal to zero for an infinite amount of x values so the graph of the cotangent has infinite vertical asymptotes. The good part is that we just need to graph one full period and in the case of the cotangent one full period is completed between two consecutive vertical asymptote. So basically we have to find two consecutive vertical asymptote and graph the function between them.
So let's begin by finding two x values that makes the denominator equal to 0. The sine is equal to 0 when its argument is equal to 0 and the next value at which the sine is equal to zero is pi so:
[tex]\sin 0=0=\sin \pi[/tex]Then we can construct two equations:
[tex]\begin{gathered} \sin (x+\frac{\pi}{6})=0=\sin 0 \\ \sin (x+\frac{\pi}{6})=0=\sin \pi \end{gathered}[/tex]The equations are:
[tex]\begin{gathered} x+\frac{\pi}{6}=0 \\ x+\frac{\pi}{6}=\pi \end{gathered}[/tex]We can substract π/6 from both sides of both equations:
[tex]\begin{gathered} x+\frac{\pi}{6}-\frac{\pi}{6}=0-\frac{\pi}{6} \\ x=-\frac{\pi}{6} \\ x+\frac{\pi}{6}-\frac{\pi}{6}=\pi-\frac{\pi}{6} \\ x=\frac{5\pi}{6} \end{gathered}[/tex]So we have a vertical asymptote at x=-π/6 and another one at x=5π/6. This means that we just need to graph f(x) between these two vertical lines. It is also important to note that f(x) reaches positive or negative values when the value of x approaches to -π/6 or 5π/6.
Now that we have the asymptotes let's find the x-intercept i.e. the point where f(x) meets with the x-axis. This happens when f(x)=0 which happens when the numerator is equal to 0. Then we get:
[tex]\cos (x+\frac{\pi}{6})=0[/tex]The cosine is equal to zero at π/2 so we have:
[tex]\begin{gathered} \cos (x+\frac{\pi}{6})=0=\cos \frac{\pi}{2} \\ x+\frac{\pi}{6}=\frac{\pi}{2} \end{gathered}[/tex]We can substract π/6 from both sides:
[tex]\begin{gathered} x+\frac{\pi}{6}-\frac{\pi}{6}=\frac{\pi}{2}-\frac{\pi}{6} \\ x=\frac{\pi}{3} \end{gathered}[/tex]So the x-intercept is located at x=π/3. So for now we have the x-intercept and two vertical asymptotes so at the moment we have the following:
The black dot is the x-intercept at (π/3,0) and the dashed lines are the asymptotes. Our function passes through the black dot and is limited by the asymptotes.
We still need to find if it reaches positive or negative infinite values when approaching to the asymptotes. As we saw the function is equal to zero at x=π/3. This means that between the first asymptote and x=π/3 the function is either entirely positive or entirely negative. The same happens with the interval between x=π/3 and the second asymptote. So we have two intervals where the function mantains its sign: (-π/6,π/3) and (π/3,5π/6). Let's evaluate f(x) in one value of each interval and see if it's positive or negative there. For example, x=0 is inside the first interval and x=2 is inside the second interval:
[tex]\begin{gathered} f(0)=1.73205>0 \\ f(2)=-1.4067<0 \end{gathered}[/tex]So f(x) is positive at (-π/6,π/3) which means that as x approaches to -π/6 from the right it reaches positive infinite values. We also have that f(x) is negative at (π/3,5π/6) so as x approaches 5π/6 from the left the function reaches negative infinite values.
Using this information and the fact that the graph must pass throug the x-intercept we can graph the function. It should look like this:
And that's the graph of f(x).
Related Questions
Evaluate the indicated function for f(x)=x^2-1 & g(x)=x-2 algebraically .
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Given:
[tex]f(x)=x^2-1\text{ ; g(x)=x-2 }[/tex][tex](\frac{f}{g})(t+2)=\frac{f(t+2)}{g(t+2)}[/tex][tex](\frac{f}{g})(t+2)=\frac{(t+2)^2-1}{(t+2)^{}-2}[/tex][tex](\frac{f}{g})(t+2)=\frac{t^2+4t+4-1}{t+2-2}[/tex][tex](\frac{f}{g})(t+2)=\frac{t^2+4t+3}{t}[/tex][tex](\frac{f}{g})(t+2)=\frac{(t+1)(t+3)}{t}[/tex]HELP I JUST WANT TO FINISH GET MY CREDIT AND GRADUATE 70 POINTS HELP ASAP What are the values of x and y in the matrix subtraction below? [[- 20, 16], [4, 0], [9, - 6]] - [[- 5, 1], [- 6, 7], [0, 11]] = [[x, y], [10, - 7], [9, - 17]]
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SOLUTION
We want to solve the question below
We can see that a matrix was subtracted from another. In Addition or subtraction of matrix, we just add or subtract the element. So
We have in the first row
[tex]\begin{gathered} -20-(-5)=x \\ -20+5=x_ \\ x=-15 \end{gathered}[/tex]And
[tex]\begin{gathered} 16-1=y \\ 15=y \\ y=15 \end{gathered}[/tex]Hence x = -15 and y = 15, the first option is correct
The oldest child in a family of four children is three times as old as the youngest. The two middle children are 19 and 23 years old. If the average age of the children is 28.5, how old is the youngest child?
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Answer:
18 years old
Solution:
Let x represent the age of the youngest child.
So the age of the oldest = 3x
If the ages of the two middle children are 19 and 23, and the average age of the four children is 28.5, let's go ahead and find x;
[tex]\begin{gathered} \frac{(x+19+23+3x)}{4}=28.5 \\ 4x+42=114 \end{gathered}[/tex]Let's go ahead and subtract 42 from both sides;
[tex]4x=72[/tex]Dividing both sides by 4, we'll have;
[tex]x=\frac{72}{4}=18[/tex]Therefore, the youngest is 18 years old.
what is 2 3/24 simplified
Answers
2 3/24
Multiply the denominator by the whole number and add the numerator to obtain the new numerator. the denominator stays the same.
(2x24)+3 /24 = 48+3 /24 = 51/24
simplify by 3
17/8
You are trying to help a friend calculate their utilization rate for their study time. They can complete a maximum of 60 HW problems per hour. In the last hour, they were a little distracted but managed to complete 20 HW problems. What is their utilization rate (in %)? Calculate as a percentage (thus .05 would be entered as 5)
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The utilization rate of friends' study time is 33%
In this question, we need to find the utilization rate for friends' study time.
They can complete a maximum of 60 HW problems per hour. In the last hour, they were a little distracted but managed to complete 20 HW problems.
We know that the formula for the utilization rate:
Utilization % = Actual Number of Hours Worked / the Total Available Hours.
So the utilization rate would be,
r = 20/60
r = 0.33
r = 0.33 × 100
r = 33%
Therefore, the utilization rate of friends' study time is 33%
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14. Construction workers are laying out the rectangular foundation for a new building.They want to check that the corner is 90°. They measure the diagonal as shown to be 9.5 m. Is the angle 90° Explain your reasoning.
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Explanation: We can see on the image that the two sides and the diagonal represent a triangle. We also know that this triangle to have a 90 degrees angle is will be called a right triangle. Finally, all right triangles obey the Pythagorean equation
[tex]h^2=a^2+b^2[/tex]NOTE:
h = hypotenuse
a and b = other sides
Step 1: Once we know the length of the two sides we can use the Pythagorean equation to find the length of the hypotenuse for the triangle to be a right triangle and consequently have an angle that measures 90 degrees.
Step 2: Let's calculate as follows
[tex]\begin{gathered} h^2=a^2+b^2 \\ h=\sqrt[]{8^2+6^2} \\ h=10 \end{gathered}[/tex]Step 3: We can see above, that to have an angle that measures 90 degrees (right triangle) the triangle have to have a hypotenuse = 10 which is different from 9.5.
Final answer: So the angle does not measure 90°.
Which of these shows the result of using the first equation to substitute for y?
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D) 9x=18
Explanationgiven
[tex]\begin{gathered} y=3x\Rightarrow equation(1) \\ 3x+2y=18\Rightarrow equation(2) \end{gathered}[/tex]Step 1
substitute the y value from equation (1) into equation(2)
so
[tex]\begin{gathered} 3x+2y=18\operatorname{\Rightarrow}equat\imaginaryI on(2) \\ replace \\ 3x+2(3x)=18 \\ 3x+6x=18 \\ add\text{ like terms } \\ 9x=18 \end{gathered}[/tex]therefore, the answer is
D) 9x=18
I hope this helps you
Evaluate 1312e 4 Sov? 3x²x3 dx (Type an exact answer.)
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We have to solve the integral:
[tex]\int ^4_03x^2e^{x3}dx[/tex]We will apply a variable substitution in order to simplify the solution. We have a hint when we see that the derivative of x^3 is 3x^2, that is part of the factors.
[tex]\begin{gathered} u=x^3\Rightarrow du=(3x^2)dx \\ x=0\Rightarrow u=0^3=0 \\ x=4\Rightarrow u=4^3=64 \end{gathered}[/tex]Then, we can write:
[tex]\int ^4_03x^2e^{x3}dx=\int ^4_0e^{x3}(3x^2)dx=\int ^{64}_0e^udu[/tex]Then, we have a simpler integral to solve:
[tex]\int ^{64}_0e^udu=e^u+C=e^{64}-e^0=e^{64}-1[/tex]The exact solution is e^64-1.
Solve for the dimensions of the rectangle. Area= length•widthThe length of a rectangle is 2cm greater than the width. The area is 80cm2. Find the length and width.
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The length of a rectangle is 2cm greater than the width. The area is 80cm2. Find the length and width.
L=W+2
W=W
[tex]\begin{gathered} A=L\cdot W \\ A=(W+2)\cdot W \\ A=W^2+2W \\ A=80\operatorname{cm} \\ Then, \\ 80=W^2+2W \\ W^2+2W-80=0 \end{gathered}[/tex][tex]\Delta=4+320=324[/tex][tex]\begin{gathered} W=\frac{-2\pm\sqrt[]{324}}{2}=\frac{-2\pm18}{2} \\ W_1=\frac{-20}{2}=-10 \\ W_2=\frac{16}{2}=8 \end{gathered}[/tex]The width should be positive, therefore W=8
L=W+2
L=8+2=10
The length is L=10
Covert1 1/4 percent to a decimal 5 bill has received a wage increased. His new hourly wage is $14.30 compared to previous wage of $12.95 find the percentage increase in bill hourly wage. Round it off to 2 decimal places
Answers
The percentage increase in bill hourly wage is 10.42%
Given,
Bill has received a wage increased.
His new hourly wage is $14.30
and, compared to old wage of $12.95
To find the percentage increase in bill hourly wage.
Now According to the question:
New bill is = $14.30
Old bill is = $ 12.95
= ($14.30 - $12.95) / $12.95
= $1.35 / $12.95
= 10.42%
Hence, The percentage increase in bill hourly wage is 10.42%
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Find the rate of change of the line represented by the table.
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Slope formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Replacing:
[tex]m=\text{ }\frac{4-6}{3-3}=-\frac{2}{0}=\text{ undefined}[/tex]since x is constant , it is a vertical line, with an undefined slope.
Graph A) -f(x) B) f(x+2) -4Then find the domain and range of each
Answers
a. Graph -f(x):
By the transformations rules for functions, the graph of -f(x) is equal to a reflection over the x-axis, and a change of the y-coordinates:
[tex](x,y)\rightarrow(x,-y)[/tex]Then, given the function:
[tex]f(x)=\sqrt[]{x}[/tex]The graph of -f(x) is:is
The domain of the function is the set of all possible x-values, then it is:
[tex]\lbrack0,+\infty)[/tex]The range is the set of all possible values of the function, then it is:
[tex]\lbrack0,-\infty)[/tex]b. Graph f(x+2)-4:
The transformation f(x+2) is an horizontal translation left 2 units.
And the transformation f(x+2)-4 is a vertical translation down 4 units.
Then, the coordinates of this graph in comparison to the given graph are:
[tex](x,y)\rightarrow(x-2,y-4)[/tex]Then for the point (1,1) the new coordinates are (1-2,1-4)=(-1,-3).
For (4,2): the new coordinates (4-2,2-4)=(2,-2)
For (9,3): the new coordinates (9-2,3-4)=(7,-1)
The graph is:
The domain of this function is:
[tex]\lbrack-2,+\infty)[/tex]And the range is:
[tex]\lbrack-4,+\infty)[/tex]Please help. I've been trying to answer this question but I haven't been successful.
Answers
Equations
It's required to find the value of x that satisfies the conditions of the figure.
We have an equilateral triangle. We know it's equilateral because all of its interior angles have the same measure (look at the tick mark on each angle).
Recall the sum of the interior angles of any triangle is 180°.
If all the interior angles have the same measure, then each angle measures 180/3 = 60°.
One of the angles is assigned an expression of x. We can equate it to 60:
5x - 18 = 60
Adding 18:
5x = 78
Dividing by 5:
x = 78/5
x = 15.6
Answer: I do believe the answer is 15.6. Hope this helps! ^w^
find the function domain and range and the slope of the graph
Answers
The line end points are (4,3) and (-5,-2).
The value of x coordinates give the domain and values of y coordinates give the range.
Since point (4,3) lies on the line and point (-5,-2) does not lie on ther line.
Domain is,
[tex](-5,4\rbrack[/tex]Range is,
[tex](-2,3\rbrack[/tex]Determine the slope of line.
[tex]\begin{gathered} m=\frac{3-(-2)}{4-(-5)} \\ =\frac{5}{9} \end{gathered}[/tex]So slope is 5/9.
Solve radical∛x²-8=4
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Let's determine the value of x on the given radical expression:
[tex]\text{ }\sqrt[3]{x^2-8}\text{ = 4}[/tex]hey there ms or mr could you help me out with this problem please?
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Similar figures have corresponding sides that are proportional.
We can set up a ratio where we are putting corresponding sides of each triangle, cross mutliply, and solve for the unknow.
Looking at answer choices:
A)
if we have x and 24 to one side, we need corresponding pair from right hand side triangle, which should be 15 and 9. But the ratio is 9 over 15. So, this isn't right.
B)
If 24 is paired with 9 [corresponding side of each triangle], then we need same correspondence.
But we have 15 and x, which is opposite of what we want. So, this isn't right.
C)
32 goes with x and 12 goes with 15. They are indeed in correspondence to each other.
This ratio is correct.
We can say Option C is the correct answwer.
There are 120 teachers. Select a sample of 40 teachers by using the systematic sampling technique.
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Given:
Total number of teachers = 120
To select a number of teachers = 40
Required:
To find a sample of 40 teachers by using the systematic sampling technique.
Explanation:
The probability formula is given as:
[tex]\begin{gathered} P=\frac{number\text{ of favourable outcomes}}{Total\text{ number of outcomes}} \\ P=\frac{40}{120} \\ P=\frac{1}{3} \end{gathered}[/tex]Final Answer:
[tex]undefined[/tex]Question 17
2(h - 6) + 20 = -4
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Alicia borrow 15000 to buy a car she borrowed the money at 8% for 6 years how much will she have to pay the bank at the end of 6 years
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Answer:
Explanation:
First, we identify the main components:
• Principal = $15,000
,• Rate = 8% =0.08
,• Time = 6 years
[tex]undefined[/tex]Which of the following is the exact value of cot(pi/4)
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We have to select the correct value of cot (pi/4).
It is known that the value of cot (pi/4) is 1.
Thus, the correct option is B.
The coordinate pairs for triangle ABC are A(1,2), B(4,5), C(2,2). It undergoes a translation of 2 units right and 1 unit 1 up. Write down the coordinates of A'
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We will have the transformation rule (x, y) -> (x+2, y+1)
Then, for A' we will have:
A'(3, 3)
B'(6, 6)
C'(4, 3)
Use the Distributive Property to rewrite each product below. Simplify your answer.
A.) 28 · 63
B.) 17 (59)
C.) 458 (15)
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As per the concept of distributive property, the values of
A.) 28 · 63 = 1768
B.) 17 (59) = 1003
C.) 458 (15) = 6870
Distributive property:
Distributive property states that, " multiplying the sum of two or more addends by a number produces the same result as when each addend is multiplied individually by the number and the products are added together."
It can be written as expression like the following,
A( B + C) = AB + AC
Given,
Here we have the expressions,
A.) 28 · 63
B.) 17 (59)
C.) 458 (15)
Now, we have to find the solution for this by using the distributive property.
Now, we have to expand the given expressions by using the distributive property then we get,
A) 28. ( 60 + 3) = (28 x 63) + (28 x 3)
=> 1680 + 84
=> 1768
Similarly, we have simplify the next expression as,
B) 17 (59) = 17 x (50 + 9)
As per the distributive property,
17 x (50 + 9) = (17 x 50) + (17 x 9)
=> 850 + 153
=> 1003
Finally, applying the distributive law, we get,
C) 458 (15) = (450 + 8) x 15
=> (450 x 15) + (8 x 15)
=> 6750 + 120
=> 6870
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Use the definition of the derivative to find the derivative of the function with respect to x. Show steps
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The derivative of the function f(x) = -5x²+3x-2 is -10x+3.
Given the function is f(x) = -5x²+3x-2
Differentiate with respect to x.
d/dx -5x²+3x-2 = d/dx (-5x²) + d/dx(3x) - d/dx(2)
using the power rule d/dx [xⁿ] = nxⁿ⁻¹
⇒ d/dx -5x²+3x-2 = -5(2)x²⁻¹ + 3(1) - 0
⇒ d/dx -5x²+3x-2 = -10x+3
The power rule states that the derivative of xn is nx(n-1) for every x if n is a positive integer, regardless of whether you are thinking of derivatives at a point (numbers) or derivatives on an interval (functions).
Hence we get the derivative as -10x+3.
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Solve the triangle with the given measures. More than one triangle may be possibletriangle ABCM
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then
[tex]undefined[/tex]Find the perimeter of the square.
Width = 4x
Length = 36 – 5x
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Answer:
The perimeter of the square is 64 units===========================
GivenA square with dimensions:
Width = 4x,Length = 36 - 5x.To findThe perimeterSolutionSquare has all sides equal:
width = length4x = 36 - 5x4x + 5x = 369x = 36x = 4Each side is:
4*4 = 16 unitsPerimeter:
P = 4*16 = 64 unitsThe perimeter of the square is found as 64 units.
What is defined as the perimeter of the square?The perimeter of such a square is indeed the total length of all of its sides. As a result, we can calculate the perimeter of the a square besides adding its four sides.A square's sides are all equal. As a result, the perimeter of such a square is determined by multiplying the side of a square by four.For the given question,
The dimension of the square are given as;
Width = 4xLength = 36 – 5xFor square, as all sides are equal.
Then,
Width = Length
Put the values.
4x = 36 – 5x
9x = 36
x = 4
Put in dimensions.
Width = 4×4 = 16 unitsLength = 36 – 5×4 = 16 units.The perimeter of square is;
Perimeter = 4 × side
Perimeter = 4 × 16
Perimeter = 64 units.
Thus, the perimeter of the square is found as 64 units.
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how much cardboard is needed to make the single slice pizza box shown
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We must find the amount of cardboard needed to make a slice of pizza box which basically means finding the surface area of the piece of box shown. This is composed of five faces divided in three groups:
- Two equal triangular faces with a base of 6.7 in and a height of 11 in.
- Two equal rectangular faces with a base of 11.5 in and a height of 1 in.
- A single rectangular face with a base of 6.7 in and a height of 1 in.
The area of the piece of box is given by the sum of the areas of the 5 faces so let's find the area of the faces of each group.
The area of a triangle is given by half the product of the length of its base and its height. Then the area of each triangular face is:
[tex]A_t=\frac{6.7\times11}{2}=36.85[/tex]So each triangular face has an area of 36.85 in².
The area of a rectangle is given by the product of its base and height. Then for the pair of equal rectangular faces we have:
[tex]A_{r1}=11.5\times1=11.5[/tex]So each of these two faces has an area of 11.5 in².
The area of the remaining rectangular face is then given by:
[tex]A_{r2}=6.7\times1=6.7[/tex]So the area of the last face is 6.7 in².
Then the total surface area is given by the sum of the areas of the 5 faces. Then we get:
[tex]A=2A_t+2A_{r1}+A_{r2}=2\times36.85+2\times11.5+6.7=103.4[/tex]AnswerThen the answer is 103.4
Which form most quickly reveals the vertex? choose one answer: a. m(x)=2(x+4)^2-8 b. m(x)=2(x+6)(x+2)c. m(x)=2x^2+16x+24what is the vertex? vertex=(___,___)
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The vertx from of the quadratic function is
[tex]f(x)=a(x-h)^2+k[/tex]Where
(h, k) are the coordinates of the vertex
a is the coefficient of x^2
By comparing this form with the answers
a.
[tex]m(x)=2(x+4)^2-8[/tex]a = 2
h = -4
k = -8
The vertex point is (-4, -8)
The quickly reveals the vertex is answer a
evaluate each limit. this is in the topic of jump discontinuities.
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we have
[tex]\begin{gathered} \lim _{x\to-2}-x^2-4x-5 \\ \lim _{x\to-2}-(-2)^2-4(-2)-5 \\ \lim _{x\to-2}-4^{}+8-5 \\ \lim _{x\to-2}--1 \end{gathered}[/tex][tex]\lim _{x\to-2}-1=-1[/tex]therefore
the answer is -1Find the y-intercept and slope of the line below. Then write the equation is slope intercept form (y=mx+b).
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The y-intercept is the value of y when x = 0
To identify y-intercept on a graph, we will check for the the value of y when the line crosses the y axis
From the graph, the line crosses the y axis at y = 6
Hence, the y-intercept is 6
To get the slope, we will pick any two points on the line.
Using points (0, 6) and (4, 0)
Applying the slope formula:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex][tex]\begin{gathered} x_1=0,y_1=6,x_2=4,y_2\text{ = }0 \\ m\text{ = }\frac{0\text{ - 6}}{4\text{ - 0}} \\ m\text{ = }\frac{-6}{4} \\ m\text{ = slope = -3/2} \end{gathered}[/tex]NOTE: the slope is negative because it is going from up to down (moving downwards)
The equation of slope in intercept form: y = mx + b
m = slope = -3/2
b = y-intercept = 6
The equation in y-intercept becomes:
[tex]y\text{ = }\frac{-3}{2}x\text{ + 6}[/tex]Subway wants to know how their customers feel about their food quality and service. When each customer pays for their food, the Subway employee hands them their receipt and tells them that they have a chance to win $500 if they go on line and answer a few questions about the restaurant. a) Experimentb) Observational Studyc) None of thesed) Survey
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From the question, we were told that a subway company decides to reward their customers if they go online and answer a few questions about the restaurant.
We are to determine what the process means.
The general view, examination, or description of something or someone in most cases for a reward is known as a survey.
So since subway wants its customers to go online and answer some question about the restaurant and get a reward, then it is a survey.
So, the process that was carried out is a survey.
Therefore, the correct option is D, which is survey.