Show why, for linear functions, a vertical translation is equivalent to a horizontal translation. For a linear function, what horizontal translation is equivalent to a vertical translation of 3 units up?

Answer:

B. g(x) = 3x^3.

Step-by-step explanation:

In general   a f(x)  stretches vertically the graph of f(x) by a factor a.

The new function after a vertical stretch of the cubic function F(x) = x^3 is G(x) = 3x³, which multiplies the output of the original function by 3.

When you apply a vertical stretch to the cubic function F(x) = x³, you multiply the output of the function, not the input. A vertical stretch by a factor of 3 would mean that each output value is tripled. Therefore, the correct equation for the new function would be G(x) = 3x³.

Option A is incorrect because (1/3x)³ represents a horizontal stretch by a factor of 3. Option C, (3x)³, represents a horizontal shrink by a factor of 1/3, and results in steeper slopes than the original function, which is the opposite effect of a vertical stretch. Option D, 1/3x³, would be a vertical compression by a factor of 1/3, not a stretch.

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Answer:

1. 90 sq. cm

2. 96 sq. cm

3. 72 sq. cm

4. 72 sq. cm

5. 92 sq. cm

6. 68 sq. cm

Step-by-step explanation:

Answer:

A. 16.12 ft  B.  9.64 ft

Step-by-step explanation:

For both of these, you need Pythagorean's Theorem because we have right triangles.  In part A, we are given 2 legs and are asked to find the length of the hypotenuse:

[tex]14^2+8^2=PR^2[/tex] and

[tex]196+64=PR^2[/tex]

and PR = 16.12 ft

In part B, we are given the length of the hypotenuse and side PQ remains the same:

[tex]14^2+QR^2=17^2[/tex] and

[tex]196+QR^2=289[/tex]

[tex]QR^2=289-196[/tex] so

QR = 9.64 ft

Answer:

  PPF, PFF

Step-by-step explanation:

There are several ways you can list all the possible combinations. A couple of my favorite are a) use a binary counting sequence; b) use a gray code counting sequence.

Using the first method, the binary numbers 000 to 111 can be listed in numerical order as 000, 001, 010, 011, 100, 101, 110, 111. Letting 0=P and 1=F, the ones missing from your list are the ones in italics in my list.

Using the second method, we change the right-most character, then the middle one, and finally the left-most character so there is one change at a time: 000, 001, 011, 010, 110, 111, 101, 100.

After you have a list of all possible combinations, it is a simple matter to compare the given list to the list of possibilities to see which are missing.

The correct outcome possibilities for the missing boxes in the given scenario are:

1. PPP (all three tires pass)

2. PPF (two tires pass, one fails)

3. PFP (two tires pass, one fails)

4. FPP (two tires pass, one fails)

5. FFP (two tires pass, one fails)

6. FPF (two tires pass, one fails)

7. FFF (all three tires fail)

To determine the missing outcomes, we must consider all possible combinations of passes (P) and failures (F) for three tires. Since each tire can either pass or fail, there are [tex]2^3 = 8[/tex] possible outcomes. We already have five of these outcomes listed, and we need to find the remaining two.

The missing outcomes must include all combinations of passes and failures that have not been listed yet. Since we have all combinations with exactly two passes and one fail, and we have the combination with three passes and three fails, the remaining combinations must have exactly one pass and two fails. These combinations are:

- PFF (one tire passes, two fail)

- FPF (one tire passes, two fail)

Therefore, the complete list of outcomes is:

1. PPP (all three tires pass)

2. PPF (two tires pass, one fails)

3. PFP (two tires pass, one fails)

4. FPP (two tires pass, one fails)

5. FFP (two tires pass, one fails)

6. FPF (two tires pass, one fails)

7. PFF (one tire passes, two fail)

8. FFF (all three tires fail)

Each of these outcomes represents a distinct possibility when testing three tires, and together they account for every possible result of the wear and tear test for the three tires.

Answer:

Option D. $1 million

Step-by-step explanation:

we know that

Reserve Ratio, it is the percentage of deposits which commercial banks are required to keep as cash

Find the 20% of $5 million

20%=20/100=0.20

0.20*5,000,000=$1,000,000

so

$1 million

The required reserve ratio is what banks must keep from a deposit. Given a 20% reserve ratio and a $5 million deposit, banks must reserve $1 million. Using the money multiplier formula, the remaining funds could theoretically create as much as $25 million in money supply.

The required reserve ratio is the percentage of deposits that a bank must hold as reserves. In this case, the required reserve ratio is 20 percent. The rest of the deposit (80 percent) can be loaned out or invested by the bank, which will create additional deposits and thus increase the money supply.

If a commercial bank receives a $5 million deposit and the required reserve ratio is 20 percent, the bank must keep $1 million (20 percent of $5 million) as reserve. The rest, $4 million, can be loaned out or invested.

However, this doesn't simply stop at $4 million. This loaned money will eventually be deposited back into the banking system (let's assume to another bank), which can then loan out 80% of that deposited money, and this cycle can continue. In an extremely simplified scenario, you can continue this process until the banks can no longer lend out money.

To simplify this scenario, the money multiplier formula can be used. The money multiplier formula is 1 divided by the reserve ratio. In this case, it's 1 / 0.20 = 5.  Therefore, a $5 million deposit can potentially lead to a $25 million increase in the money supply, so the answer is (a) $25 million.

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Answer:

[tex]14\ candles[/tex]

Step-by-step explanation:

step 1

Find the volume of the cylindrical mold

The volume is equal to

[tex]V=\pi r^{2} h[/tex]

we have

[tex]r=3.4\ cm[/tex]

[tex]h=6\ cm[/tex]

assume

[tex]\pi=3.14[/tex]

substitute

[tex]V=(3.14)(3.4)^{2}(6)[/tex]

[tex]V=217.79\ cm^{3}[/tex]

step 2

Find the volume of the wax

The volume is equal to

[tex]V=(15)(12)(18)[/tex]

[tex]V=3,240\ cm^{3}[/tex]

step 3

Divide the volume of the wax by the volume of the cylindrical mold, to calculate the number of candles

[tex]3,240/217.79=14.9\ candles[/tex]

Round down

[tex]14\ candles[/tex]

let's use engineering notation for the sake of brevity.

1 trillion is 1,000,000,000,000, or just 1E12, twelve zeros.

1 million is then 1E6, six zeros.

we know the discharge for one day is 3E12 gallons, and that'd do just fine for country A for 5 months.  How many gallons in 1 month only?

[tex]\bf \begin{array}{ccll} gallons&months\\ \cline{1-2} 3E12&5\\ x&1 \end{array}\implies \cfrac{3E12}{x}=\cfrac{5}{1}\implies 3E12=5x \\\\\\ \cfrac{3E12}{5}=x\implies 6E11=x\implies 600000000000=x[/tex]

if there are 200million inhabitants in A, namely 200E6 or 2E8 inhabitants, how many gallons per inhabitant from all those 6E11 gallons?

[tex]\bf \begin{array}{ccll} gallons&households\\ \cline{1-2} 6E11&2E8\\ x&1 \end{array}\implies \cfrac{6E11}{x}=\cfrac{2E8}{1}\implies 6E11=2E8x \\\\\\ \cfrac{6E11}{2E8}=x\implies \cfrac{600000000000}{200000000}=x\implies 3000=x[/tex]

Answer:

156.25% =P

Step-by-step explanation:

Is means equals and of means multiply

50 = P * 32

Divide each side by 32

50/32 = 32P/32

1.5625= P

Now we need to change it to percent form by multiplying by 100%

1.5625 * 100% = P

156.25% =P

Answer:

Part a) The polynomial for the area including the frame is

[tex](4x^{2}+72x+320)\ ft^{2}[/tex]

Part b) The polynomial for the perimeter including the frame is

[tex](72+8x)\ ft[/tex]

Part c) The perimeter of the picture including the frame when the width of the frame is 2 inches is equal to [tex]88\ ft[/tex]

Part d) The area of the picture including the frame when the width of the frame is 2 inches is equal to [tex]480\ ft^{2}[/tex]

Step-by-step explanation:

Let

x-----> the width of the frame

L-----> the length of the picture

W-----> the width of the picture

Part a) What is a polynomial for the area including the frame?

we have

The dimensions of the picture are

[tex]L=20\ in[/tex]

[tex]W=16\ in[/tex]

The area including the frame is equal to

[tex]A=(20+2x)(16+2x)\\ \\A=320+40x+32x+4x^{2}\\ \\A=(4x^{2}+72x+320)\ ft^{2}[/tex]

Part b) What is a polynomial for the perimeter including the frame?

we have

The dimensions of the picture are

[tex]L=20\ in[/tex]

[tex]W=16\ in[/tex]

The perimeter including the frame is equal to

[tex]P=2[(20+2x)+(16+2x)]\\ \\P=2[36+4x]\\ \\P=(72+8x)\ ft[/tex]

Part c) What is the perimeter of the picture including the frame when the width of the frame is 2 inches

we have

[tex]P=(72+8x)\ ft[/tex]

For x=2 in

substitute

[tex]P=72+8(2)=88\ ft[/tex]

Part d) What is the area of the picture including the frame when the width of the frame is 2 inches

we have

[tex]A=(4x^{2}+72x+320)\ ft^{2}[/tex]

For x=2 in

substitute

[tex]A=(4(2)^{2}+72(2)+320)=480\ ft^{2}[/tex]

The perimeter of a two-dimensional figure is the distance covered around it.

The polynomial for the area including the frame is [tex]\rm 4x^2+72x +320[/tex].

The polynomial for the perimeter including frame is [tex]\rm 72+8x[/tex].

The perimeter of the picture including the frame when the width of the frame is 2 inches is 88 feet.

The area of the picture including the frame when the width of the frame is 2 inches is 480 feet.

You are framing a picture with a frame of equal width on each side.

The width of the frame is 2 inches.

The length is 20 inches and the width is 16 inches.

The width of the frame is 2 inches.

The perimeter of a two-dimensional figure is the distance covered around it.

Let the width of the frame be x.

The length of the picture be L.

The width of the picture is W.

1. What is a polynomial for the area including the frame?

[tex]\rm Area \ of \ the \ frame = length \times width\\\\ Area \ of \ the \ frame = (20+2x) (16+2x)\\\\ Area \ of \ the \ frame = 320+40x+32x+4x^2\\\\ Area \ of \ the \ frame = 4x^2+72x +320[/tex]

The polynomial for the area including the frame is [tex]\rm 4x^2+72x +320[/tex].

2. What is a polynomial for the perimeter including the frame?

[tex]\rm Perimeter \ of \ the \ picture = 2 (length + width)\\\\ Perimeter \ of \ the \ picture = 2(20+2x+16+2x)\\\\ Perimeter \ of \ the \ picture = 2(36+4x)\\\\ Perimeter \ of \ the \ picture = 72+8x[/tex]

The polynomial for the perimeter including frame is [tex]\rm 72+8x[/tex].

3. What is the perimeter of the picture including the frame when the width of the frame is 2 inches.

[tex]\rm Perimeter \ of \ the \ picture = 72+8x\\\\Perimeter \ of \ the \ picture = 72+8(2)\\\\ Perimeter \ of \ the \ picture = 72+16\\\\Perimeter \ of \ the \ picture = 88\\\\[/tex]

The perimeter of the picture including the frame when the width of the frame is 2 inches is 88 feet.

4. What is the area of the picture including the frame when the width of the frame is 2 inches.

[tex]\rm Area \ of \ the \ frame = 4x^2+72x +320\\\\ Area \ of \ the \ frame = 4(2)^2+72(2)+320\\\\ Area \ of \ the \ frame=480[/tex]

The area of the picture including the frame when the width of the frame is 2 inches is 480 feet.

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Answer:

55 degrees.

Step-by-step explanation:

We can use the postulate if two angles are congruent in both triangles than the triangle is congruent.

That means we can plug in 80 and 45 for triangle QRS.

Then, we can find angle 1 by subtracting the sum of those numbers from 180.

80 + 45 = 125

180 - 125 = 55

Answer:

D. An exponential function, because exponential functions increase at a nonconstant rate

Step-by-step explanation:

Each month Tai's brother grows more than he grew the previous month.

We can't model his growth by a linear function, because linear functions increase at a constant rate.

The model must be an exponential function.

For example, if the boy's height at Month 0 were 100 units, a model like h = 100(1.1)ⁿ would give the following results.

[tex]\begin{array}{ccc}\textbf{Month} & \textbf{Height} & \textbf{Diff.}\\0 & 100 & \\1 & 110 & 10\\2 & 121 & 11\\3 & 133 & 12\\4 & 146 & 13\\\end{array}[/tex]

An [tex]\boxed{\textbf{ exponential function }}[/tex] is consistent with a monthly change in height that increases each month.

Answer:

541.8 cm³

Step-by-step explanation:

Volume of a prism is the height times the area of the base.

V = hA

Area of a rectangle is width times length, so:

V = hwl

Given h = 8.8 cm, w = 4.7 cm, and l = 13.1 cm:

V = (8.8 cm) (4.7 cm) (13.1 cm)

V = 541.8 cm³

Answer:

[tex]541.8cm^{3}[/tex]

Step-by-step explanation:

V = Bh or V = lwh

substitute given measurements, then simplify.[tex]13.1cm * 4.7cm * 8.8cm = 541.8cm^{3}[/tex]

Answer:

21x^2 - 41x + 10

Step-by-step explanation:

The area of a rectange is length x width.

L = 7x-2

W = 3x-5

So, you would do (7x-2)x(3x-5).

To do this, you can do FOIL.

F (first times first) - (7x)(3x)=21x^2

O (outside times outside) - (7x)(-5)= -35x

I (inside times inside) - (-2)(3x)= -6x

L (last times last) - (-2)(-5) = 10

So, it is 21x^2 - 35x - 6x +10

Then you combine like terms giving you:

21x^2 - 41x + 10

Answer: 21x^2 - 41x + 10

Step-by-step explanation: If the lengths are even then the number would be angle.

Answer:

  a) monthly payment: $177.50

  b) total amount paid: $31,950

  c) toward principal: $17,500; toward interest: $14,450

Step-by-step explanation:

a) The amount of the monthly payment (A) is computed from the principal (P), the annual interest rate (r) and the number of years (n) using the formula ...

  A = P·(r/12)/(1 -(1 +r/12)^(-12n))

Filling in your numbers, we can use r/12 = 0.09/12 = 0.0075, and 12n = 12·15 = 180:

  A = $17500·0.0075/(1 - 1.0075^-180) ≈ $177.50

__

b) The total payment over the term of the loan is 180 of these monthly payments:

  180·$177.50 = $31,950

__

c) $17,500 is paid toward the principal.

 $14,450 is paid toward interest.

Answer:

1-C, 2-B, 3-A, 4-E, 5-D

Step-by-step explanation:

It is convenient to let a graphing calculator or spreadsheet compute the standard deviations for you. (Some will compute sample standard deviation; some will compute population standard deviation. It makes no difference to the ordering, as long as the same computation is used for all data sets.)

The correct order of the data sets from smallest to largest standard deviation is: C, B, A, E, D.

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Answer:

b.

Step-by-step explanation:

x-intercepts are found by factoring.  We will use standard factoring here since this one is straightforeward and has real zeros as its solutions.  

In our equation,

a = 2

b = 2

c = -4

The rules are to take a * c and then find the factors that number, determine which combination of those factors will give you the linear term (the term with the single x on it), and rearrange those signs accordingly.  Let's start with that:

Our a * c is 2 * -4 = -8.

We need the factors of |-8|:  1,8 and 2,4

Some combination of those factors needs to give us a +2x.  2,4 will work as long as the 4 is positive and the 2 is negative.

Now we put them back into the equation, the absolute value of the larger number first:

[tex]2x^2+4x-2x-4=0[/tex]

Now group the terms in sets of 2 without moving any of them around:

[tex](2x^2+4x)-(2x-4)=0[/tex]

In each set of parenthesis, pull out what is common to both terms.  In the first set, the 2x is common, and in the second set, the 2 is common:

[tex]2x(x+2)-2(x+2)[/tex]

Now what is common between both terms is the (x + 2), so pull that out, grouping what is remaining in its own set of parenthesis:

[tex](x+2)(2x-2)=0[/tex]

To find the zeros, remember that the Zero Product Property tells us that for that equation above to equal zero, one of those factors has to equal zero, so:

x + 2 = 0 or 2x - 2 = 0.  Solve both for x:

x = -2 so the coordinate is (-2, 0)

2x - 2 = 0 and

2x = 2 so

x = 1 so the coordinate is (1, 0)

Answer:

D

Step-by-step explanation:

Congruent means the same. Translating it just moves it somewhere else.

Answer: D) The two triangles are congruent because a translation does not change size and shape.

Step-by-step explanation:

Since all rigid motions create congruent figures , it means it do not change the shape and size of the figure.

So, translation does not change size and shape.

If a triangle is drawn and then translated, then they are congruent because a translation does not change size and shape.

3.75 miles or 3 3/4 would be your answer

Answer:

Option C is correct.

Step-by-step explanation:

The given equation or function is linear if the variables x and y have degree zero or 1.

The given equation is 5x^4+5y =9

here x has degree =4 and y has degree = 1, but for linear equation both x and y variables must have degree zero or 1.

So, Option C is correct.

Answer:

A. The graph of the function increases and decreases over its domain.

Step-by-step explanation:

The graph is attached. Some places, it has positive slope (is increasing); other places it has negative slope (is decreasing).

Answer:

1. 22 in

2. 11.304 units

Step-by-step explanation:

Formulas for the circumference are

C = 2πr

C = πd

___

1. Put the given numbers in the appropriate formula:

C = 2·(22/7)·(7/2 in) = 22 in

__

2. Put the given numbers in the appropriate formula:

C = 3.14·3.6 = 11.304 . . . units

Answer:

Domain: -6≤x<0 or 0<x≤2

Range: 1<x≤6

Step-by-step explanation:

Domain = The set of starting numbers - the x values.

Range = The set of numbers it becomes - the y values.

For the domain, it starts at -6, on the left and ends at 0; however, it doesn't include 0. Then is starts at 0, exclusive, and ends at 2.

For the range, it starts at 1, but doesn't include 1, and ends at 6, inclusive.

The domain of the given piecewise-defined function is -6≤x≤0 or 0. The domain of a function refers to the set of all possible input values or x-values of the function, while the range refers to the set of all possible output values or y-values of the function. In a piecewise-defined function, each piece has its own domain and range.

The domain of a function refers to the set of all possible input values or x-values of the function, while the range refers to the set of all possible output values or y-values of the function. In a piecewise-defined function, each piece has its own domain and range.

To determine the domain of the given piecewise-defined function, we need to identify the intervals on the x-axis where each piece of the function is defined. From the graph, we can see that the function is defined from -6 to 0, and also from 0 to 2. So the domain is -6≤x≤0 or 0

The range of the function is the set of all y-values that correspond to the x-values in the domain. Looking at the graph, we see that the function takes on values between 1 and 6, inclusive. So the range is 1≤y≤6.

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Could you please elaborate? Is there an equation?

Answer: What do you need to know?

Step-by-step explanation:

Is there more to the question,

Answer:

It is D

Step-by-step explanation:

Answer:

[tex]f(x)=x^2+16[/tex]

Step-by-step explanation:

By the conjugate rule, if -4i is a root, then so is +4i.  So we have 2 roots, thus, we have a second degree polynomial (namely, a quadratic).  If

x = -4i, then

x + 4i is a root.

If

x = 4i, then

x - 4i is a root.  

Having (x - 4i)(x + 4i) as roots, we can now FOIL them together to get a polynomial of least degree.

FOILing gives us

[tex]x^2+4ix-4ix-16i^2[/tex]

Notice that the +4ix and the -4ix cancel each other out, leaving you with

[tex]x^2-16i^2[/tex]

Since

[tex]i^2=-1[/tex]

we can make the substitution:

[tex]x^2-16(-1)[/tex]

which simplifies to

[tex]x^2+16[/tex]

In function notation form:

[tex]f(x)=x^2+16[/tex]

Final answer:

The polynomial function in standard form with the root -4i is f(x) = x² + 16. We derive this by realizing that the conjugate, 4i, is also a root and multiplying the factors (x + 4i)(x - 4i), simplifying to the quadratic polynomial.

Explanation:

To write a polynomial function with the given root -4i, we must take into consideration that complex roots come in conjugate pairs for polynomials with real coefficients. Hence, if -4i is a root, then its conjugate 4i must also be a root. The factors of the polynomial corresponding to these roots are (x + 4i) and (x - 4i).

We can find the polynomial by multiplying these two factors:

(x + 4i)(x - 4i)

Applying the difference of squares, we get:

x² - (4i)²

x² - (-16)

x² + 16

The polynomial function in standard form with the given root -4i is:

f(x) = x² + 16

Answer:

x = 31

Step-by-step explanation:

The sides of an isosceles trapezoid are the same length, so ...

5x -32 = 2x +61

3x = 93 . . . . . . . . add 32-2x

x = 31 . . . . . . . . . . divide by 3

Answer:

y=1/4x-1

Step-by-step explanation:

First step to determine the equation of a line is is to determine its slope.

We see the line passes through points (4,0) and (0,-1), that means its slope is:

S = (0 - -1) / (4 - 0) = 1/4

Since there's only one choice with a slope of 1/4, the choice is easy :-)

But we can also verify the equation by checking if it validates the given points. So, what's the value of y if x = 0?

y = (1/4)0 - 1 = 0 - 1 = -1  Validated.

And when x = 4?

y = (1/4)4 - 1 = 1 - 1 = 0  Validated too.

Answer:

[tex]x-4y=4[/tex]

Step-by-step explanation:

The given function passes through: (4,0) and (0,-1).

The equation is of the form;

y=mx+b

where b=-1 is the y-intercept.

The equation now becomes:

y=mx-1

We substitute the point (4,0) into the function to obtain;

0=m(4)-1

0+1=4m

1=4m

[tex]m=\frac{1}{4}[/tex]

Therefore the equation is:

[tex]y=\frac{1}{4}x-1[/tex]

Multiply through by 4 to get;

[tex]4y=x-4[/tex]

The equation in standard form is;

[tex]x-4y=4[/tex]

Answer:

x=4,-2/3

Step-by-step explanation:

To solve for x, we first subtract 8 from both sides.

So 3x^2-10x-8=0

Using the quadratic formula, we get

X=(10+sqrt(100+96))/6 or x = (10-sqrt(100+96))/6

So x=(10+14)/6 or x=(10-14)/6

So x=4 or x=-2/3.

Answer:

x=4 or x=-2/3.

Step-by-step explanation:

Answer:

15%

Step-by-step explanation:

29,400-25,000 = 4,400

4,400/29,400 =0.14965

0.14965 x 100 = 14.965%

or round up to 15%

Final answer:

The rate of depreciation for Hugo's car is approximately 14.97%, calculated using the formula for the rate of depreciation and the given values of the original and the selling price.

Explanation:

To solve for the rate of depreciation of Hugo's car, we can use the following equation:

R = ((P - S) / P) × 100

Where:
R = rate of depreciation (%)
P = original price of the car
S = selling price of the car after one year

Given:
P = $29,400
S = $25,000

Substituting the values into the equation:

R = (($29,400 - $25,000) / $29,400) × 100

R = ($4,400 / $29,400) × 100

R = 0.14966 × 100

R = 14.966%

Hence, the annual rate of depreciation for Hugo's car is approximately 14.97%.