Answers
Step-by-step explanation:
The equation of a trigonometric function is
[tex]y = a \sin(bx + c) + d[/tex]
or
[tex]y = a \: cos(bx + c) + d[/tex]
Let define some variables,
D is the midline, this refers to the midpoint of the highest y value and lowest y value. Some textbooks call it the vertical translations but it is the same thing.
A is the amplitude. The amplitude is the distance from the midline to the highest y value. Some distance is non negative, the formula for the amplitude is
[tex] |a| [/tex]
The period is how often the wave repeats itself on a interval.
Period can't be negative so the formula for period is
[tex] \frac{2 \pi}{ |b| } [/tex]
To find the period, look at the extreme points.
The phase shift tells us if the sinusoid have been shifted to the right or left. The formula for the phase shift
[tex]bx + c = 0[/tex]
Solving for x gives us
[tex]x = \frac{ - c}{b} [/tex]
If our x is negative, we have a phase shift to the right
If our x is Positve, we have a phase shift to the left.
Let solve this equation, Let use Sine since sin(0)=0,
The smallest y value here is 0, and the highest is 2, so the amplitude is 1.
[tex]d = 1[/tex]
Next, the distance from the max to the midline is 1, as well the min to the midline is also 1.
So
[tex]a = 1[/tex]
We have minimum at 0, and 8, so our period is 8.
[tex] \frac{2\pi}{b} = 8[/tex]
Solve for b,
[tex] \frac{\pi}{4} = b[/tex]
Plug this in the equation.
[tex]1 \sin( \frac{\pi}{4} x) + [/tex]
[tex]y = \sin( \frac{\pi}{4} x) + 1[/tex]
The graph passes through (4,2) so let see if that holds true for our equation
[tex]2 = \sin( \frac{\pi}{4} (4)) + 1[/tex]
[tex]2 = \sin(\pi) + 1[/tex]
[tex]2 = 0 + 1[/tex]
[tex]2 = 1[/tex]
This doesn't hold true, so we must have a phase shift
Notice that
[tex] \sin( \frac{\pi}{2} ) = 1[/tex]
So if we shift this pi/2 to the right, we can get our equation to be true.
[tex]y = \sin( \frac{\pi}{4} x - \frac{\pi}{2} ) + 1[/tex]
This equation works so our equation is
[tex]y = \sin( \frac{\pi}{4} x - \frac{\pi}{2} ) + 1[/tex]
Related Questions
Solving Quadratic Inequalities
Nathan is a product manager for a home security company. Through testing and focus groups, he has modeled the equation P = -(x - 32)2 + 1000 to represent the profit (in thousands) of the price x for a new home security camera. Find the prices in dollars at which Nathan can make a profit of at least $471, 000. Show your work as it is done in the lesson content and include units of measurement.. HINT: Use 471 as part of the inequality you set up, not 471,000.
Answers
The prices in dollars at which Nathan can make a profit of at least $471, 000 are $55 and $9
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given the profit (P) in thousands for security camera with price of x, hence for a profit of at least $471, 000:
-(x - 32)² + 1000 ≥ 471
-(x² - 64x + 1024) + 1000 ≥ 471
-x² + 64x - 1024 + 1000 - 471 ≥ 0
-x² + 64x - 495 ≥ 0
x = 55 or x = 9
The prices in dollars at which Nathan can make a profit of at least $471, 000 are $55 and $9
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Following the recommended steps for solving equations, what should be your first step in solving the given
equation?
2x-3-4x-2x = -6
A. add 3 to both sides
B. subtract 4x from 2x
C. add 6 to both sides
D. add 4x to 2x
Answers
Answer:
The first step in solving the equation should be to add 3 to both sides.
Step-by-step explanation:
Adding 3 to both sides will cancel out the -3 on the left hand side of the equation, leaving only the 2x term. On the right hand side, adding 3 will cancel out the -6, leaving only 0. This will give you the equation 2x = 0, which is much easier to solve.
A 16 -foot ladder is leaning on a tree. The bottom of the ladder on the ground at a distance of 6 feet from the base of the tree. The base of the tree and the ground form a right angle. What is the distance, in feet, between the ground and the top of the ladder? Round your answer to the nearest tenth.
Answers
Answer:
14.8 feet
Step-by-step explanation:
since this is creating a right triangle you can use the Pythagorean theorem which is a^2 + b^2 = c^2, and by plugging in the one leg length and the hypotenuse length you get 14.8324 which rounds up to 14.8
The distance between the ground and the top of the ladder is 14.8 feet.
What is Pythagoras theorem?The relationship between the three sides of a right-angled triangle is explained by the Pythagoras theorem, commonly known as the Pythagorean theorem. The Pythagoras theorem states that the square of a triangle's hypotenuse is equal to the sum of its other two sides' squares.
Given:
Hypotenuse= 16 foot,
Base = 6 feet
Using Pythagoras theorem,
H² = B² + P²
16² = 6² + P²
256 = 36 +P²
P² = 256-36
P= 14.83 feet
Hence, the distance between the ground and the top of the ladder is 14.8 feet.
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A total of 316 tickets were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was three times the number of adult tickets sold. How many adult tickets were sold?
Answers
Step-by-step explanation:
ATTACHED IS THE SOLUTION!Answer:
Adults: 79 tickets
Students: 237 tickets
Step-by-step explanation:
a + b = 316 Eq. 1
b = 3a Eq. 2
a = adults
b = students
Replacing Eq. 2 in Eq. 1
a + 3a= 316
4a = 316
a = 316/4
a = 79
from the Eq. 2
b = 3a
b = 3*79
b = 237
Check:
from the Eq. 1
a + b = 316
79 + 237 = 316
If you rolled two dice, what is the probability
Give
that you would roll a sum of 5?
your answer as a simplified fraction. Enter
the
number that belongs in the green box.
second
first
1 2
3 456
1 1,1 1,2 1,3 1,4 1,5 1,6
2 2,1 2,2 2,3 2,4 2,5 2,6
3 3,1 3,2 3,3 3,4 3,5 3,6
§
4 4,1 4,2 4,3 4,4 4,5 4,6
5 5,1 5,2 5,3 5,4 5,5 5,6
6 6,1 6,2 6,3
6,4 6,5 6,6
Answers
Answer:
Probability of rolling a sum of 5 is 1/9.
There are 36 different combinations because 6(first roll)*6(second roll) = 36. The only possible combinations are {1,4},{2,3}{2,3}and{4,1}
4/36 = 1/9
factor the following expression
28w^(4)x^(2)y^(7)+12w^(9)x^(8)
Answers
The expression after factorisation is 4w⁴x² (7y⁷ + 3w⁵x⁶)
What is an Expression ?An expression is defined as a mathematical statement consisting of variables , constants and mathematical operators.
The expression given is
28w⁴x²y⁷+12w⁹x⁸
Factors of the given expression will be the common terms in both the terms
w⁴x² is common in the variable term
The HCF of 28 and 12 is 4
4w⁴x² (7y⁷ + 3w⁵x⁶)
The expression after factorisation is 4w⁴x² (7y⁷ + 3w⁵x⁶)
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The function below models the correlation between the number of hours a plant is kept in sunlight (x) and the height(y), in mm, to which it grows:
Y=2+4x
What does the y- intercept of this function represent?
Answers
the original height of the plant was 2mm. The correct option is the second one.
What does the y-intercept of the function represent?
We know that the function:
y = 2 + 4x
Represents the height of a plant in mm, as a function of x, the hours kepts in sunlight.
The y-intercept is the value that y takes when we evaluate in x = 0.
So, the y-intercept is the height of the plant when it is exposed to zero hours of light.
In this case:
y = 2 + 4*x
Evaluating in x = 0
y = 2 + 4*0 = 2
This means that the original height of the plant was 2mm. The correct option is the second one.
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10 + (-8) - 2
Answer the equation
Answers
Answer:
0
Step-by-step explanation:
[tex]10 + ( - 8) - 2 \\ = 1 0 - 8 - 2 \\ = 2 - 2 \\ = 0[/tex]
Answer: 0
Step-by-step explanation: -8+-2 is -10 and -10+10 is 0
In the diagram of AABC shown below, DE || BC.
B
A
3) 3
4) 15
E
If AB= 10, AD = 8, and AE = 12, what is the
length of EC?
1) 6
2) 2
C
dd or create
k as done
nments
As
to Susan Guha
Answers
Answer:
3
Step-by-step explanation:
AD/AB = AE/AC
8/10 = 12/AC
AC = 12(10/8)=15
EC = 15 - 12 = 3
Use the function below to find F(4).
Answers
Answer:
A. 81
Step-by-step explanation:
F(x)=[tex]3^{x}[/tex]
if x=4, then F(4) = [tex]3^{4}[/tex] = 3×3×3×3=81
Calculate the volume of a cylinder of radius 5cm and height 14cm (take pie 22/7)
Answers
Answer:
Volume of a cylinder = 1,100cm³
Step-by-step explanation:
Volume of a cylinder = πr²h
Given
π = 22/7
r = 5cm
h = 14cm
Substitute the given information in the problem.
Volume of a cylinder = 22/7 × (5cm)² × 14cm
Volume of a cylinder = 22/7 × 25cm² × 14cm
Volume of a cylinder = 1,100cm³
Which expression is equivalent to -1/6y+1/3?
Answers
An expression is defined as a set of numbers, variables, and mathematical operations. The correct option is C.
What is an Expression?In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.
The expression that is equivalent to -(1/6)y + 1/3 is,
-(1/6)y + 1/3
Take 1/6 common from both first and the last term,
1/6 (-y + 2)
Hence, the correct option is C.
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Which of the following segments is a radius of OK?
M
K
OA. KM
B. MR
C. RT
D. MT
SUB
Answers
Answer:
A. KM
Step-by-step explanation:
A circle is named by its center. Circle K has its center at point K.
A radius is a line segment that extends from the center of the circle to a point on the circle. Point K must be one end of any radius.
ApplicationSegment KM is a radius.
Segments between points on the circle are chords. The other named segments in the answer choices are all chords of the circle.
Find f(5) for f(x)=1/4(2)^x
A. 2
B. 4
C. 32
D. 8
Answers
Step-by-step explanation:
The value of X is given so put the value of X in 1/4 (2)^x
Which expression is equivalent to (3x² + 4x-7)(x-2)?
OA. (3x2 + 4x-7)(x) + (3x² + 4x-7)(-2)
OB. 2x(3x² + 4x-7)
OC. x(3x² + 4x-7) - 2
OD. x(3x² + 4x-7)+2(3x² + 4x-7)
Answers
Answer: A
Step-by-step explanation:
The expressions are equivalent by the distributive property.
Using the model from the example above, where f(x)=10001+999e−0.6030x, estimate the number of cases of flu on day 15. Remember that answers should be rounded to the nearest whole number.
Answers
The number of cases of flu on day 15 will be 1001 flus
Exponential functions
Exponential functions are expressed as y = ab^x
Given the expression below
[tex]f(x)=10001+999e^{-0.6030x}[/tex]
In order to determine the number of cases of flu on day 15, we will substitute x = 15 into the expression to have:
[tex]f(x)=10001+999e^{-0.6030(15)}\\f(x)=10001+999e^{-9.045}\\f(15) = 10001+0.11786\\f(15) \approx 10001[/tex]
Hence the number of cases of flu on day 15 will be 1001 flus
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Answer:
Step-by-step explanation:
Evaluate the function for the value x=15.
f(x)f(15)f(15)=10001+999e−0.6030x=10001+999e−0.603(15)≈894.565217
Rounded to the nearest whole number, there will be 895 cases on day 15.
A house has increased in value by 37% since it was purchased. If the original value was $720, 000, what is the value after the increase? A house has increased in value by 37 % since it was purchased . If the original value was $ 720 , 000 , what is the value after the increase ?
Answers
Answer:$986,4000
Step-by-step explanation:
The function g(x)=8(4x) is reflected across the x-axis to create f(x) What is the equation of f(x)?
Answers
Answer:
g(x)=-32x
Step-by-step explanation:
When something is reflected over the x-axis. The x doesn't change, it's the y that changes. So let's just generally say you have the point (x, y). Reflecting this over the x-axis means the y is changing, so it becomes (x, -y). And since the g(x) represents the y, you multiply the entire thing by -1. So the new equation becomes: g(x) = -(8(4x)) = -8(4x) = -32x. So the equation is g(x) = -32x or -8(4x) if you want to leave it like how it was before.
Use the function below to find F(3).
Answers
Answer: B
Step-by-step explanation:
Plug in x for 3. Then solve.
4*(1/3)^3
4*1/9
4/9
plug in x for 3
4*(1/3)^3
4*(1/27)
4/27
choose the equation below that represents the line passing through the point (-2,-3) with a slope of -6
Answers
The equation of the line is: y + 3 = -6(x + 2) [point-slope form] or y = -6x - 15 [slope-intercept form].
How to Write the Equation of a Line?Given a point (-2, -3) and a slope (m) of -6, you can write the equation of the line in point-slope form by substituting (a, b) = (-2, -3) and m = -6 into y - b = m(x - a):
y - (-3) = -6(x - (-2))
y + 3 = -6(x + 2) [point-slope form]
Rewrite in slope-intercept form as:
y + 3 = -6(x + 2)
y + 3 = -6x - 12
y = -6x - 12 - 3
y = -6x - 15 [slope-intercept form]
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Using the image, find the distance between the points given on the graph. Distance =√
Answers
Answer:
I think 5m is the correct answer
Answer:
Hello! The answer to your question is [tex]\sqrt{29}[/tex], or approximately 5.39.
Step-by-step explanation:
(Two methods)
Method 1:
To solve this problem, we will use the Pythagorean theorem. To create the triangle, you can connect all the points with lines. This will leave you with the base of the triangle as 2 and the leg as 5. We do not know the hypotenuse. Now, we can use the Pythagorean theorem formula:
[tex]a^2 + b^2 = c^2[/tex]
[tex]\sqrt{2^2 + 5^2 } \\= \sqrt{4 + 25} \\= \sqrt{29}\\or 5.39[/tex]
Now, we can substitute our values:
Method 2:
To solve this problem, we will use the distance formula:
[tex]\sqrt{(x2 - x1)^2 + (y2 - y1)^2}[/tex]
Our points are:
A(-3, 1) and B(-1, -4)
Now, we substitute our values into the distance formula provided:
[tex]\sqrt{(-1 - (-3))^2 + (-4 - 1)^2} \\= \sqrt{(-1 + 3))^2 + (-5)^2} \\= \sqrt{(2)^2 + (-5)^2} \\= \sqrt{4 + 25} \\= \sqrt{29} \\or 5.39[/tex]
write an equation in slope- intercepty form of the line that passes through the point (5,9) with slope 7
Answers
m=slope
y=7x+b
Plug in points (5,9) to x&y to get
9=7(5)+b
b=-26
then plug both b and m in to get
y=7x-26
Hope this helped :)
If f(x) = x..........
Answers
Answer:
b.) 0
Explanation:
⇒ f(g(-1))
go from right to left while solving such function questions.
⇒ f(-1 + 2)
⇒ f(1)
⇒ (1)² -2(1) + 1
⇒ 0
Step-by-step explanation:
[tex]f(x) = {x}^{2} - 2x + 1[/tex]
[tex]f(x) = {(x - 1)}^{2} [/tex]
[tex]g(x) = x + 2[/tex]
[tex] \: [/tex]
[tex] = f(g( - 1))[/tex]
[tex] = {(x - 1)}^{2} [/tex]
[tex] = {((x + 2) - 1)}^{2} [/tex]
[tex] = {(( - 1 + 2) - 1)}^{2} [/tex]
[tex] = {(1 - 1)}^{2} [/tex]
[tex] = {0}^{2} [/tex]
[tex] = 0[/tex]
The answer is B.
The number is between 8,500 and 8,800.
When multiplied by 8, the result is a whole number.
The digit in the hundreds place is ¾ the digit in the thousands place.
The sum of all digits in the number is 26.
The digit in the hundredths place is 200% of the digit in the tenths place.
There are no zeros in the decimal places.
Your first job is to post at least two numbers that do NOT work.
Explain why they do not work.
Post your best guess for a number that works (or a number that almost works)
Answers
We can use variables ABCD to Stand in place of the numbers.
A + B + C + D = 26
Since the number is between 8,500 and 8,800, A is 8.
The digit in the hundreds place is 3/4 of the thousands place, so:
8 x 3/4 = 24/4
24/4 reduces to 6
B = 6
The digit in the hundreds place is 200% of the digit in the tens place, so:
6 / 2 = 3
C = 3
The last step is simple! Since we have our values, we can use inverse operations to solve.
8 + 6 + 3 + D = 26
17 + D = 26
26 - 17 = 9
D = 9
Hence, your number is 8,639.
Two numbers that will not work:
7 in the thousands place would not work. It would not work because the problem says that the number is between 8,500 and 8,800, and would therefore mean the number can be 8,500 at least.
Another number that would not work would be 2 in the hundreds place. This number would not work because a 1 would be in the tens place, and 3 + 8 = 11. This seems right, but it isn’t because 11 + 15 = 26. That would mean that in order to add up, the digits would have to expand higher than 8 in the thousands place and we would possibly need to use the ten thousands place.
If you need help, let me know and I will gladly assist you.
The expression to find the perimeter of a rectangle is 2(length + width). What is the perimeter, in units, of a rectangle of length fraction 1 over 7 unit and width fraction 1 over 2 unit?
fraction 1 and 1 over 7 units
fraction 1 and 2 over 7 units
fraction 4 over 9 unit
fraction 2 over 9 unit
Answers
The perimeter of the rectangle is 1 1/7 units
Perimeter of a rectangleThe formula for calculating the perimeter of a rectangle is given as:
P = 2(l + w)
Given the following
length. = 1/7 units
width = 1/2 units
Substitute
P = 2(1/7 + 1/2)
P = 2(9/14)
P = 9/7 = 1 1/7
Hence the perimeter of the rectangle is 1 1/7 units
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A table has an area of x² + 12x + 35. Find the possible dimensions of the table.
Answers
Answer:
x + 7 and x + 5 are possible dimensions
Step-by-step explanation:
x² + 12x + 35 ← factor the quadratic
consider the product of the factors of the constant term (+ 35) which sum to give the coefficient of the x- term (+ 12)
the factors are + 7 and + 5 , since
7 × 5 = + 35 and 7 + 5 = + 12 , then
x² + 12x + 35 = (x + 7)(x + 5) ← in factored form
now the area A = length × breadth , that is
A = x² + 12x + 35 = (x + 7)(x + 5) , then
length = x + 7 or x + 5
breadth = x + 5 or x + 7
Practice Problems
1. The absolute value of a number is its distance from 0 on the
number line (no sign, just value). Compare the absolute value of
7 and -7.
Answers
Answer: The absolute values of 7 and -7 are equal because they are both a distance of 7 units from 0 on the number line.
write this ratio as a fraction in simplest form without any units. 3 pounds to 54 ounces
Answers
The ratio of the measurements written as a fraction is 8/9.
What are the ratios written as fraction?A fraction is a quantity that is not a whole number. In maths, a fraction usually has a numerator and a denominator.
In order to convert the ratio to fraction, the units of measurements have to be the same. So, pounds would be converted to ounces.
3 x 16 = 48 ounces
48 / 54 = 8/9
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The table below shows which hand is favored by each person from a sample of 100 people (50 male and 50 female). Use the information in the table to answer the following questions
Determine the probability that a randomly selected person from the sample will be female or left-handed.
Answers
The probability that a randomly selected person from the sample will be female or left-handed will be 0.05.
What is the probability?Probability is synonymous with possibility. It is concerned with the occurrence of a random event.
Probability can only have a value between 0 and 1. Its simple notion is that something is very likely to occur. It is the proportion of favorable events to the total number of events.
Given data;
Total number of sample,n(S) = 100 person
No of female or left-handed is,n(E) =5
P(E) is the probability that a randomly selected person from the sample will be female or left-handed
The probability that a randomly selected person from the sample will be female or left-handed is found as;
P(E) = n(E)/n(S)
P(E) = 5 / 100
P(E) = 0.05
Hence the probability that a randomly selected person from the sample will be female or left-handed will be 0.05.
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The information below explains the transformations and translations of a logarithmic function. What is the equation for this function?
Shift 4 units left
Shift 3 units up
Vertical Compress by 1/3
Horizontal stretch by 3
Fill in the table below for the information of the function f(x) = log[-(x – 5)] + 4
Function
Log[-(x - 5)] + 4
Domain
Range
Interval of Increase/Decrease
Equation of Asymptote
Answers
The transformed function is y = 1/3(log(x/3 + 4)) + 1
How to transform the logarithmic function?The parent logarithmic function is
y = log(x)
When shifted left by 4 units, we have:
y = log(x + 4)
When shifted up by 3 units, we have:
y = log(x + 4) + 3
When compressed vertically by 1/3, we have:
y = 1/3(log(x + 4) + 3)
This gives
y = 1/3(log(x + 4)) + 1
When stretched horizontally by 3, we have:
y = 1/3(log(x/3 + 4)) + 1
Hence, the transformed function is y = 1/3(log(x/3 + 4)) + 1
The transformation of function f(x)We have:
f(x) = log[-(x – 5)] + 4
Set the radicand greater than 0
-(x - 5) > 0
Divide by -1
x - 5 < 0
Add 5 to both sides
x < 5 -- this represents the domain
A logarithmic function can output any real number.
So, the range is [tex]-\infty < f(x) < \infty[/tex]
In the domain, we have:
x < 5
This means that the interval of decrease is [tex]-\infty < x < 5[/tex]
Rewrite as an equation
x = 5 --- this represents the equation of asymptote
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