Use the definition of a logarithm to solve the equation. ln ( − 5 z ) = ln ( z^ 2 − 7 z )

The equation for the circle O, given the diameter, the secant K and the common point of tangent I, is:

(x - 8)² + (y - 7)² = 15²

(x - 8)² + (y - 7)² = 225

To determine the equation of the circle O with center S(8, 7) and point M(-5, -2) on the secant k, we first need to find the radius of the circle.

Given that the distance from point S to secant k is 5, we can use the Pythagorean theorem to find the length of MA, which is equal to the radius of the circle.

MA² + AS² = MS²

MA² + 5² = MS²

Calculate the distance between points M and S:

MS = √((8 - (-5))² + (7 - (-2))²) = √(13² + 9²) = √(169 + 81) = √250

Now we can find the length of MA:

MA² + 5² = 250

MA² = 250 - 25 = 225

MA = √225 = 15

The equation of a circle with center (h, k) and radius r is given by:

(x - h)² + (y - k)² = r²

Using the given center S(8, 7) and radius 15, the equation of the circle O is:

(x - 8)² + (y - 7)² = 15²

(x - 8)² + (y - 7)² = 225

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Answer:

Step-by-step explanation:

because 110

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2 \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{26}\\ a=\stackrel{adjacent}{2x}\\ o=\stackrel{opposite}{2x-14} \end{cases} \\\\\\ (26)^2= (2x)^2 + (2x-14)^2\implies 676 = (4x^2)+(4x^2-56x+14^2) \\\\\\ 676=4x^2+4x^2-56x+14^2\implies 676=8x^2-56x+196 \\\\\\ 0=8x^2-56x-480\implies 0=8(x^2-7x-60) \\\\\\ 0=x^2-7x-60\implies 0=(x-12)(x+5)\implies x= \begin{cases} ~~ 12 ~~ \checkmark\\ -5 ~~ \bigotimes \end{cases}[/tex]

now, -5 is a valid value for "x", however in this case we can't use it, because that makes one of our sides negative and all sides must be a positive value.

[tex]\stackrel{ 2(12) }{\text{\LARGE 24}}\hspace{5em}\stackrel{ 2(12)-14 }{\text{\LARGE 10}}\hspace{5em}\text{\LARGE 26}[/tex]

Answer:

Step-by-step explanation:

Given:-

Area = 72

width = 8

Area = length  × width

[tex]\bf 72=length \times 8[/tex]

Divide 72 / 8:-

[tex]\bf length=9\; cm[/tex]

If you're asking to find the length, this's your answer.

____________________________

Hope this's what you're looking for!

Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio. This is also known as Thales theorem.

Given:

[tex]\angle Q= 90^\circ , XY \ || \ QR, PQ = 6 \ \text{cm}, PY = 4 \ \text{cm} \ \text{and} \ PX : XQ = 1 : 2[/tex]

Since, [tex]XY \ || \ QR[/tex],

[tex]PX/XQ = PY/YR[/tex]

[ By Thales theorem (BPT)]

[tex]\dfrac{1}{2} = PY/YR[/tex]      [tex][PX : XQ = 1 : 2][/tex]

[tex]\dfrac{1}{2} = 4 /(PR - PY)[/tex]

[tex][YR= PR - PY][/tex]

[tex]\dfrac{1}{2} = 4 /(PR - 4)[/tex]

[tex]PR - 4 = 2 \times 4[/tex]

[tex]PR - 4 = 8[/tex]

[tex]PR = 8 +4[/tex]

[tex]PR = 12 \ \text{cm}[/tex]

In right [tex]\Delta PQR[/tex],

[tex]PR^2 = PQ^2 + QR^2[/tex]

[ By Pythagoras theorem]

[tex]12^2 = 6^2 + QR^2[/tex]    [tex][\text{Given} : PQ= 6 \ \text{cm}][/tex]

[tex]144 = 36 + QR^2[/tex]

[tex]144 - 36 + QR^2[/tex]

[tex]108= QR^2[/tex]

[tex]QR =\sqrt{108} =\sqrt{3\times36} = 6\sqrt{3} \ \text{cm}[/tex]

Hence, the lengths of PR and QR is 12 cm and [tex]6\sqrt{3}[/tex] cm.

Answer:

there are 41 white marbles in the bag.

Step-by-step explanation:

Let's use the variable w to represent the number of white marbles and y to represent the number of yellow marbles.

From the problem, we know that:

w + y = 49 (since there are 49 marbles in total)

And we also know that:

w = 5y + 1 (since the number of white marbles is 1 more than 5 times the number of yellow marbles)

Now we can use substitution to solve for w:

w + y = 49

(5y + 1) + y = 49 (substitute w = 5y + 1)

6y + 1 = 49 (combine like terms)

6y = 48 (subtract 1 from both sides)

y = 8 (divide both sides by 6)

Now we know there are 8 yellow marbles. We can use this information to find the number of white marbles:

w = 5y + 1

w = 5(8) + 1

w = 41

We can conclude after answering the presented question that Hence the equation slower train's speed is 13.25 miles per hour, while the faster train's speed is 27.25 miles per hour (13.25 + 14).

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the number "9". The purpose of equation solving is to determine the value or readings of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. In the calculation "x2 + 2x - 3 = 0," for example, the variable x is raised to the second power. Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

Because we know the two trains are going in the same direction, their relative speed equals the total of their speeds, which is (x + x+14) = 2x + 14.

We also know that the distance between them is shrinking at a pace of 648 miles in 4 hours, or at a rate of 648/4 = 162 miles per hour.

[tex]162 = (2x + 14) x 4 = distance = rate x time\\162 = 8x + 56\\106 = 8x\sx = 13.25\\[/tex]

Hence the slower train's speed is 13.25 miles per hour, while the faster train's speed is 27.25 miles per hour (13.25 + 14).

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Consequently, the retirement account's income rate is roughly 3.8%. (rounded to one decimal place).

Consider borrowing $1,000 at a 10% interest rate for seven years. Your interest for the first year would be $100. Your interest payment for the following year would be made up of the original sum plus interest, or $1,100. As a result, your income for the following year would be $110 ($1,100 multiplied by 0.10).

The interest rate can be calculated using the continuous compounding formula:

[tex]A=P e^{r t}[/tex]

where:

A = final balance = $9,483.80

P = initial investment = $6,398

r = rate

t = time in years = 16

Substituting the given values, we have:

$9,483.80 = $6,398[tex]e^{r16}[/tex]

Dividing both sides by $6,398, we get:

1.4829 = [tex]e^{r16}[/tex]

Using the simple logarithm of both parts, the following is obtained:

ln(1.4829) = r × 16

Solving for r, we get:

r = ln(1.4829)/16

r ≈ 0.038

Therefore, the interest rate of the retirement account is approximately 3.8% (rounded to one decimal place).

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Answer:

[tex]a_{n}[/tex] = 7n - 6

Step-by-step explanation:

there is a common difference between consecutive terms , that is

8 - 1 = 15 - 8 = 22 - 15 = 29 - 22 = 7

this indicates the sequence is arithmetic with nth term

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

here a₁ = 1 and d = 7 , then

[tex]a_{n}[/tex] = 1 + 7(n - 1) = 1 + 7n - 7 = 7n - 6

Answer:

7n-6

Step-by-step explanation:

Work out the difference of the sequence:

8-1=7

Now you have the first part of the equation: 7n

n is the number that the integer is on the sequence

In this case:

1 = 1 as 1 is the first number of the sequence

And 2 = 8 as 8 is the 2nd number of the sequence

To find the full equation:

Do 7x1 to get you 7

Now see how far the 1st number is from 7

In this case you would do:

7-1 which gives you 6

Since you subtracted it to find the difference, it would be:

- 6

Therefore your answer would be 7n-6

To check it:

Times 7 by let's say 3 to get you 21

Then subtract 6 to get 15.

This is proven right as the 3rd number of the given sequence is 15.

Hope this helped

Answer:

The answer would be 9 and 1/3

Step-by-step explanation:

3 and 1/2 times 2 and 2/3 gives you 9.3333333 which is rounded to 9 and 1/3.

The number of girls in the gym must be: g ≥ 16

Let's define the variable "g" to be a representation of the number of girls in the gym.

We know that there are 30 students in total. Therefore, the number of boys in the gym will be:

b = 30 - g

We also know that there are at least 16 girls in the gym. So, we can write the inequality:

g ≥ 16

This inequality means that the number of girls in the gym must be greater than or equal to 16.

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Answer: 1. (8b + 5)

2. (22p - 9)

HAVE A GREAT DAY!!!!

Step-by-step explanation:

The range of the difference in means' 99% confidence level is from -1.42 to 10.02. The true mean difference between Treatments 1 and 2 falls between these two numbers, we can claim with 99% certainty for t-distribution.

We can use a t-distribution to determine a confidence interval for the difference in means using paired data. Prior to determining the mean difference and standard deviation of the differences, we first compute the difference between the paired observations.

Because we are only interested in the mean difference between Treatments 1 and 2, we compute the differences for each pair and get the following outcomes:

6 -3 8 2 6

The sample mean and sample standard deviation of these differences are then computed. The average of these variations is the sample mean.

(6 - 3 + 8 + 2 + 6)/5 = 3.8

The square root of the sum of squared differences divided by the degrees of freedom yields the sample standard deviation.

[tex]\sqrt{[(6 - 3.8)^2 + (-3 - 3.8)^2 + (8 - 3.8)^2 + (2 - 3.8)^2 + (6 - 3.8)^2]/(5-1)) } = 3.06[/tex]

The 99% confidence interval for the mean difference can then be determined using the t-distribution. Our sample size is tiny (n=5), so we utilise a t-distribution with four degrees of freedom.

The sample mean, which is 3.8, provides the most accurate approximation of the mean difference.

We must determine the crucial value of t for a 99% confidence interval with 4 degrees of freedom in order to determine the margin of error. The crucial value, which we determine using a t-table, is 4.604.

The margin of error is:

[tex]4.604 * (3.06/\sqrt{5}) = 5.22[/tex]

Lastly, by deducting and adding the margin of error from the sample mean, we can determine the confidence interval:

3.8 - 5.22 = -1.42

3.8 + 5.22 = 10.02

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In the given right angle, the required value of x is 10° respectively.

Right-angled shapes can be any polygon, ranging from triangles to figures with numerous sides.

Right-angled shapes like squares and rectangles have exactly 4 right angles that add up to 360 degrees.

Right angles can also be found in other shapes like trapezoids, pentagons, and hexagons.

An angle with a measure of exactly 90 degrees (or /2) is referred to as a right angle.

The proper angles are frequently demonstrated in everyday life. For instance, the edges of the cardboard or the corner of a book.

So, we know that the given angles together make a right angle as:

2x + 8 + 62 = 90

Then, solve for x as follows:

2x + 8 + 62 = 90

2x + 70 = 90

2x = 90 - 70

2x = 20

x = 20/2

x = 10

Therefore, in the given right angle, the required value of x is 10° respectively.

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Answer:

Step-by-step explanation:

Total no. of possible outcomes = 6 (1,2,3,4,5,6,)

As given we have only one prime even no. from the above given set, that is 2.

Let's call this event A.

Total no. of favourable outcomes =P(A)= 1

Therefore the probability is  P(A)= 1/6

The value of the operation 6 of 1/4 is 3/2

Fractions are simply described as part of a whole number, element or variable.

There are different types of fractions in mathematics. They include;

Some examples of simple fractions are; 1/2 , 2/3

Some examples of mixed fractions are; 2 1/3, 4 1/2

Some examples of improper fractions are; 3/2, 4/3

From the information given, we have that;

6 of 1/4,

'of' in this sense means multiplication, then, we get;

6 × 1/4

Multiply the values

6/4

Divide

3/2

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From the expression, the form of the d⁵ is provided by the stated assertion.

A group of words joined with the actions +, -, x, or  form an expression, such as 4 x 3 or 5 x 2  3 x y + 17. A statement containing the equals symbol, such as 4 b 2 = 6, says that two formulas are equivalent in value and is known as an equation.

As an illustration, the expression x + y is one where both x and y have words with an addition function in between. There are two kinds of expressions in mathematics: numerical expressions, which only comprise integers, and algebraic expressions, which also include variables.

[tex]d^{(8-3)} = d^5[/tex]

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Enter an expression equivalent to (d^(8))/(d^(3)) in the form, d^(n).

$26.40

ten percent is 88 divided by 10= 8.8

8.8 multiplied by 3 is 26.40

The value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010 is approximately -2.326.

With its bell-shaped structure and heavier tails, the t-distribution, commonly referred to as the Student's t-distribution, is a kind of probability distribution that resembles the normal distribution. When there are insufficient samples or unknown variances, it is used to estimate population parameters. T-distributions have broader tails than normal distributions because they are more likely to contain extreme values.

To find the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010, we can use a t-table or a calculator. Using a calculator, we can use the inverse t-distribution function. The inverse t-distribution function gives us the value of t for a given probability and degrees of freedom.

Using this function, we have:

t = invT(0.010, 45) ≈ -2.326

Rounding this to three decimal places gives us the answer:

t ≈ -2.326

Therefore, the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010 is approximately -2.326.

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The independent variable and the dependent variable are the number of dollars spent and the number of tickets bought

Given that we have the following statement:

The independent variable is the input value

i.e. the number of dollars spent

Similarly, the dependent variable is the output value

i.e. the number of tickets bought

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Answer:

Therefore, the shuffleboard disk will travel a distance of 4.71 meters before coming to a stop, which is less than the length of the court (14.3 meters).

Step-by-step explanation:

We can start by using the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy. In this case, we can assume that the initial kinetic energy of the disk is entirely converted to work done by friction, which causes the disk to come to a stop. The equation can be written as:

Work done by friction = Change in kinetic energy

The work done by friction can be calculated using the formula:

Work = force x distance

The force of friction can be found using the formula:

Force of friction = coefficient of friction x normal force

The normal force is equal to the weight of the disk, which can be found using the formula:

Weight = mass x gravity

Substituting the values given in the problem, we get:

Weight = mass x gravity = 0.75 kg x 9.81 m/s^2 = 7.3575 N

Force of friction = coefficient of friction x normal force = 0.34 x 7.3575 N = 2.4985 N

Work done by friction = Force of friction x distance

We can solve for the distance by rearranging the equation as:

Distance = Work done by friction / Force of friction

The initial kinetic energy of the disk can be found using the formula:

Kinetic energy = 0.5 x mass x velocity^2

Substituting the values given in the problem, we get:

Kinetic energy = 0.5 x 0.75 kg x (5.6 m/s)^2 = 11.76 J

Using the work-energy principle, we know that the work done by friction is equal to the change in kinetic energy, which is:

Work done by friction = Kinetic energy = 11.76 J

Substituting this value and the force of friction into the distance formula, we get:

Distance = Work done by friction / Force of friction = 11.76 J / 2.4985 N = 4.71 m

Therefore, the shuffleboard disk will travel a distance of 4.71 meters before coming to a stop, which is less than the length of the court (14.3 meters).

Answer:

Ask away by typing or recording your messageAdam, Ben and Carly work out the mean of their ages.Adam is 4 years older than the mean. Ben is 1 year younger than the mean.Is Carly older or younger than the mean?By how many years?Let's start by finding the mean of their ages. We can do this by adding their ages and dividing by the number of people: Mean = (Adam's age + Ben's age + Carly's age) / 3 Let's call the mean "M" for now. We can use this to create two equations based on the information given: Adam = M + 4 Ben = M - 1 We can substitute these equations into the mean equation to get: M = (M + 4 + M - 1 + Carly's age) / 3 Simplifying this equation gives us: 3M = 2M + 3 + Carly's age Carly's age = M - 3 So Carly's age is younger than the mean by 3 years.Hey ✋, are you still around? anything else on your mind?To the nearest tenth of a second, how long after the pebble falls will it hit the ground?sTo answer this question, we need to know the height from which the pebble was dropped. We can use the formula: time = sqrt(2 * height / acceleration due to gravity) Assuming the pebble was dropped from a height of 10 meters, and taking the acceleration due to gravity as 9.8 m/s^2, we get: time = sqrt(2 * 10 / 9.8) = 1.43 seconds (to two decimal places) Therefore, the pebble will hit the ground approximately 1.43 seconds after it was dropped.

0 / 350

No, the process for multiplying polynomials is different from adding and subtracting.

What exactly are polynomials?

Polynomials are algebraic expressions made up of variables and coefficients that are joined using the addition, subtraction, and multiplication operations. The variables in a polynomial can be raised to non-negative integer powers.

For example, the expression 3x² - 2x + 1 is a polynomial, where 3, -2, and 1 are the coefficients, and x², x, and 1 are the variables with their respective powers.

Now,

The process for adding and subtracting polynomials involves combining like terms. To add or subtract two polynomials, we simply combine the coefficients of the same degree terms.

For example, to add the polynomials 2x² + 3x + 4 and 4x² - 2x - 1, we group the like terms and add the coefficients:

(2x² + 4x²) + (3x - 2x) + (4 - 1) = 6x² + x + 3

To subtract the polynomial 4x² - 2x - 1 from the polynomial 2x² + 3x + 4, we change the sign of the second polynomial and then combine the like terms:

(2x² + 3x + 4) - (4x² - 2x - 1) = 2x² + 3x + 4 - 4x² + 2x + 1 = -2x² + 5x + 5

The process for multiplying polynomials is different from adding and subtracting. When we multiply two polynomials, we need to distribute each term of polynomials, and then combine the like terms.

For example, to multiply the polynomials (x + 2) and (x - 3), we use the distributive property:

(x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x² - 3x + 2x - 6 = x² - x - 6

As we can see, the process for multiplying polynomials is different from adding and subtracting, but all three operations involve combining like terms in some way.

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Answer:

Nathan would have to pay 20 dollars if he parked for 6 hours.

2t+8

Step-by-step explanation:

Let's call Michelle's speed "M" and John's speed "J". We know that John's speed is 10 miles per hour faster than Michelle's speed, so we can express this as:

J = M + 10

We also know that Michelle is 50 miles ahead of John, so we can express this as:

Distance = 50 miles

Now we can use the formula:

Distance = Rate x Time

We want to know how long it will take John to catch up to Michelle, so we can call this time "t". We can use the formula for both Michelle and John, and set their distances equal to each other since they will meet at the same point:

M * t + 50 = J * t

Now we can substitute J with M + 10, and simplify:

M * t + 50 = (M + 10) * t

M * t + 50 = M * t + 10t

50 = 10t

t = 5

Therefore, John will catch up to Michelle in 5 hours (answer choice B).

Coli present in my sandwich increases by 3.5265% per minute at room temperature. After 90 minutes, the amount of E. Coli present in my sandwich is 85.02 cfu/g, which is much higher than the acceptable limit of 100 cfu/g.

Amount refers to the total sum of money or value of goods, services, or resources. It's usually the result of a calculation or the total of several different things added together. Amounts can also refer to the size, quantity, or degree of something. For example, one might say the amount of time it takes to complete a task.

a. After 1 minute, the amount of E. Coli in my sandwich is 10.3533 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.0352 (3.5265% increase).

b. After 2 minutes, the amount of E. Coli in my sandwich is 10.716 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.0716 (3.5265% increase).

c. After 3 minutes, the amount of E. Coli in my sandwich is 11.0861 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.1086 (3.5265% increase).

d. After 10 minutes, the amount of E. Coli in my sandwich is 14.14 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.4140 (3.5265% increase).

e. After 60 minutes, the amount of E. Coli in my sandwich is 56.68 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 5.668 (3.5265% increase).

f. After 90 minutes, the amount of E. Coli in my sandwich is 85.02 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 8.502 (3.5265% increase).

From the above calculations, it is evident that the amount of E. Coli present in my sandwich increases by 3.5265% per minute at room temperature. After 90 minutes, the amount of E. Coli present in my sandwich is 85.02 cfu/g, which is much higher than the acceptable limit of 100 cfu/g. This is why it is important to store and consume food with E. Coli count below the acceptable limit and refrigerating food can help in reducing the growth of bacteria.

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Hence, the rectangular prism has a surface area of 280 and 1/4 [tex]cm^2[/tex] as the  rectangular prism has the measurements .

A hexagon having four right angles (90º angles) and opposing sides of equal length is called to as a rectangle. It is a particular kind of rectangle in which every angle is a right angle. A rectangle has parallel opposed sides and diagonals that cut it in half. The length (l) and breadth (w), which have been perpendicular to one another, define a rectangle. A rectangle's area is equal to the product of the its width and length, or lw, and its perimeter is equal to the sum of all of its sides, or 2(l+w). In geometry, math, and daily life, such as in doors, frames, and picture frames, rectangles are frequently occurring forms.

given

The rectangular prism has the following measurements: 5 centimetres by 4 centimetres by 11 and three-quarters centimetres.

The region of each face is thus:

Face 1: 5 3/4 cm x 4 cm equals 23 [tex]cm^2[/tex]

Face 2: 5 and 3/4 inches by 11 and 3/4 inches is 70 and 1/8 centimetres square.

Face 3: 4 cm x 11 3/4 cm equals 47 [tex]cm^2[/tex]

Total surface area = 23 cm2 + 70 eighths of a cm2 + 47 eighths of a [tex]cm^2[/tex] + 23 eighths of a cm2

Surface area total = 280 and 1/4 [tex]cm^2[/tex]

Hence, the rectangular prism has a surface area of 280 and 1/4 [tex]cm^2[/tex] as the  rectangular prism has the measurements .

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[tex]\textit{Law of sines} \\\\ \cfrac{\sin(\measuredangle A)}{a}=\cfrac{\sin(\measuredangle B)}{b}=\cfrac{\sin(\measuredangle C)}{c} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\sin(x)}{32}=\cfrac{\sin(50^o)}{40}\implies \sin(x)=\cfrac{32\sin(50^o)}{40} \\\\\\ x=\sin^{-1}\left[ \cfrac{32\sin(50^o)}{40} \right]\implies x\approx 37.79^o[/tex]

Make sure your calculator is in Degree mode.

B. The graph οf functiοn f is stretched hοrizοntally by a scale factοr οf 3 tο create the graph οf functiοn g.

A linear transfοrmatiοn is a functiοn that mοves frοm οne vectοr space tο anοther while maintaining the underlying (linear) structure οf each vectοr space.

The rules fοr linear transfοrmatiοns are that

g(x) = a·f(b·(x-c)) +d

stretches the graph vertically by a factοr οf "a" (befοre the shift)

cοmpresses the graph hοrizοntally by a factοr οf "b" (befοre the shift)

shifts it tο the right by amοunt "c"

shifts it up by amοunt "d".

Yοur equatiοn has b=1/3, sο the graph is cοmpressed by a factοr οf 1/3, which is equivalent tο a stretch by a factοr οf 3.

The apprοpriate chοice οf descriptiοn is ...

b) the graph οf g(x) is hοrizοntally stretched by a factοr οf 3

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Complete Question:

Select the correct answer. If , which statement is true? if g(x) = f(1/3x)

A. The graph of function f is stretched vertically by a scale factor of 3 to create the graph of function g.

B. The graph of function f is stretched horizontally by a scale factor of 3 to create the graph of function g.

C. The graph of function f is compressed horizontally by a scale factor of to create the graph of function g.

D. The graph of function f is compressed vertically by a scale factor of to create the graph of function g.