The image of the function after vertically stretched by a factor of 3 is f'(x) = 3|x + 1| - 3
How to determine the equation of f(x) after the transformation?From the question, the given parameters are:
f(x) = |x + 1| - 1
The transformation is given as
Vertical stretch by a factor of 3
This implies that we stretch the function f(x) by a factor of 3 across the origin
Mathematically, this transformation can be represented as
(x, y) = (x, 3y)
When represented as a function, we have
f'(x) = 3 * f(x)
Substitute the equation f(x) = |x + 1| - 1 in the equation f'(x) = 3 * f(x)
So, we have the following equation
f'(x) = 3 * (|x + 1| - 1)
Remove the bracket
So, we have
f'(x) = 3 * |x + 1| - 3 * 1
Evaluate the products
So, we have
f'(x) = 3|x + 1| - 3
Hence, the equation of f'(x) is f'(x) = 3|x + 1| - 3
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Prove that if x and y are real numbers, then max(x, y) +
min(x, y) = x + y. [Hint: Use a proof by cases, with
the two cases corresponding to x ≥ y and x
Real numbers contain the number zero and can be either positive or negative.
If x > y, then max (x, y) = x and min (x, y) = y.
So, max (x, y) + min (x, y) = x + y
What is meant by real numbers?Real numbers contain the number zero and can be either positive or negative. Because they exists not imaginary, which exists a separate type of number system, they are directed to as real numbers. Unquantifiable quantities, like the square root of -1, are referred to as imaginary numbers.
A "actual" and practical idea, infinity. Infinity, on the other hand, is not a number on the real number line since it is not a part of the mathematically defined set of "real numbers."
If x > y, then max (x, y) = x and min (x, y) = y.
So, max (x, y) + min (x, y) = x + y
If x ≤ y , then max (x, y) = y and min (x, y) = x.
And we have, max (x, y) + min (x, y) = y + x = x + y
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This month, a vegetarian restaurant used 9,168 ounces of spinach. That is 20% more than last month, when the restaurant had a different menu. How much spinach did the restaurant use last month?
Consider 9,168 ounces of spinach is 20% more than the amount of spinach of the last month. If x is such unknown amount of spinach, you can write:
[tex]x+(\frac{20}{100})x+9,168[/tex]where (2/100)x factor represents the 20% of x.
Simplify the pervious equation and solve for x:
[tex]\begin{gathered} x+0.2x=9,168 \\ 1.2x=9,168 \\ x=\frac{9,168}{1.2} \\ x=7,640 \end{gathered}[/tex]Hence, last month the restaurant used 7,640 amount of spinach.
B. -3/4x-4=-6C. Is 0 solution to -3/4x-4>-6Select answerYes. No.D. Using interval Notation, solve: -3/4x-4>-6
Explanation:
Part A:
The question is given below as
[tex]\begin{gathered} whenx=0,evaluate \\ -\frac{3}{4}x-4= \end{gathered}[/tex]By putting x=0, we will have that
[tex]\begin{gathered} \begin{equation*} -\frac{3}{4}x-4 \end{equation*} \\ -\frac{3}{4}(0)-4 \\ =-4 \end{gathered}[/tex]Hence,
The final answer for part A is
[tex]\Rightarrow-4[/tex]Part B:
[tex]\begin{gathered} solve, \\ -\frac{3}{4}x-4=-6 \end{gathered}[/tex]add 4 to both sides
[tex]\begin{gathered} -\frac{3}{4}x-4=-6 \\ -\frac{3}{4}x-4+4=-6+4 \\ -\frac{3}{4}x=-2 \\ coess\text{ multiply, we will have} \\ -3x=-2\times4 \\ -3x=-8 \\ divide\text{ both sides by -3} \\ \frac{-3x}{-3}=-\frac{8}{-3} \\ x=\frac{8}{3} \end{gathered}[/tex]Hence,
The final answer for part B is
[tex]\Rightarrow x=\frac{8}{3}[/tex]Part C:
[tex]-\frac{3}{4}x-4>-6[/tex]Add 4 to both sides, we will have
[tex]\begin{gathered} -\frac{3}{4}x-4\gt-6 \\ -\frac{3}{4}x-4+4\gt-6+4 \\ -\frac{3}{4}x>-2 \\ cross\text{ multiply} \\ -3x>-2\times4 \\ -3x>-8 \\ divide\text{ bth sides by -3} \\ \frac{-3x}{-3}>-\frac{8}{-3}(the\text{ sighn will be reversed\rparen} \\ x<\frac{8}{3}(0\text{ is a solution\rparen} \end{gathered}[/tex]Hence,
The final answer for part C is YES
Part D:
[tex]\begin{gathered} -\frac{3}{4}x-4\gt-6 \\ -\frac{3}{4}x-4+4\gt-6+4 \\ -\frac{3}{4}x>-2 \\ cross\text{ multiply} \\ -3x>-2\times4 \\ -3x>-8 \\ divide\text{ bth sides by -3} \\ \frac{-3x}{-3}>-\frac{8}{-3}(the\text{ sighn will be reversed\rparen} \\ x<\frac{8}{3} \\ hence,in\text{ interval notation we will have the final answer be} \\ (-\infty,\frac{8}{3}) \end{gathered}[/tex]Hence,
The final answer for part D is given below as
[tex]\Rightarrow(-\infty,\frac{8}{3})[/tex]
Find the length of the hypotenuse of the triangle pictured below. Give your answer accurate to at least 2 decimal places. 8. 7 hypotenuse =
We can use the Pythagoras Theorem:
[tex]\begin{gathered} 8^2+7^2=hypotenusa^2 \\ \text{hypotenusa}=\sqrt[\square]{8^2+7^2}=\sqrt[]{64+49}=\sqrt[]{113}\approx10.63 \end{gathered}[/tex]The hypotenuse is 10.63
helppppppppppppppppppppppppp
Answer:
No
Step-by-step explanation:
V/3 is -35, which is smaller than 5
can someone pls help with this problem?
Answer:
(6√13)/13
Step-by-step explanation:
You want the exact value of cot(arcsin(√13/7)).
Trig ratiosThe relevant trig ratios are ...
Sin = Opposite/Hypotenuse
Tan = Opposite/Adjacent
Cot = 1/Tan = Adjacent/Opposite
Pythagorean theoremThe Pythagorean theorem can be used to find the side adjacent to the angle whose sine is √13/7. Using the sine ratio, we can take the opposite side to be √13, and the hypotenuse to be 7. Then the adjacent side is ...
adjacent² +opposite² = hypotenuse²
adjacent² +(√13)² = 7²
adjacent² = 49 -13 = 36
adjacent = √36 = 6
CotangentThen the cotangent of the angle is ...
cot(arcsin(√13/7)) = adjacent/opposite = 6/√13
cot = (6√13)/13
Jason and Whitney deposit $700.00 into a savings account which earns 11% interest compounded quarterly. They want to use the money in the account to go on a trip in 3 years. How much will they be able to spend?
To solve this problem we need to use the compound interest formula:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Where P is the initial deposit:
[tex]P=700[/tex]r is the interest rate in decimal form, since the interest rate is 11%, in the decimal form we have:
[tex]r=0.11[/tex]n is the number of times that the interest is compounded in a year, in this case, it is compounded quarterly, and since there are 3 quarters in a year, the interest will be compounded 3 times per year:
[tex]n=3[/tex]and t is the total time, in this case, 3 years:
[tex]t=3[/tex]Substituting all of these values into the formula to find the Amount "A" they will have after 3 years:
[tex]A=700(1+\frac{0.11}{3})^{3\cdot3}[/tex]Solving the operations:
[tex]A=700(1+0.03666)^9[/tex][tex]A=700(1.03666)^9[/tex][tex]A=700(1.382777)[/tex][tex]A=967.944[/tex]Answer: They will have $967.944 to spend
I don’t understand these questions. Could you please help me? I need clearly explain.
We need to sketch the graph of V(t), with V in ml and t in seconds. Notice that:
[tex]V(0)=2200[/tex]because for t = 0, her lungs start with a volume of 2200 ml.
Since the period (of a complete breath) is 4 seconds, and the function is sinusoidal, the graph is:
b. Now, a sinusoidal function has the form:
[tex]V(t)=A\sin \lbrack\frac{2\pi}{T}(t-b)\rbrack+c[/tex]where:
• A is the amplitude (half the difference between the smallest and the largest values of V);
,• T is the period;
,• b is the horizontal shift (the difference between the initial value of t and the value of t for which V(t) has the mean value);
,• c is the vertical shift (the mean value of V(t)).
So, observing the graph, we see that:
[tex]A=\frac{2800-2200}{2}=300[/tex][tex]T=4[/tex][tex]b=1[/tex][tex]c=\frac{2200+2800}{2}=2500[/tex]Therefore, the equation representing the volume V, in ml, after t seconds is:
[tex]\begin{gathered} V(t)=300\sin \lbrack\frac{2\pi}{4}(t-1)\rbrack+2500 \\ \\ V(t)=300\sin \lbrack\frac{\pi}{2}(t-1)\rbrack+2500 \end{gathered}[/tex]c. If Aira breathed more rapidly, the period and the horizontal shift of the function V(t) would be smaller. Thus, the graph would change more rapidly also, i.e., we would see more variations in the 20 seconds.
For example, if she breathed 2 times faster, the period would be 2 instead of 4, and the equation would be:
[tex]\begin{gathered} V(t)=300\sin \lbrack\frac{2\pi}{2}(t-0.5)\rbrack+2500 \\ \\ V(t)=300\sin \lbrack\pi(t-0.5)\rbrack+2500 \end{gathered}[/tex]And the graph would be:
And if Aire took bigger breaths, the amplitude and the vertical shift of the function V(t) would increase. For example, if her lungs could reach the volume of 3000 ml, assuming the same period of 4 seconds, the equation would be:
[tex]V(t)=400\sin \lbrack\frac{\pi}{2}(t-1)\rbrack+2600[/tex]And the graph would be:
-3×+9=-3(2×+3)+3(×-4)+1
The expression we have is:
[tex]-3x+9=-3(2x+3)+3(x-4)+1[/tex]Step 1. To solve for x, we need to first apply the distributive property on the right-hand side of the equation. The distributive property is to multiply the number outside each pair of parenthesis by the terms inside it. We get the following:
[tex]-3x+9=-3\cdot2x-3\cdot3+3\cdot x+3\cdot(-4)+1[/tex]Solving the multiplications on the right-hand side:
[tex]-3x+9=-6x-9+3x-12+1[/tex]Step 2. Combine the like terms.
We start by combining the terms that contain x:
[tex]-3x+9=-3x-9-12+1[/tex]And then, combine the independent terms:
[tex]-3x+9=-3x+4[/tex]Step 3. Add 3x to both sides of the equation:
[tex]-3x+3x+9=4[/tex]On the left side, we get that -3x+3x cancel each other:
[tex]9=4[/tex]As we can see, this is not true, which means that there is no solution for x.
Answer: There is no solution for x.
A cab company charges a $11 boarding fee and a meter rate of $2 per mile. The equation is y = 2x + 11 where x represents the number of miles to your destination. If you traveled 5 miles to your destination, how much would your total cab fee be?
We are given a function, y = 2x + 11, where x is the number of miles driven. If we are told that the cab was driven 5 miles, x = 5. So, substitute 5 for x.
y = 2x + 11
= 2(5) + 11
= 10 + 11
= 21
y = 21
Be Allison has been to the large muffin sale table Chris wants to pay more signs
By proportion, Chris should mix 8/3 parts of the red paint in 2 fluid ounces of blue paint.
Purple paint is made using a combination of red paint and blue paint.
Now Allison made purple paint by mixing 3 parts red [aint and 4 parts blue paint.
Therefore, the ratio for the purple paint is :
= 4 parts blue paint / 3 red paint
Chris has 2 fluid ounces of blue paint.
Let the amount of red paint Chirs have to be x.
Then the proportion should be equal.
So,
4 / 3 = x / 2
By cross multiplication,
4 × 2 = 3 × x
8 = 3x
Dividing each side by 3,
x = 8/3
Chirs should mix 8/3 parts of the red paint with blue paint.
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EX #5: Based on the definition above, identify the following: A. Relative Maximum is __ where x = __ B. Relative Minimum is __ where x = __ C. Relative Minimum is __ where x = __
Based on the given graph, you can conclude:
A. Relative Maximum is f(x) = 1 where x = 2
(Because the relative maximum of a function is given by a point of the graph with a slope m = 0, but this point is not necessarily the absolute maximum. Furthermore, the concavity of the curve at this point is negative).
B. Relative minimum is f(x) = -8 where x = -3
C. Relative minimum is f(x) = 0 where x = 4
Both relative minima correspond to points in which the slope of the line is 0 and the concavity of the line is positive.
this morning the temperature was "-12" degrees. the temperature will rise 5/8 degree every hour for 3 hours before dropping 1/2 degree each hour for 5 hours. what is the temperature after 8 hours
Using proportions, it is found that the temperature after 8 hours is of -12.63 degrees.
What is a proportion?A proportion is a fraction of a total amount, and equations can be built to find the desired measures in the problem using arithmetic operations such as multiplication or division considering the given rates in the function.
The temperature is of -12 degrees, and rises by 5/8 every hour for 3 hours, hence the temperature in 3 hours will be given by:
-12 + 3 x 5/8 = -12 + 15/8 = -10.13 degrees.
Then, for each of the next 5 hours, the temperature drops by 0.5 degrees, hence the temperature after 8 hours will be given by:
-10.13 degrees - 5 x 0.5 = -10.13 - 2.5 = -12.63 degrees.
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Help with algebra 2 question.12). In a circle with radius of 18cm, a central angle of 5pi/6 radians intercept an arc. Find length of the arc in cm.
Given:
Radius of circle r is 18 cm.
angle
[tex]\emptyset=\frac{5\pi}{6}=5*\frac{180}{6}=150\degree[/tex]Required:
We need to find the arc length in cm
Explanation:
The formula to find arc length is
[tex]l=2\pi r*\frac{\emptyset}{360}[/tex]Substitute the values in the formula
[tex]l=2*3.14*18*\frac{150}{360}=47.1\text{ cm}[/tex]Final answer:
Arc length of given circle is 47.1 cm
Which of the following inequalities would have the solution set graphed below?
t - 2 ≤ 0
t - 3 ≤ 5
t + 2 ≤ 0
t + 3 ≤ 1
Answer:
t - 2 ≤ 0
Step-by-step explanation:
Given inequality
t- 2 ≤ 0
Add 2 on both sides of the inequality without changing the meaning
t - 2 + 2 ≤ 0 + 2
or
t ≤ 2
Looking at the number line given, we can see that all numbers to the left of 2 including 2 are part of the solution set
Hence the solution set graphed is t- 2 ≤ 0
I am an odd number. When you multiply me by 6, then
divide the product by 3, the quotient is 10. What number am I?
Answer:
20 hope this is helpful and also enjoy your day
I am 5 when you multiply me by 6, then divide the product by 3, the quotient is 10.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Assuming I am 'n'.
∴ When you multiply me by 6, which is 6n, then divide the product by 3, which is 6n/3 the quotient is 10 which is 6n/3 = 10.
6n/3 = 10.
2n = 10.
n = 5.
So, I am an odd number 5.
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Necesito la fórmula y una breve explicación
The value of the height of the rectangle is 80 m.
Rectangles
The rectangle is a quadrilateral. The classification for quadrilaterals is given by the length of sides and angles. For a rectangle, the opposite sides are equal and parallel and their interior angles are equal to 90°.
A rectangle is a geometric figure that has 4 sides called: length and height. Due to its characteristics, the rectangle presents two length (L) and two heights (H).
The perimeter is the sum of all sides of a geometric figures. For a rectangle, the perimeter is 2L+2H.
The question gives:
length (base) = 20 + H width (height) = H perimeter = 360 mFrom the perimeter, you can find the height, see below.
P=2L+2H
360=2*(20 + H) +2H
360= 40 +2H +2H
360-40 = 4H
320= 4H
320/4 = H
80=H
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PLEASE HELP ASAP!!!!! DUE SOON!!! FULL ANSWER!!!!!
A woman can bicycle 44 miles in the same time as it takes her to walk 8 miles. She can ride 9 mph faster than she can walk. How fast can she walk?
She can walk _____________ mph
Answer: 4 mph
Step-by-step explanation:
If woman can walk xmph, then she can ride {x + 10} mph (because she can ride 10 mph faster than she can walk)
Woman ride 28 miles by 28 hours, and can walk 8 miles by 8 hours.
x+10 x
28 =8 [tex]28x=8%28x%2B10%2928x=8x%2B8020x=80x=80%2F20x=4[/tex]
x+10 x
Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.A pair of kids and a pair of adults decided to compete in a three-legged race. The kids got to start 34 meters ahead of the adults, since they had shorter legs. When they were told to start, the kids hobbled forward at a rate of 1 meter per second, and the adults hobbled after them at a rate of 2 meters per second. Soon they were side-by-side. How long did that take? How far did the adults go?It took ____ seconds for the adults to go ____ meters and catch up to the kids.
From the question,
The equation for the distance covered by the kids is d = 1 t + 34
The equation for the distance covered by the adults is d = 2t
Now, when the adults caught up with the kids was when : 2 t = 1 t + 34
Solving this equation, we have 2 t - 1 t = 34
t = 34
since d= 2 t , when t = 34
d = 2 x 34 = 68
It took ___34_ seconds for the adults to go _68___ meters and catch up to the kids.
Solve each system by graphing. If the lines are parallel, write no solution. If the lines coincident, write infinitely many solutions.
Answer:
No Solution
Explanation:
Given the system of equations:
[tex]\begin{gathered} y-2x=4 \\ y=5+2x \end{gathered}[/tex]First, graph each equation using the x and y-intercepts.
Equation 1
[tex]\begin{gathered} y-2x=4 \\ \text{When }x=0,y=4\implies\text{Point (0,4)} \\ \text{When y}=0,x=-2\implies\text{Point (-2,0)} \end{gathered}[/tex]Join the points (0,4) and (-2,0) as done below:
Equation 2
[tex]\begin{gathered} y=5+2x \\ \text{When }x=0,y=5\implies\text{Point (0,5)} \\ \text{When y}=0,x=-2.5\implies\text{Point (-2.5, 0)} \end{gathered}[/tex]Join the points (0,5) and (-2.5, 0) as done below:
We observe that the two lines are parallel.
Therefore, the system of equations has No Solution.
Do all proofs have the same number of steps?
4. What is the value of y in the system of equations shown below?
z+y+z=22
y+z=16
z-2=7
Answer:
Step-by-step explanation:
Y = 22 - 2Z
22 - 2Z + Z = 16
22 - Z = 16
Z = 6
Y + Z = 16
Y + 6 = 16
Y = 10
What is 100,000 written as a power of 10 in exponent. Help
If we have the following number:
[tex]100,000[/tex]and we want to write it as a power of 10 in exponent, we just have to count the zeros of the number, in this case there are 5, and that would be our exponent, then, we have:
[tex]100,000=10^5[/tex]how do I write a function that describes the given transformation
1) Considering that this is our parent function:
[tex]f(x)=2^x[/tex]2) Then let's visualize what happens when we transform it to :
[tex]g(x)=2^{x-3}+1[/tex]3) So now let's reflect that across the y-axis, and vertically shrink that by 1/3:
Find the maximum value of the objective function and the values of x and y for which it occurs.
F = 2x + y
3x + 5y ≤ 45 x 20 and y20
2x + 4y ≤ 32
In linear equation, value of x and y = ( 10 , 3 ).
What is linear equation with example?
Ax+By=C is the typical form for two-variable linear equations. A linear equation in standard form is, for instance, 2x+3y=5. Finding both intercepts of an equation in this format is rather simple (x and y).3x + 5y = 45 ............1
2x + 4y =32 .........2
Multiply by 2 in equation 1 and by 3 in 2
6x + 10y = 90
6x + 12y = 96
equation 2 - equation 1
2y = 6
y = 3
put value of y put in equation 1
3x + 5y = 45
3x + 5 * 3 = 45
3x = 45 - 15
3x = 30
x = 10
value of x and y = ( 10 , 3 )
the value of the objective function at each of the vertices to find the maximum.
I'll do one vertex, you do the rest.
(0,8)
F = 2x + y
F = 2 * 0 + 8
F = 8
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NO LINKS!! Part 4: Please help me with this Similarity Practice
Answer:
C' = (10, 1)
Dilation by a scale factor of 2 with the origin (0, 0) as the center of dilation, followed by a translation of 3 units down.
Step-by-step explanation:
If two triangles are said to be similar, their corresponding angles are congruent and their corresponding sides are in the same ratio.
Given vertices of triangle ABC:
A = (0, 0)B = (1, 4)C = (5, 2)To maintain similarity but not maintain congruence, dilate triangle ABC (since dilation keeps the corresponding angles of both triangles the same).
Given vertex of triangle A'B'C':
B' = (2, 5)If ΔABC is dilated by a scale factor of 2, with the origin as the center of dilation, B' = (2, 8). If the triangle is then translated 3 units down, B' = (2, 5), which matches the given coordinate of point B'.
Therefore, the series of transformations is:
Dilation by a scale factor of 2 with the origin (0, 0) as the center of dilation.Translation of 3 units down.Mapping Rule: (x, y) → (2x, 2y - 3)
Therefore, the coordinates of point C' are:
⇒ C' = (2(5), 2(2) -3) = (10, 1)
a number from 15 to 23 is drawn at random p(a number divisable by 3)
Solution:
Given:
[tex]\begin{gathered} Sample\text{ space}=\lbrace15,16,17,18,19,20,21,22,23\rbrace \\ Total\text{ n}umber\text{ of possible outcomes}=9 \\ \\ \\ Numbers\text{ divisible by 3}=\lbrace15,18,21\rbrace \\ Number\text{ of required outcome}=3 \end{gathered}[/tex]Therefore, the probability of picking a number divisible by 3 is;
[tex]\begin{gathered} P(a\text{ number divisible by 3\rparen}=\frac{3}{9} \\ \\ Simplifying\text{ further,} \\ P(a\text{ number divisible by 3\rparen}=\frac{1}{3} \end{gathered}[/tex]A) In how many years will both companies have the same profit ?
B) Approximately what will that profit be ?
C) Which company’s profits are growing more quickly
Answer: 3 years
$60,000
Company B
Step-by-step explanation: Analyze the data on the graph
Explain what it means to solve a formula for a variable
The meaning of solving a formula for a variable is rewriting the formula in a way that the variable is "isolated" in one side of the formula, that is, one side of the equation has only the variable alone, without coefficients or other expressions.
For example, let's solve the formula below for the variable y:
[tex]\begin{gathered} 2x+3y+4=0\\ \\ 3y=-2x-4\\ \\ y=\frac{-2x-4}{3} \end{gathered}[/tex]We can see that in the formula above, the left side of the equation has only "y", this way it is solved for y.
Write an equation for the scenario. Solve the equation to answer the question. Show all steps.
5. Ben is signed up for drivers ed. He will spend 7.5 hours driving with an instructor and will also attend
a 2.5 hour class each week. When Ben completes the 30 total hours of instruction, he will take his
driver's test. How many weeks will he be required to attend class?
He will require 9 weeks to attend class, Based on given condition, 7.5 + 2.5x = 30.
What is Equation?Algebraically speaking, an equation is a statement that shows the equality of two mathematical expressions. For instance, the two equations 3x + 5 and 14, which are separated by the 'equal' sign, make up the equation 3x + 5 = 14.
According to question,
The time he spend as instructor is 7.5 and 2.4 hour class.
total hours = 30
7.5 +2.5x = 30
2.5x = 30- 7.5
2.5x = 22.5
x= 22.5/2.5
x = 9
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