Which expressions are equivalent to the one below? Check all that apply

64^x

A. 8^2*8^x

B. (8*8)^x

C. 8*8^2x

D. 8*8^x

E. 8^2x

F. 8^x*8^x

Answer:

Step-by-step explanation:

Please use " ^ " to denote exponentiation:  f(x) = x^2 + 4x - 21.

This function has a graph which is a parabola that opens up.

Its vertex is found by completing the square:

x² + 4x + 4 - 4 - 21, or

 (x + 2)² - 25

Comparing this to the standard equation

 (x - h)² + k, we see that h = -2 and k = -25.

Thus, the vertex (and the minimum of this function) is (-2, -25).

Thus, the range is [-25, ∞ ).  This being a polynomial function, it has no restrictions on the domain:  the domain is (-∞, ∞ )

Answer:

Counting aces: 30%.

Not counting aces: 23%

Step-by-step explanation:

If you're counting aces, it's 30%. But if you're only counting Queens, Kings and Jokers then it's about 23%.

I hope this helps

Answer:

FALSE

Step-by-step explanation:

What we have here is experimental probability:   12 heads out of 20 tosses.

The fraction 12/20 reduces to  6/10, or 0.60, which corresponds to 60%.

The answer to this question is FALSE, because this is not theoretical probability.

Answer:

false

Step-by-step explanation:

Answer:

The ratio of chocolate covered caramels to solid chocolates is 1 to 2. Or 1 : 2

Step-by-step explanation:

Answer: 1/2

Step-by-step explanation: We can write a ratio using the word "to," using a colon, or using a fraction bar. I would personally write the ratio using a fraction bar since it's easier to write in lowest terms.

So we need to compare the number of chocolate covered caramels to the number of solid chocolates.

We know that we have 4 chocolate covered caramels so we write 4 in the numerator of our ratio. We know that we have 8 solid chocolates so we put an 8 in the denominator of our ratio.

Now we have the fraction 4/8.

Notice however that 4/8 is not in lowest terms so we need to divide the numerator  and denominator by the greatest common factor of 4 and 8 which is 4 to get 1/2. So the ratio of the number of chocolate covered caramels to the number of solid chocolates is 1/2.

Answer:

[tex]P (M\ and\ C) = \frac{1}{4}=25\%[/tex]

Step-by-step explanation:

We call M the event in which a student gets an A in math and we call C the event in which a student gets an A in chemistry. As the events are not mutually exclusive, then.

[tex]P (M\ or\ C) = P (M) + P (C) - P (M\ and\ C)[/tex]

In this case we know that:

[tex]P (M\ or\ C) = \frac{7}{10}[/tex]

[tex]P (M) =\frac{17}{20}[/tex]

[tex]P (C) =\frac{1}{10}[/tex]

So

[tex]P (M\ and\ C) = P (M) + P (C) - P (M\ or\ C)[/tex]

[tex]P (M\ and\ C) = \frac{17}{20} + \frac{1}{10} -\frac{7}{10}[/tex]

[tex]P (M\ and\ C) = \frac{1}{4}[/tex]

Answer:

B

Step-by-step explanation:

You are given the inequality

[tex]8a-15>73[/tex]

Add 15 to both sides:

[tex]8a-15+15>73+15\\ \\8a>88[/tex]

Now divide both sides by 8:

[tex]a>11[/tex]

You should choose option B, because this number line shows all values of x which are greater than 11.

Hold on I’m figuring it out right now

Answer:

x = 5

Step-by-step explanation:

The line segment divides the sides in proportion, that is

[tex]\frac{14}{10}[/tex] = [tex]\frac{2x-3}{x}[/tex] ( cross- multiply )

10(2x - 3) = 14x ← distribute left side

20x - 30 = 14x ( subtract 14x from both sides )

6x - 30 = 0 ( add 30 to both sides )

6x = 30 ( divide both sides by 6 )

x = 5

Answer:

B:  down payment

Step-by-step explanation:

An increase in the down payment decreases the amount of money that has to be paid over time.

The monthly payment decrease with the increase in down payment of the loan amount. Option B is correct.

Monthly payment is the payment which has to paid against the loan amount or the borrowed money calculated with interest rate.

The monthly payment is the amount which is required to pay each month to pay off the borrowed principal amount.

It can be calculated with the following formula.

[tex]M=P\left(\dfrac{r}{1-(1+r)^{-nt}}\right)[/tex]

Here, (P) is the principal amount, (r) is the interest rate and (t) is time.

The monthly payment is inversely proportional to the down payment. When the money paid as down payment is more, then the monthly payment of loan is less.

Thus, the monthly payment decrease with the increase in down payment of the loan amount. Option B is correct.

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On the left side, you can condense the logarithms into one:

[tex]\ln(1-x)-\ln(3x+5)=\ln\dfrac{1-x}{3x+5}[/tex]

Then

[tex]\ln\dfrac{1-x}{3x+5}=\ln(1-6x)\implies e^{\ln((1-x)/(3+5))}=e^{\ln(1-6x)}\implies\dfrac{1-x}{3x+5}=1-6x[/tex]

From here it's a purely algebraic equation. Multiply both sides by [tex]3x+5[/tex] to get

[tex]1-x=(1-6x)(3x+5)[/tex]

[tex]1-x=5-27x-18x^2[/tex]

[tex]18x^2+26x-4=0[/tex]

[tex]9x^2+13x-2=0[/tex]

By the quadratic formula,

[tex]x=\dfrac{-13\pm\sqrt{241}}{18}[/tex]

or about [tex]x\approx-1.5847[/tex] and [tex]x\approx0.14023[/tex].

Before we finish, first note that in order for the original equation to make sense, we need [tex]x[/tex] to satisfy 3 conditions:

[tex]-x+1>0\implies x<1[/tex]

[tex]3x+5>0\implies x>-\dfrac53\approx-1.67[/tex]

[tex]-6x+1>0\implies x<\dfrac16\approx0.17[/tex]

or taken together,

[tex]-\dfrac53<x<\dfrac16[/tex]

so both solutions found above are valid.

To solve the logarithmic equation, combine the logarithms, set the arguments equal to each other, solve for x, and check the solution.

To solve the given equation ln(–x + 1) – ln(3x + 5) = ln(–6x + 1), we can use the properties of logarithms to simplify it.

1. Combine the logarithms using the quotient rule: ln((–x + 1)/(3x + 5)) = ln(–6x + 1).

2. Set the arguments equal to each other: (–x + 1)/(3x + 5) = –6x + 1.

3. Solve for x by cross-multiplying and simplifying the equation.

4. Check the solution in the original equation for validity.

The solution to the equation is x = -1.

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Answer:

The total volume of the pieces of fruit in the can is [tex]640\ cm^{3}[/tex]

Step-by-step explanation:

step 1

Find the volume of the cylindrical can

The volume of the can  is equal to

[tex]V=BH[/tex]

where

B is the area of the base of the can

H is the height of the can

we have

[tex]B=75\ cm^{2}[/tex]

[tex]H=10\ cm[/tex]

substitute

[tex]V=(75)(10)=750\ cm^{3}[/tex]

step 2

Find the volume of the pieces of fruit in the can

The volume of the pieces of fruit in the can is equal to subtract  the volume of syrup from the volume of the can

[tex]750\ cm^{3}-110\ cm^{3}=640\ cm^{3}[/tex]

The volume of the cylinder is defined as the product of the base or height.

The total volume of the pieces of fruit in the can is 640 cubic cm.

The base of the can has an area of 75 cm2, and the height of the can is 10cm.

If 110 cm3 of syrup is needed to fill the can to the top then the total volume of the pieces of fruit.

The volume of the cylinder is defined as the product of the base or height.

The volume is the cylinder is given by;

[tex]\rm Volume \ of \ the \ cylinder = Base \times Height[/tex]

Substitute all the values in the formula;

[tex]\rm Volume \ of \ the \ cylinder = Base \times Height\\\\\rm Volume \ of \ the \ cylinder = 75 \times 10\\\\\rm Volume \ of \ the \ cylinder = 750[/tex]

Therefore,

The total volume of the pieces of fruit in the can is,

[tex]= 750 -110\\\rm \\=640 \ cm^3[/tex]

Hence, the total volume of the pieces of fruit in the can is 640 cubic cm.

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Answer:

h/g

Step-by-step explanation:

we know that

The cotangent of angle G is equal to divide the adjacent side angle G to the opposite side angle G

so

cot(G)=h/g

Answer:

[tex]l=\frac{A-2wh}{2w+2h}[/tex]

Step-by-step explanation:

We were given that; the formula for the surface area, A, of a prism is given by

[tex]A=2lw +2lh +2wh[/tex]

where is the length of the prism, w is the width, and h is the height.

We want to solve this formula for l,

Group the l terms;

[tex]A-2wh=2lw +2lh [/tex]

Factor l on the right;

[tex]A-2wh=(2w +2h)l [/tex]

Divide both sides by 2w +2h

[tex]\frac{A-2wh}{2w+2h}=l[/tex]

Therefore:

[tex]l=\frac{A-2wh}{2w+2h}[/tex]

Answer:

What he said.

Step-by-step explanation:

Answer:

210

Step-by-step explanation:

There are calculators online that can help you with questions like this, and give you a step by step explaination to show the work

The Combination to elect four representatives out of 10 students is 210.

The correct option is (A).

permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.

Firstly, n=10 and r=4

Combination: C(n,r)=C(10,4)

                                 = [tex]\frac{10!}{4!(10-4)!}[/tex]

                                 = [tex]\frac{10!}{4!* 6!}[/tex]

                                 = [tex]\frac{10*9*8*7}{4*3*2*1}[/tex]

                                 = 210

and permutation: P(n,r)=P(10,4)

                                      = [tex]\frac{10!}{(10-4)!}[/tex]

                                      = 10*9*8*7

                                      = 5040

Hence, the combination of data is 210 and permutation is 5040.

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Answer:

The answer is: yes; k = -3 and y = -3x ⇒ the 3rd answer

Step-by-step explanation:

* Lets revise how to know the relation is direct proportion

- If all the ratios of x/y are proportion (equal ratios), then they

 are varies directly

* Now lets check the relation between x and y

∵ x = 1 and y = -3

∴ y/x = -3/1 = -3

∵ x = 3 and y = -9

∴ y/x = -9/3 = -3

∵ x = 5 and y = -15

∴ y/x = -15/5 = -3

∵ All ratios are equal -3 (they are proportion)

∴ y varies directly with x

* y ∝ x

∴ y = k x

∴ The constant of variation k is -3

∴ y = -3x

* The answer is: yes; k = -3 and y = -3x

Answer:

(B) The correct answer is B: m = n, so y = am / bn is the horizontal asymptote.

The second part is The horizontal asymptote is y = 5

The horizontal asymptote is a horizontal line that guides the graph for values of x, but is not part of the graph

The correct option that defines the horizontal asymptote is the option;

Reason:

The possible function of the question is [tex]f(x) = \dfrac{20 + 5 \cdot x}{x}[/tex]

The general form of the rational function is presented as follows;

[tex]f(x) = \dfrac{x^m+...+ a \cdot x + c}{x^n + ...+b\cdot x + d}[/tex]

The power or degree of the numerator and denominator of a rational function  determine the nature of the horizontal asymptote

Where highest power in numerator is less than the highest power or degree of the denominator, the horizontal asymptote is at y = 0

Therefore;

m < n the horizontal asymptote is y = 0

Where the power of the numerator is larger than the power of the denominator by one, the asymptote is slant, and the graph has no asymptote

m > n, there is no horizontal asymptote

In a rational function where the power of the numerator is equal to the power of the denominator, the horizontal asymptote occurs at the ratio of the leading zeros, [tex]y = \dfrac{a_m}{b_n}[/tex]

m = n, the horizontal asymptote is [tex]y = \dfrac{a_m}{b_n}[/tex]

Therefore;

In the given function, [tex]f(x) = \dfrac{20 + 5 \cdot x}{x}[/tex], the power of the numerator is equal to the power of the denominator, therefore, we have;

The horizontal asymptote of the function is [tex]y = \dfrac{5}{1} = 5[/tex]

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Answer:

6.05 units

Step-by-step explanation:

We are given that radius,r=3 units

Volume of cone=57 cubic units

We have to find the height of cone.

We know that

Volume of cube=[tex]\frac{1}{3}\pi r^2 h[/tex]

Where [tex]\pi=3.14[/tex]

Using the formula

[tex]57=\frac{1}{3}\times 3.14\times (3)^2\times h[/tex]

[tex]h=\frac{3\times 57}{3.14(3)^2}[/tex]

h=6.05 units

Hence, the height of cone=6.05 units

Answer:

$80

Step-by-step explanation:

If $100 is 25% more than Dwayne's original amount (a), you have ...

a + 0.25a = 100

1.25a = 100

a = 100/1.25 = 80

Dwayne originally had $80.

Answer:

[tex]\frac{6x-14}{x^{4} +4x}[/tex]

Step-by-step explanation:

I have to [tex]\frac{\frac{3x-7}{x^{2} } }{\frac{x^{2} }{2}+\frac{2}{x}}[/tex]

Let's start by joining the macro denominator with a common denominator. So, by applying a minimum common multiple [tex]\frac{x^{2} }{2} +\frac{2}{x}=\frac{x^{3}+ 4 }{2x}[/tex]

Now I can write the expression as

[tex]\frac{\frac{3x-7}{x^{2}}}{\frac{x^{3}+ 4 }{2x}}[/tex]

Now to convert both fractions into one, I multiply the numerator of the one above by the denominator of the one below, and the denominator of the one above with the numerator below, remaining that way.

[tex]\frac{\frac{3x-7}{x^{2}}}{\frac{x^{3}+4}{2x}}=\frac{(3x-7)(2x)}{(x^{2})(x^{3}+ 4)}[/tex]

Having the fraction in this way, I could simplify the x of the "2x" of the numerator with an x^2 (x^2=x*x) of the denominator

[tex]\frac{(3x-7)(2x)}{(x^{2})(x^{3}+4)}=\frac{2(3x-7)}{x(x^{3}+ 4)}[/tex]

finally, applying distributive property, I have to

[tex]\frac{(6x-14)}{(x^{4}+ 4x)}[/tex]

Done

Combine fractions with common denominator [tex]\(2x^2\),[/tex] then simplify numerator and denominator separately: [tex]\(\frac{12x^2 - 28x^3}{x^4 + 8x^3}\).[/tex]

let's simplify the complex fraction step by step:

1. Start by finding a common denominator for all the fractions involved. In this case, the least common denominator is [tex]\(2x^2\).[/tex]

2. Rewrite each fraction with the common denominator:

  - [tex]\(\frac{3x - 7}{x^2}\)[/tex]  becomes [tex]\(\frac{3x - 7}{x^2} \cdot \frac{2}{2}\)[/tex] to match the denominator [tex]\(2x^2\).[/tex] So, it becomes[tex]\(\frac{6 - 14x}{2x^2}\).[/tex]

  -[tex]\(\frac{x^2}{2}\)[/tex] remains the same.

  - [tex]\(\frac{2}{x}\)[/tex]  becomes [tex]\(\frac{2}{x} \cdot \frac{2x}{2x}\)[/tex] to match the denominator [tex]\(2x^2\).[/tex] So, it becomes [tex]\(\frac{4x}{2x^2}\).[/tex]

3. Now, combine all the fractions:

[tex]\[ \frac{\frac{6 - 14x}{2x^2}}{\frac{x^2}{2} + \frac{4x}{2x^2}} \][/tex]

4. Combine the terms in the numerator and denominator:

  - The numerator remains the same: [tex]\(6 - 14x\).[/tex]

  - The denominator becomes[tex]\(\frac{x^4 + 8x^3}{2x^2}\).[/tex]

5. Divide the numerator by the denominator:

[tex]\[ \frac{6 - 14x}{\frac{x^4 + 8x^3}{2x^2}} \][/tex]

6. Multiplying the numerator by the reciprocal of the denominator:

[tex]\[ (6 - 14x) \cdot \frac{2x^2}{x^4 + 8x^3} \][/tex]

7. Distribute the numerator:

 [tex]\[ \frac{12x^2 - 28x^3}{x^4 + 8x^3} \][/tex]

So, the simplified form of the complex fraction is [tex]\(\frac{12x^2 - 28x^3}{x^4 + 8x^3}\).[/tex]

Use the chain rule:

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\cdot\dfrac{\mathrm dt}{\mathrm dx}[/tex]

So we have

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]

[tex]x=t^2+4\implies\dfrac{\mathrm dx}{\mathrm dt}=2t[/tex]

[tex]y=t^3+3t\implies\dfrac{\mathrm dy}{\mathrm dt}=3t^2+3[/tex]

[tex]\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{3t^2+3}{2t}[/tex]

Now write [tex]f(t)=\dfrac{\mathrm dy}{\mathrm dx}[/tex]. Then by the chain rule,

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{\mathrm dy}{\mathrm dx}\right]=\dfrac{\mathrm df}{\mathrm dx}=\dfrac{\mathrm df}{\mathrm dt}\cdot\dfrac{\mathrm dt}{\mathrm dx}=\dfrac{\frac{\mathrm df}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]

so that

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\frac{\mathrm d}{\mathrm dt}\left[\frac{3t^2+3}{2t}\right]}{2t}=\dfrac{3(t^2-1)}{4t^3}[/tex]

The curve is concave upward when the second derivative is positive:

[tex]\dfrac{3(t^2-1)}{4t^3}>0\implies t^2>1\implies\sqrt{t^2}>\sqrt1\implies|t|>1[/tex]

or equivalently, when [tex]t<-1[/tex] or [tex]t>1[/tex].

The dy/dx is equal to 1 + 3/2t, and the d2y/dx2 is 3/2t. The curve is concave upward for t values in the (0, Infinity) interval.

To answer your question, we first need to take the derivatives of x and y with respect to t. The derivatives of x = t^2 + 4 and y = t^2 + 3t by t yield dx/dt = 2t and dy/dt = 2t + 3. Now, dy/dx = (dy/dt) / (dx/dt) = (2t + 3) / 2t = 1 + 3/2t.

Next the second derivative d2y/dx2 (concavity), is obtained as the derivative of dy/dx = 1 + 3/2t with respect to t, which is 3/2t. Now, the curve will be concave upward whenever d2y/dx2 > 0. Solving 3/2t > 0 gives the interval for t as (0, Infinity), which indicates the values of t for which the curve is concave upward.

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Answer:

True

Step-by-step explanation:

True.

The solution to a system of equations is the point where two equations intercept. In both cases, we can see that the interception of the two systems shown occurs at (0, 1). So the statement is correct.

Answer:

It's True

Step-by-step explanation:

Answer:The answer is B

Step-by-step explanation:Because the exponent is 6 for Silver Town and 8 for Lake City

Answer:

Step-by-step explanation:

10^6=1000000

1000000x8.2

silver town 8200000

lake city 164000000

Answer:

The volume is equal to [tex]320\pi\ cm^{3}[/tex]

Step-by-step explanation:

we know that

The volume of the cone is equal to

[tex]V=\frac{1}{3}\pi r^{2} h[/tex]

we have

[tex]r=8\ cm[/tex]

[tex]h=15\ cm[/tex]

substitute the values

[tex]V=\frac{1}{3}\pi (8)^{2}(15)[/tex]

[tex]V=320\pi\ cm^{3}[/tex]

Final answer:

volume of 320π cm³.

Explanation:

To calculate the volume of the oblique cone with the given dimensions, use the formula for the volume of a cone,

The volume of the oblique cone can be calculated using the formula V = 1/3 * π * r^2 * h.

Answer:

Option A

Domain = Range = (-∞,∞)

Step-by-step explanation:

We can easily solve this question by using a graphing calculator or any plotting tool.

The function is

f(x) = -2x + | 3 sin(x) |

Which can be seen in the picture below

We can notice that f(x) is a line with periodical ups and downs thanks to the sinusoidal term, but there are no restrictions over the domaoin or range of the function.

It can take any real value as an input, and can produce any real value as an output

Answer:

Domain is set of all real numbers

Range is set of all real numbers

Step-by-step explanation:

Domain is the set of x values for which the function is defined

to find domain we look at the graph and check if there is any restriction for x

WE have line graph for all x values

So there is no restriction for x. Hence, Domain is set of all real numbers

Range is the set of y values for which the function is defined

to find range we check the continuity on graph

The graph is contininious and there is no break . The graph is continious for all y values

So range is set of all real numbers

Answer:

its d.

i just took it

Step-by-step explanation:

ANSWER

a=53°

b=106°

c=74°

d=37°

EXPLANATION

A semicircle creates a right angle on the circumference.

d+53°=90°

d=90°-53°

d=37°

Angle subtended at the center by an arc is twice the angle subtended at the circumference by the same arc.

c=2d

c=2(37)

c=74°

Adjacent angles on a straight line add up to 180°

c+b=180°

b=180°-c

b=180°-74°

b=106°

The sum of interior angles in an isosceles triangle is 180°

a+a+c=180°

2a+74=180

2a=180-74

2a=106

a=53°

Answer:

a) the temperature is y = 16° C at 9 AM.

b) The probability of selecting 2 vowels at the same time is 2.8%

Step-by-step explanation:

a)

Temperature = y=?

Time = x = 9 A.M

The given equation is:

[tex]y = 19 + 6 sin (\frac{\pi }{12}(x-11))[/tex]

Putting x = 9 and solving

[tex]y = 19 + 6 sin (\frac{\pi }{12}(9-11))\\y = 19 + 6 sin (\frac{\pi }{12}(-2))\\y = 19 + 6 sin (\frac{-\pi }{6})\\y = 19 + 6(-0.5)\\y = 19 - 3\\y = 16[/tex]

So, the temperature is y = 16° C at 9 AM.

b)

WORD CLEMSON

Total Words = 7

Vowels = e, o = 2

Probability of selecting 2 vowels at the same time = P(2 Vowels) =?

We will use the formula of combinations nCr = n! / r! (n-r)!

As we are selecting three letter out of 7, the sample space will be 7C3

And

As there are only two vowels in the given word, and we have to find the probability of selecting two vowels so our event space will be 2C2.

P(2 Vowels) = 2C2 / 7C3

                    = 1/35

                    = 0.028 = 2.8%

So, The probability of selecting 2 vowels at the same time is 2.8%

Answer:

4 cups

Step-by-step explanation:

Answer:

15 or 16 times

Step-by-step explanation:

The ball will land on 6 or 29 about 14 times.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.

The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.

The degree to which something is likely to happen is basically what probability means.

Given:

There are 2 slots that have a "6" or a "29" on them. The other 36 slots have different numbers.

The probability of landing on "6" or "29"

= 2/38

= 1/19

Now, Multiply this probability by the number of trials to get

= 266 x (1/19)

= 14

Hence, we expect the ball will land on 6 or 29 about 14 times.

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Answer:

The approximate population of the state of California in 2010 was [tex]37,293,600\ people[/tex]

Step-by-step explanation:

Let

x----->the estimated population of Los Angeles in 2010

y----->the estimated population of the state of California in 2010

we know that

[tex]y=9.84x[/tex] ------> equation A

we have

[tex]x=3.79*10^{6}\ people[/tex]

Substitute the value of x in the equation A and solve for y

[tex]y=9.84(3.79*10^{6})=37.2936*10^{6}\ people[/tex]

[tex]37.2936*10^{6}=37,293,600\ people[/tex]

Answer:

3.73 × 10^7 people

Step-by-step explanation:

The original equation 3.79 × 10^6 = 3790000. 3790000× 9.84=37293600. 37293600 estimated is about 37300000. That would make 3.73 × 10^7 the expression that represents the amount of people in California.

Answer:

y = tan(1/2 x + π/2) ⇒ answer c

Step-by-step explanation:

* Lets revise some fact of y = tanx

- The domain of tanx is all x(≠ π/2) + nπ, where n is the number of cycle

- The range is all real numbers

- The period of tanx is π ÷ coefficient of x

* Lets revise some transformation

- A horizontal stretching is the stretching of the graph away from

 the y-axis

• if 0 < k < 1 (a fraction), the graph is f (x) horizontally stretched by

 dividing each of its x-coordinates by k (x × 1/k)

- A horizontal compression is the squeezing of the graph toward

 the y-axis.

• if k > 1, the graph f (x) horizontally compressed by dividing each

 of its x-coordinates by k. (x × 1/k)

* Look to the graph of y = tanx ⇒ red graph

- the graph of tanx intersect x-axis at the origin

- The period of tanx is π

* Look to the blue graph (the problem graph)

∵ The graph intersect x-axis at points (-π , 0)

- That means the graph of tanx moved to the left by π units

∴ y = tan(x + π)

- The period of the graph is 2π

∵ The period = π/coefficient of x

∴ 2π = π/coefficient of x ⇒ using cross multiplication

∴ Coefficient of x = π/2π = 1/2

- That means the graph stretched horizontally

∴ y = tan1/2(x + π)

* y = tan(1/2 x + π/2)