Answers
Answer:
Option C. [tex]g(x)=(\frac{1}{3}x)^2[/tex]
Step-by-step explanation:
Stretch transformations of functionsIn which direction is the transformation happening?Given a function to start with, extra operations that are done outside of the given function, cause vertical transformations, whereas operations that are done inside of the function cause horizontal transformations.
Transformations on the outside
Ex. [tex]g(x)=(x^2)-3[/tex] is subtracting 3 outside, so its transformation is vertical.
Transformations on the inside
Ex. [tex]g(x)=(x-7)^2[/tex] is subtracting 7 inside, so its transformation is horizontal.
Stretch/compression transformationsFor any function, stretches or compressions occur by multiplying by positive numbers larger or smaller than 1.
Multiplying by numbers larger than one, quantities get larger, and multiplying by a positive number less than one, quantities get smaller.
For example, [tex]\$10*2 = \$20[/tex], but [tex]\$10*\frac{1}{2} = \$5[/tex]
For transformations, this intuition needs a small modification:
Operations outside: transformations happen "normally" as you would expect.Operations inside: transformations happen "backwards" from the natural way one might expect.Multiplying on the outside
When multiplying outside of the function, things outside happen normally, and since it is happening outside, it is in a vertical direction.
Ex. Multiplying outside by 3, things get larger (stretch) vertically to 3 times as much as (300% of) normal (away from a height of zero).
Multiplying outside by [tex]\frac{1}{2}[/tex] , things get smaller (compress) vertically down to [tex]\frac{1}{2}[/tex] as much as (50% of) normal (toward a height of zero).
"g" is lower than "f", so it may have been compressed vertically (see alternative solution at end).
Looking at the only options with multiplication outside (A & B), option B multiplies by 3 (greater than 1), so it would stretch "f" even taller.
Option A, does compress "f" vertically (by 1/3), but doesn't compress it enough to arrive at the point (3,1) defined on function "g". Note ordered pair (3,1) on "g", meaning when you input "3", you get out "1".
Putting 3 into the "f" function, f(3)=9. Since one-third (the transformation in Option A) of 9 is 3, not 1, Option A doesn't compress "f" enough.
Multiplying on the inside
When multiplying inside, transformations happen horizontally, and inside things happen "backwards".
So, if multiplying inside by 4, (...normally things get bigger...) things actually get smaller (compressed) horizontally, reduced to 1/4 (the reciprocal of 4) the size (toward a horizontal distance of zero).
If multiplying by a number less than one (but positive), like [tex]\frac{1}{3}[/tex] , (...things normally get smaller...) things will actually get larger (stretch) in the horizontal direction out to triple (the reciprocal of [tex]\frac{1}{3}[/tex]) as much as normal.
Looking options C & D (where multiplication happens inside), both have multiplication by a positive number less than one (... normally would make things smaller...), which will stretch "f" out horizontally.
How far has the function been stretched out horizontally?Looking at the red point (3,1), the blue function does have a height-matched point at (1,1).
Measuring horizontal distances, (3,1) is 3-units from the y-axis, whereas (1,1) is only 1-unit away. Thus, the "g" is 3 times as far as "f", meaning a horizontal stretch outward by 3.
Answer C multiplies inside by [tex]\frac{1}{3}[/tex], so it actually makes things 3 times bigger horizontally.
The correct answer is option C.
Verifying algebraicallyTo verify, from (3,1), put 3 into the option C g(x) -- it gives "1" as an output.
[tex]g(x)=(\frac{1}{3}x)^2\\g(3)=(\frac{1}{3}(3))^2\\g(3)=(1)^2\\g(3)=1\\[/tex]
An alternative solutionThis function could have been compressed vertically to obtain the red graph.
Note that f(3)=9.
The point (3,1), on the red function, is only at a height of 1 ... 1/9th the height of the blue function. We could compress the "f" vertically by 1/9th to transform the blue function into the red function.
Vertical transformations come from operations outside, and outside things behave "normally", so to vertically compress by 1/9th, just multiply on the outside by 1/9th.
Thus, an alternative answer to transform f(x) to g(x) is [tex]g(x)=\frac{1}{9}(x^2)[/tex]
Two last things:
Simplifying this function we just obtained we get [tex]g(x)=\frac{1}{9}x^2[/tex]
Returning to the function that we chose for our answer in option C, [tex]g(x)=(\frac{1}{3}x)^2[/tex]
[tex]g(x)=(\frac{1}{3}x)(\frac{1}{3}x)[/tex]
[tex]g(x)=\frac{1}{9}x^2[/tex]
Note that the transformation answer for this problem (Option C) and our alternative solution both simplify and match perfectly, so they both represent the same end result.
Related Questions
whats the answer? :(
Answers
Answer:
D) 8[tex]\frac{1}{2}[/tex]
Step-by-step explanation:
5[tex]\frac{2}{3}[/tex] ÷ [tex]\frac{2}{3}[/tex] = 8[tex]\frac{1}{2}[/tex]
If the domain (or pre-image) is (-4, -2), (5, 9), (-6, 2) and the range (or image) is
(-2, 4), (9,-5), (2, 6), what rotation was applied?
Answers
Answer: 90 degrees clockwise
This is equivalent to 270 degrees counterclockwise
The rule for either rotation is [tex](x,y) \to (y,-x)[/tex]
The x and y coordinates swap places, and the new second coordinate flips from positive to negative (or vice versa).
The diagram below shows an example of this for the point (-4,-2) rotating to (-2, 4). The center of the rotation is the origin (0,0).
what is the fraction 2500 of 3600
Answers
Answer:
25/36
Step-by-step explanation:
put 2500 over 3600 and you get 2500/3600 but you can simplify to get 25/36 but removing the 0's
f(a)=3a^2+2a+7 find the value of f(a)
Answers
Answer:
Step-by-step explanation:
If a = 0, f(a) = f(0) = 0 + 0 + 7 = 7;
If a = 1, f(1) = 3(1^2) + 2(1) + 7 = 12;
and so on. Normally an 'a' value would be given as part of this problem.
Actually, the value of f(a) is given and is f(a) = 3a^2+2a+7 for input value a.
Step 1: Choose the lowest common denominator.
02 03 04 05 06 08 010 012
3/4
50/80
O
Find the sum: -and
The expression written in equivalent form with a
common denominator is
The sum is
Answers
The expression written in equivalent form with a common denominator is -1/6
Adding fractionsFractions are written as ratio of two integers. For instance a/b is a fraction.
Given the sum of the fractions shown;
-3/4 and 5/8
Sum = -3/4 + 5/8
Sum = 5/8 - 3/4
Sum = 5-6/8
Sum = -1/8
Hence the sum of the given fraction -1/3 and 5/8 is -1/6
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Given that (1, 0, 1) is a solution to a system of three linear equations, which of the following is true about the system?
Answers
The true option about the system of linear equation is that the system can only be independent and consistent.
How to illustrate the equation?It should be noted that the system is consistent as it can illustrate more than a solution.
Also, it can't be independent and dependent at the same time as depicted in some of the option given.
In this case, true option about the system is that the system can only be independent and consistent.
The complete question is:
Given that (1, 0, 1) is a solution to a system of three linear equations, which of the following is true about the system?
The system can be either inconsistent or consistent.
The system can be either independent or dependent.
The system can only be independent and consistent.
The system can only be dependent and inconsistent.
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Help is needed with this math problem!
Answers
Answer:
expression for (g + h)([tex]x\\[/tex]) = [tex]2x[/tex] + [tex]3x^2\\[/tex] and for (g - h)([tex]x[/tex]) = [tex]2x[/tex] - [tex]3x^2[/tex]
value of (g .h)(2) = 24
Step-by-step explanation:
function operations: Functions with overlapping domains can be added, subtracted, multiplied and divided. If f(x) and g(x) are two functions, then for all x in the domain of both functions we can find the sum, difference, product and quotient
given :
g([tex]x[/tex]) = [tex]2x[/tex]
h([tex]x[/tex]) = [tex]3x^2[/tex]
we know that ;
(g + h)([tex]x[/tex]) = g([tex]x[/tex]) + h([tex]x[/tex])
= [tex]2x[/tex] + [tex]3x^2[/tex]
similarly;
(g - h)([tex]x[/tex]) = g([tex]x[/tex]) - h([tex]x[/tex])
= [tex]2x[/tex] - [tex]3x^2[/tex]
now,
(g . h)([tex]x[/tex]) = 2([tex]3x^2\\[/tex])
= 6[tex]x^2\\[/tex]
for (g .h)(2) = 6([tex]2^2[/tex])
= 6 × 4
= 24
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If y = 2x-3, then which of the following ordered pairs lies on the graph?
O (-3,0)
O (1,-1)
O (5,4)
Answers
Plug in the points and see if they satisfy the equation.
y = 2x - 3
(1, -1)
-1 = 2(1) - 3
-1 = 2 - 3
-1 = -1
So (1, -1) satisfies the equation.
(-3, 0)
0 = 2(-3) - 3
0 = -6 - 3
0 = -9
So (-3, 0) does not satisfy the equation.
(5, 4)
4 = 2(5) - 3
4 = 10 - 3
4 = 7
So (5, 4) does not satisfy the equation.
Therefore your answer is (1, -1).
Answer:
the answer is 5,4
Write each of the following numerals in base 10. For base twelve, T and E represent the face values ten and eleven, respectively.
a. 421 five
b. 11101 two
c. 13Ttwelve
Answers
(a)
421₅ = 4×5² + 2×5¹ + 1×5⁰
… = 100 + 10 + 1
… = 111
(b)
11101₂ = 1×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰
… = 16 + 8 + 4 + 0 + 1
… = 29
(c)
13T₁₂ = 1×12² + 3×12¹ + 10×12⁰
… = 144 + 36 + 10
… = 190
What does the ratio 6 to 8 represent in the situation?
Answers
Answer:
its 80
Step-by-step explanation:
the ratio is 80 hope this helps!
Rick bought some tetra fish and goldfish for his fish tank. He bought twice as many goldfish as he did tetras. Each goldfish costs $1.50, and each tetra fish costs $2.00. He spent $20 on the fish. Which system can be used to represent x, the number of tetra fish, and y, the number of goldfish, he purchased?
x = 2 y. 2 x + 1.5 y = 20
2 x = y. 2 x + 1.5 y = 20
x = y. 2 x + 1.5 y = 20
x + y = 20. 2 x + 1.5 y
Answers
Answer:
B. 2x = y; 2x +1.5y = 20
Step-by-step explanation:
The two given relations can be written as two equations, one for the numbers of fish, and one for their cost.
__
number of fishThe number of goldfish (y) is twice the number of tetras (x), so one equation can be ...
2x = y
cost of fishThe tetras (x) are $2 each, and the goldfish (y) are $1.50 each. Their total cost is the sum of products of the number of fish and the cost of each. That total is given as $20, so the other equation could be ...
2x +1.5y = 20
Then the appropriate system of equations is ...
2x = y2x +1.5y = 20_____
Additional comment
The solution can be found by substituting for 2x to get ...
y +1.5y = 20 . . . . . . substitute for 2x
y = 20/2.5 = 8 . . . . divide by the coefficient of y
x = y/2 = 4 . . . . . . . find the number of tetras
Rick bought 4 tetras and 8 goldfish.
__
The wording "twice as many goldfish as tetras" easily causes confusion. It is helpful to reword it as "goldfish were twice the number of tetras."
What is the value of the expression
(12+x) +
4+x
when X= 4 and y=-8?
OA) 4
OB)
OC)
OD 10
Answers
Step-by-step explanation:
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The value of the expression x
4
−8x
3
+18x
2
−8x+2, when x=2+
3
, is:
Medium
Solution
verified
Verified by Toppr
Correct option is B)
Let f(x)=x
4
−8x
3
+18x
2
−8x+2
Substituting x=2+
3
, we get
f(2+
3
)=(2+
3
)
4
−8(2+
3
)
3
+18(2+
3
)
2
−8(2+
3
)+2
=(2+
3
)
2
((2+
3
)
2
+18)−8(2+
3
)((2+
3
)
2
+1)+2
=(7+4
3
)(4
3
+25)−8(2+
3
)(8+4
3
)+2
=175+48−128−96+2
=1
Hence, option B is correct.
David throws a ball up in the air. The graph below shows the height of the ball hh in feet after tt seconds. What is the ball’s greatest height?
HELP ASAP!
Answers
The greatest height is 9 feet.
What is projectile?
A projectile is any object thrown into space upon which the only acting force is gravity.
As, from the graph we can see that
The highest point the ball reaches shown by the coordinates (0.75, 9)
As, the maximum height is shown by y- axis(vertical axis).
Hence, the greatest height is 9 feet.
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Based on the circle graph,how many pounds of milk and eggs does the average American consume each year?PLEASE EXPLAIN IMAGE IS PROVIDED
E.300
F.368
G.460
H.552
Answers
About 492 pounds of milk and egg were consumed each year
Pie charts and percentagesGiven the following parameters from the chart
Total consumption = 1640 pounds per person
The percentage of milk and egg = 100 -(38 + 10 + 10 +8 + 4)
The percentage of milk and egg = 30%
Determine the amount consumed each year
Amount of milk and egg consumed = 30% of 1640
Amount of milk and egg consumed = 492 pounds
Hence about 492 pounds of milk and egg were consumed each year
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A school bus is moving 26.8 m/s on flat ground when it begins to accelerate at 4.73 m/s^2. How much time does it take to travel 95.1 m?
Answers
The school bus will take 5.34 seconds to travel 95.1 m if the bus is moving 26.8 m/s on flat ground when it begins to accelerate at 4.73 m/s².
What is a quadratic equation?Any equation of the form [tex]\rm ax^2+bx+c=0[/tex] where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.
As we know, the formula for the roots of the quadratic equation is given by:
[tex]\rm x = \dfrac{-b \pm\sqrt{b^2-4ac}}{2a}[/tex]
We have:
A school bus is moving 26.8 m/s on flat ground when it begins to accelerate at 4.73 m/s².
From the second equation of motion:
[tex]\rm s = u + \dfrac{1}{2}at^2[/tex]
We have given:
s = 95.1 m
u = 26.8 m/s
a = 4.73 m/s²
[tex]\rm 95.1 = 26.8 + \dfrac{1}{2}(4.73)t^2[/tex]
The above equation is a quadratic equation:
After simplification:
[tex]\rm \dfrac{4.73}{2}t^2=68.3[/tex]
[tex]\rm t^2=28.879[/tex]
t = 5.373 seconds ≈ 5.34 seconds
Thus, the school bus will take 5.34 seconds to travel 95.1 m if the bus is moving 26.8 m/s on flat ground when it begins to accelerate at 4.73 m/s².
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A rental car agency charges $230 per week plus $0.25 per mile to rent a car. How many miles can you travel in one week for $415?
The number of miles you can travel in one week for $415 is
Answers
Answer:
740 miles
Step-by-step explanation:
$415-$230=$185
$185÷0.25 per miles =740 miles
Find the prime factorization of the composite number 10692
Answers
When decomposing into prime factors, it is when a number is taken out of its half, third, fifth and seventh, and it is also possible for the same number number.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \left.\begin{matrix} \blue{10692 }\\ \ 5346\\ \ 2673\\ \ \ \ 891\\ \ \ \ 297\\ \ \ \ \ 99\\ \ \ \ \ 33\\ \ \ \ \ 11\\ \ \ \ \ 1 \end{matrix}\right|\begin{matrix} 2\\ 2\\ 3\\ 3\\ 3\\ 3\\ 3\\ 11\\ \: \end{matrix} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf 10692=2\cdot2\cdot3\cdot3\cdot3\cdot3\cdot3\cdot11=\bf{2^{2}\cdot3^{5}\cdot11 } \end{gathered}$}[/tex][tex]\huge \red{\boxed{\green{\boxed{\boldsymbol{\purple{Pisces04}}}}}}[/tex]
find the measure of angle ACB
Answers
Answer:
angle ACB=45º
Step-by-step explanation:
angle 135º lies on a straight line
a straight line = 180º
thus, angle ACB= 180-135=45º
Given the data set (4,85),(7,92),(14,110), which of the following equation best represents a line of best fit
Answers
You buy a CD for $9000 that earns $172.92 in interest every 2 months. What is the
annual yield on your investment?
Answers
Answer:
$4,000.343
Step-by-step explanation:
Divided the months of the year and caculate the intrest.
Consider the following hypothesis test: 0 : 12 a : 12 HH A sample of 25 provided a sample mean x = 14 and a sample standard deviation s = 4.32. a. Compute the value of the test statistic. b. Use the t distribution table to compute a range for the p-value. c. At a = 0.05, what is your conclusion? d. What is the rejection rule using the critical value? What is your conclusion?
Answers
a) The test statistic is given as t = [tex]\dfrac {14-12}{\dfrac{4.32}{\sqrt25}}}[/tex]= 2.315
b) The pa value will be P = P( t₂₄ > 2.315 ) = 0.015
c) If we compare the p-value and the significance level is given we see that so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is higher than 12 at 1% of significance
What is a null hypothesis?In null hypotheses, there is no relationship between the two phenomena under the assumption that it is not associated with the group. And in alternative hypotheses, there is a relationship between the two chosen unknowns.
X= represents the sample mean
s = represent the sample standard deviation
n = 25 sample size
[tex]\mu_o[/tex] = 12 represent the value that we want to test
[tex]\alpha[/tex] = 0.05 represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p-value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the mean is higher than 12, the system of hypothesis would be:
Null hypothesis: ≤ 12
Alternative hypothesis: > 12
If we analyze the size for the sample is > 30 but we don't know the population deviation so is better to apply a t-test to compare the actual mean to the reference value, and the statistic is given by:
[tex]t=\dfrac{X-\mu_o}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to a specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]\dfrac {14-12}{\dfrac{4.32}{\sqrt25}}}[/tex]= 2.315
P-value:-
The first step is calculate the degrees of freedom, on this case:
df = n - 1 =25 - 1 = 2
Since is a one side right tailed test the p value would be:
[tex]p_v[/tex] = P( t₂₄ > 2.315 ) = 0.015
Conclusion
If we compare the p-value and the significance level is given [tex]\alpha[/tex]= 0.05 we see that [tex]p_v < \alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is higher than 12 at 1% of significance.
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Briana monitors the number of E. coli infections reported in a certain
neighborhood in a given week. The recent numbers are shown in
this table:
According to her reports, the reported infections are growing at a
rate of 50%.
If the number of infections continues to grow exponentially,what
will the number of infections be in week 6?
Answers
An equation is formed of two equal expressions. The number of infections in 6 weeks is 456.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
The general equation of an exponential function is written as,
y = A(B)ˣ
Since the infection is increasing at a rate of 50%, therefore, the rate of growth will be,
B = 1.50
Substitute any value then we will get,
40 = A(1.50)⁰
A = 40
Thus, the exponential equation of the growth will be,
y = 40(1.50)ˣ
Now, the number of infections in 6 weeks will be,
y = 40(1.50)⁶
y = 455.625 ≈ 456
Hence, the number of infections in 6 weeks is 456.
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Someone explain the equation
Answers
Answer:
An equation connects two expressions with an equal sign. It shows that the two expressions are equal to each other, like 5 = 5, or for variables, 5x = 2x, which means that x = 0.
Brainliest, please :)
Answer:
An equation is a problem consist sting of many numbers mostly division or mulitiplication
Step-by-step explanation:
Study each of the growing geometrical patterns. Then do the following for each pattern. (I) Give a verbal description of how the pattern is growing. (ii)Set up a table of values.(III) Write down a numerical description for each pattern (you must relate the number of the frame to the number of figures in the frame). (IV) For each pattern, write down a formula that can be used to calculate the number of building blocks in any term. (v) Write down the value of the 10th frame. Use your formula to determine how many building blocks there are in the 100th frame of each growing pattern. M M Δ Δ Δ 2 (6) (6) (8) [40]
Answers
A description of how the pattern is growing is that the pattern is growing by the addition of two blocks every time.
How to illustrate the pattern?A numerical description of the pattern is:
1 - 2 blocks.
2 - 4 blocks.
3 - 6 blocks
4 - 8 blocks.
The formula that can be used to calculate the number if building blocks in any term will be 2n. For example, the number of blocks for the 5th term will be:
= 2n = 2 × 5 = 10
The value for the 10th frame will be:
= 2n
= 2 × 10 = 20
The value for the 100th frame will be:
= 2n
= 2 × 100 = 200
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dollars per pint
pints per dollar.
▼
Find the unit rates. If necessary, round your answers to the nearest hundredth.
$6.79 for 16 pints
Answers
Answer:
0.42 dollar per pint
2.36 pints per dollar
Solve the inequality graphically. 3^-8 > 6^-8
a. x < 1
b. x > 1
c. x > 0
d. x < 0
Answers
THE ANSWER IS -> b. x > 1
hope it helps
thanks
Answer: c. x>0
Step-by-step explanation:
Emma bought 1305 bags of carrot for 16910 dollars. Find the unit rate in terms of bags per dollar.
Answers
Answer: $12.96
Given:
1305 bags of carrots purchased
Total cost = $16,910
Step-by-step explanation:
To find the unit cost, divide 16910 by 1305 to find the cost of one bag of carrots:
[tex]\frac{16910}{1305} = 12.96[/tex] (rounded to nearest hundredth)
Here are the first five numbers in a sequence.
47 41
(a) Find the first negative number in the sequence.
35
29
23
Answers
Answer:
- 1
Step-by-step explanation:
note there is a common difference of 6 between consecutive terms , that is
23 - 29 = 29 - 35 = 35 - 41 = 41 - 47 = - 6
by subtracting 6 from the previous term we get
23 - 6 = 17
17 - 6 = 11
11 - 6 = 5
5 - 6 = - 1
the first negative number in the sequence is - 1
Identifying an Error
Identify the error in the student solution shown
below. Find the correct answer.
2ln(x)= In(3x) - [In(9) - 2ln(3)]
In(x²) In(3x)-[In(9) - In(9)]
In(~2) 1/25)
Answers
The incorrect step is ln(x²) = ln(3x/0) because ln(x/y) = lnx - lny and the solutions are x = 0, 3
What is a logarithm?It is another way to represent the power of numbers, and we say that 'b' is the logarithm of 'c' with base 'a' if and only if 'a' to the power 'b' equals 'c'.
[tex]\rm a^b = c\\log_ac =b[/tex]
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
We have:
2ln(x) = In(3x) - [In(9) - 2ln(3)]
In(x²) = In(3x) - [In(9) - In(9)]
ln(x²) = ln(3x) - 0
ln(x²) = ln(3x/0) (incorrect step)
ln(x²) = ln(3x)
x² = 3x (correct step)
x² - 3x =0
x(x - 3) = 0
x = 0, 3
Thus, the incorrect step is ln(x²) = ln(3x/0) because ln(x/y) = lnx - lny and the solutions are x = 0, 3
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Olympic Enterprises has the following inventory data: Assuming average cost, what is the cost of goods sold for the June 14 sale?
Answers
Assuming average cost of $55.5, the cost of goods sold by Olympic Enterprises for the June 14 sale is equal to $444.
How to calculate cost of goods?In Financial accounting, the cost of goods sold for a business firm can be calculated by multiplying the total quantity of goods by an average cost. Mathematically, this is given by:
Cost of goods = quantity × average cost
Note: We would assume an average cost of $55.5.
Substituting the parameters into the formula, we have:
Cost of goods sold = 8 × 55.5
Cost of goods sold = $444.
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