3^x+3^(4-2x)=1+3^(4-x)

The expression 3x² + 6x + 4x + 8 is factorized as (x + 2)(3x + 4).

To factorize the algebraic expression 3x² + 6x + 4x + 8, you can follow these three steps:

Step 1: Grouping

Group the terms in pairs based on their common factors:

(3x² + 6x) + (4x + 8)

Step 2: Factor out common factors

Factor out the greatest common factor from each group of terms:

3x(x + 2) + 4(x + 2)

Now, we have a common factor of (x + 2) in both terms.

Step 3: Combine the factored terms

Combine the factored terms using the common factor:

(x + 2)(3x + 4)

The expression 3x² + 6x + 4x + 8 is factorized as (x + 2)(3x + 4).

In this process, we grouped the terms with similar variables and then factored out the greatest common factor from each group. Finally, we combined the factored terms using the common factor to obtain the fully factorized expression.

It's important to note that factoring algebraic expressions requires practice and familiarity with common factoring techniques. In some cases, you may encounter expressions that require additional methods such as factoring by grouping, using special factoring formulas, or applying quadratic factoring techniques.

By following these three steps, you can factorize the given expression by identifying common factors and combining terms accordingly.

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The simplified fraction of 350/120 is 35/12

from the question, we have the following parameters that can be used in our computation:

350/120

Divide the zeros

So, we have

350/120 = 35/12

Divide the fractions by 3

This is impossible

Hence, the simplified fraction is 35/12

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Answer:

3

Step-by-step explanation:

parallel lines have the same gradient

Answer: [tex](-\infty, 6) \cup (6, \infty)[/tex]

Step-by-step explanation:

Polynomials are continuous for all reals, so we only need to consider the rational part. Since the denominator is undefined at [tex]x=6[/tex], there is a vertical asymptote at [tex]x=6[/tex].

The limit of the expression lim (x² - √x⁴ + 3x²) as x approaches any value is indeterminate (∞ - ∞), except when x approaches zero, where the limit is 0.

To find the limit of the expression lim (x² - √x⁴ + 3x²) as x approaches a certain value, we can simplify the expression and evaluate the limit.

First, let's simplify the expression:

lim (x² - √x⁴ + 3x²)

= lim (4x² - x² - √x⁴)

= lim (3x² - √x⁴)

Now, let's consider the behavior of the expression as x approaches a value.

As x approaches any finite value, the term 3x² will approach a finite value.

For the term √x⁴, as x approaches a finite value, the square root of x⁴ will approach the absolute value of x².

Therefore, the limit becomes:

lim (3x² - √x⁴) = lim (3x² - |x²|)

Next, let's consider the different cases as x approaches positive infinity, negative infinity, and zero.

1. As x approaches positive infinity, the term 3x² will tend to positive infinity, and |x²| will also tend to positive infinity. Thus, the expression becomes:

lim (3x² - |x²|) = lim (∞ - ∞)

In this case, the limit is indeterminate (∞ - ∞).

2. As x approaches negative infinity, the term 3x² will tend to positive infinity, and |x²| will also tend to positive infinity. Thus, the expression becomes:

lim (3x² - |x²|) = lim (∞ - ∞)

Again, in this case, the limit is indeterminate (∞ - ∞).

3. As x approaches zero, the term 3x² will tend to zero, and |x²| will also tend to zero. Thus, the expression becomes:

lim (3x² - |x²|) = lim (0 - 0) = 0

Therefore, the limit of the expression lim (x² - √x⁴ + 3x²) as x approaches any value is indeterminate (∞ - ∞), except when x approaches zero, where the limit is 0.

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Answer:

arc LKF = 208°

Step-by-step explanation:

the angle FLX between the tangent and the secant is half the measure of the intercepted arc LKF , then intercepted arc is twice angle FLX , so

arc LKF = 2 × 104° = 208°

The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;

1. ΔAJK ~ ΔSWY according to the SAS similarity postulate

2. ΔLMN ~ ΔLPQ according to the AA similarity postulate

3. ΔPQN ~ ΔLMN

LM = 12, QP = 8

4. ΔLMK~ΔLNJ

NL = 21, ML = 14

Similar triangles are triangles that have the same shape but may have different sizes.

1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;

24/16 = 3/2

18/12 = 3/2

The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.

2. The ratio of the corresponding sides in each of the triangles are;

MN/LN = 8/10 = 4/5

PQ/LQ = 12/(10 + 5) = 12/15 = 4/5

The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule

3. The alternate interior angles theorem indicates;

Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;

ΔPQN ~ ΔLMN by the AA similarity postulate

LM/QP = (x + 3)/(x - 1) = 18/12

12·x + 36 = 18·x - 18

18·x - 12·x = 36 + 18 = 54

6·x = 54

x = 54/6 = 9

LM = 9 + 3 = 12

QP = x - 1

QP = 9 - 1 = 8

4. The similar triangles are; ΔLMK and ΔLNJ

ΔLMK ~ ΔLNJ by AA similarity postulate

ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)

ML/NL = LK/LJ = (24 - 8)/24

(24 - 8)/24 = (6·x + 2)/((7·x + 7)

16/24 = (6·x + 2)/(7·x + 7)

16 × (7·x + 7) = 24 × (6·x + 2)

112·x + 112 = 144·x + 48

144·x - 112·x = 32·x = 112 - 48 = 64

x = 64/32 = 2

ML = 6 × 2 + 2 = 14

NL = 7 × 2 + 7 = 21

MN = 2 + 5 = 7

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Answer:

35

Step-by-step explanation:

18-4 = 14 minutes

number of words typed in 14 minutes is 765-275= 490 words

490 words/14 mins= appox 35

Answer:

acute triangle

Step-by-step explanation:

for the given sides to form a triangle

the sum of any 2 sides must be greater than the third side

given 56 , 90 , 100

56 + 90 = 146 > 100

56 + 100 = 156 > 90

90 + 100 = 190 > 56

the condition is met so the 3 lengths will form a triangle.

let a = 56 , b = 90 and c = 100 , where c represents the longest of the 3 sides , then

• if a² + b² = c² , then triangle is right

• if a² + b² > c² , then triangle is acute

• if a² + b² < c² , then triangle is obtuse

a² + b² = 56² + 90² = 3136 + 8100 = 11,236

c² = 100² = 10, 000

since a² + b² > c² , then triangle is acute

(a) The average cost in 2010 is $2088.82.

(b) A graph of the function g for the period 2006 to 2015 is: D. graph D.

(c) Assuming that the graph remains accurate, its shape suggest that: B. the average cost increases at a slower rate as time goes on.

Based on the information provided, we can logically deduce that the average annual cost (in dollars) for health insurance in this country can be approximately represented by the following function:

g(x) = -1736.7 + 1661.4Inx

where:

x = 6 corresponds to the year 2006.

For the year 2011, the average cost (in dollars) is given by;

x = (2010 - 2006) + 6

x = 4 + 6

x = 10 years.

Next, we would substitute 10 for x in the function:

g(10) = -1736.7 + 1661.4In(10)

g(10) = $2088.82

Part b.

In order to plot the graph of this function, we would make use of an online graphing tool. Additionally, the years would be plotted on the x-axis while the average annual cost would be plotted on the x-axis of the cartesian coordinate as shown below.

Part c.

Assuming the graph remains accurate, the shape of the graph suggest that the average cost of health insurance increases at a slower rate as time goes on.

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The measure of the indicated arc angle is 134 degrees.

The angle subtended by the arc at the centre of the circle is the angle of the arc.

Therefore, the central angle of an arc is the angle at the centre of the circle between the two radii subtended by the arc.

Hence, the central angle is also known as the arc's angular distance.

Therefore, the measure of an arc is the measure of its central angle.

Hence, the measure of the indicated arc is 134 degrees.

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The conditions that prove that the triangles ΔABC and ΔXYZ are similar, ΔABC ~ ΔXYZ, based on the order of the lettering are;

BA/YX = BC/YZ = AC/XZ

AC/XZ = BA/YX, ∠A ≅ ∠C

Similar triangles are triangles that have the same shape (the same two or all three interior angles in each triangle) but in which may have different sizes.

Triangles are similar if they equivalent ratio for their sides and and if the angles in each triangle are congruent to the angles in the other triangle

Therefore, the conditions that indicates that two triangles are that one triangle is a scaled image of the other triangle, therefore, the ratio of the corresponding sides of the triangles are equivalent, which indicates;

ΔABC is similar to ΔXYZ if we get;

A condition that indicates that two triangle are similar is the Side Angle Side, SAS, similarity postulate, which states that two triangles are congruent if the ratio of two corresponding sides are equivalent and the angle between the two sides in the two triangles are congruent, therefore;

ΔABC is similar to ΔXYZ if we get;

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Answer:

Step-by-step explanation:

Let's revisit the equation and the solution using the values x = -4, 0, and 5.

The given equation is: 5 + x - 12 = x - 7

Let's go through the steps again to solve it:

5 + x - 12 = x - 7

Combine like terms:

x - 7 = x - 7

Now, subtract x from both sides:

x - x - 7 = x - x - 7

Simplify:

-7 = -7

The equation simplifies to -7 = -7, which is true.

Now, let's use the values x = -4, 0, and 5 to verify the solution.

For x = -4:

5 + (-4) - 12 = (-4) - 7

-11 = -11

The equation is satisfied for x = -4.

For x = 0:

5 + 0 - 12 = 0 - 7

-7 = -7

The equation is satisfied for x = 0.

For x = 5:

5 + 5 - 12 = 5 - 7

-2 = -2

The equation is satisfied for x = 5.

Therefore, the solution x = 0 is correct, and it satisfies the equation for the given values. My earlier statement that x = -4 and x = 5 do not satisfy the equation was incorrect. I apologize for the confusion caused.

Answer:

Step-by-step explanation:

The correct answer is A. A rectangle the same size as the base.

When a rectangular pyramid is sliced parallel to the base, the resulting cross-section is a rectangle that is the same size as the base. The parallel slicing ensures that the cross-section maintains the same dimensions as the base of the pyramid. Therefore, option A, a rectangle the same size as the base, represents the cross-section.

Step-by-step explanation:

a. f(g(0)) = f(0^2 - 2) = f(-2) = -2 + 3 = 1

b. g(f(0)) = g(0+3) = g(3) = 3^2 - 2 = 7

c. f(g(x)) = g(x) + 3 = x^2 - 2 + 3 = x^2 + 1

d. g(f(x)) = f(x)^2 - 2 = (x+3)^2 - 2 = x^2 + 6x + 7

e. f(f(-2)) = f(-2+3) = f(1) = 1+3 = 4

f. g(g(4)) = g(4^2 - 2) = g(14) = 14^2 - 2 = 194

g. f(f(x)) = f(x+3) = (x+3)+3 = x+6

h. g(g(x)) = g(x^2 - 2) = (x^2 - 2)^2 -2 = x^4 - 4x^2 + 2

Step-by-step explanation:

in the formula

y = Ae^rt

y is 675,000

A is 250,000

r is 0.08

to get the value of t

y = Ae^rt

y/A = e^rt

ln(y/A) = rt

[ln(y/A)]/r = t

The sentence that contains no errors in subject-verb agreement is: "Those people whose long lives are about to end feel reassured by this belief."

This sentence correctly matches the plural subject "Those people whose long lives are about to end" with the plural verb "feel." The verb "feel" agrees in number with the subject "people."

The other sentences have subject-verb agreement errors:

"Those people whose long lives are about to end feels reassured by this belief."

The verb "feels" does not agree with the plural subject "Those people." It should be "feel."

"A person whose long life is about to end feel reassured by this belief."

The verb "feel" does not agree with the singular subject "A person." It should be "feels."

"A person whose long life are about to end feels reassured by this belief."

The verb "are" does not agree with the singular subject "A person." It should be "is."

Therefore, the sentence with no errors in subject-verb agreement is: "Those people whose long lives are about to end feel reassured by this belief."

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1/3 of a full rotation: (-0.5, √3/2)

1/2 of a full rotation: (-1, 0)

2/3 of a full rotation: (0.5, -√3/2)

These are the coordinates of point P after the corresponding rotations around the unit circle's center.

To find the coordinates of point P after the unit circle rotates a certain amount counter-clockwise around its center, we can use the properties of the unit circle and the trigonometric functions.

1/3 of a full rotation:

A full rotation in the unit circle corresponds to 360 degrees or 2π radians. Therefore, 1/3 of a full rotation is equal to (1/3) * 360 degrees or (1/3) * 2π radians.

When the unit circle rotates 1/3 of a full rotation, point P will end up at an angle of (1/3) * 2π radians or 120 degrees from the positive x-axis.

In the unit circle, the x-coordinate of a point on the circle represents the cosine of the angle, and the y-coordinate represents the sine of the angle.

At an angle of 120 degrees or (1/3) * 2π radians, the cosine is -0.5 and the sine is √3/2.

Therefore, the coordinates of point P after rotating 1/3 of a full rotation are (-0.5, √3/2).

1/2 of a full rotation:

Similarly, 1/2 of a full rotation is equal to (1/2) * 360 degrees or (1/2) * 2π radians.

When the unit circle rotates 1/2 of a full rotation, point P will end up at an angle of (1/2) * 2π radians or 180 degrees from the positive x-axis.

At an angle of 180 degrees or (1/2) * 2π radians, the cosine is -1 and the sine is 0.

Therefore, the coordinates of point P after rotating 1/2 of a full rotation are (-1, 0).

2/3 of a full rotation:

Again, 2/3 of a full rotation is equal to (2/3) * 360 degrees or (2/3) * 2π radians.

When the unit circle rotates 2/3 of a full rotation, point P will end up at an angle of (2/3) * 2π radians or 240 degrees from the positive x-axis.

At an angle of 240 degrees or (2/3) * 2π radians, the cosine is 0.5 and the sine is -√3/2.

Therefore, the coordinates of point P after rotating 2/3 of a full rotation are (0.5, -√3/2).

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You will need to purchase approximately 1/42 of a tube, which is less than a full tube. In practical terms, you would need to purchase at least one tube to meet your requirement of 42 ounces of capsaicin arthritis rub.

To determine the number of tubes of capsaicin arthritis rub you will need to purchase, we can divide the total required quantity by the quantity in each tube.

Given that the cost of capsaicin arthritis rub is $21 and you need to buy 42 ounces, we need to find out how many ounces are in each tube.

Let's assume that each tube contains x ounces of capsaicin arthritis rub.

Now we can set up a proportion to solve for x:

42 ounces / x tubes = 1 tube / x ounces

Cross-multiplying gives us:

42x = 1 * x

Simplifying the equation:

42x = x

Dividing both sides of the equation by x (since x cannot be zero):

42 = 1

Since this equation is not true, it means that there is an error in our assumption. We need to revise our assumption.

Let's assume that each tube contains 1 ounce of capsaicin arthritis rub.

Now we can set up a new proportion:

42 ounces / x tubes = 1 tube / 1 ounce

Cross-multiplying gives us:

42x = 1 * 1

Simplifying the equation:

42x = 1

Dividing both sides of the equation by 42:

x = 1/42

As a result, you will need to buy less than a full tube—roughly 1/42 of a tube. In order to get the 42 ounces of capsaicin arthritis rub you need, you would essentially need to buy at least one tube.

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The linear equation from the given table is expressed as:

y =  -1500x + 8350

The general form for the equation of a line in slope intercept form is:

y = mx + c

where:

m is slope

c is y-intercept

The formula for the equation of a line through two coordinates is:

(y - y₁)/(x - x₁) = (y₂ - y₁)/(x₂ - x₁)

We will take the two coordinates (1, 6850) and (2, 5350)

Thus:

(y - 6850)/(x - 1) = (5350 - 6850)/(2 - 1)

(y - 6850)/(x - 1) = -1500

y - 6850 = -1500x + 1500

y = -1500x + 1500 + 6850

y =  -1500x + 8350

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Answer:

AB = √3

Step-by-step explanation:

Since ABCD is a rectangle, all angles are 90°

∠CDA = 90°

⇒ ∠CDB + ∠BDA = 90

⇒ ∠BDA = 60

In ΔABD,

sin(∠BDA) = opposite/ hypotenuse = AB / BD

⇒ sin(60) = AB/2

⇒ AB = 2 sin(60)

⇒ AB = 2 (√3)/2

AB = √3

a. There are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.

b. There are 14,950 ways to form a 4-person committee from a class of 26 members.

a. To select a president, a vice president, and a secretary from a class of 26 members, we can use the concept of permutations.

For the president position, we have 26 choices. After selecting the president, we have 25 choices remaining for the vice president position. Finally, for the secretary position, we have 24 choices left.

The total number of ways to select a president, a vice president, and a secretary is obtained by multiplying the number of choices for each position:

Number of ways = 26 * 25 * 24 = 15,600

Therefore, there are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.

b. To form a 4-person committee from a class of 26 members, we can use the concept of combinations.

The number of ways to choose a committee of 4 people can be calculated using the formula for combinations:

Number of ways = C(n, r) = n! / (r!(n-r)!)

where n is the total number of members (26 in this case) and r is the number of people in the committee (4 in this case).

Plugging in the values, we have:

Number of ways = C(26, 4) = 26! / (4!(26-4)!)

Calculating this expression, we get:

Number of ways = 26! / (4! * 22!)

Using factorials, we simplify further:

Number of ways = (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1) = 14,950

Therefore, there are 14,950 ways to form a 4-person committee from a class of 26 members.

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a) Option a) 9 - 1/2 is equal to 17/2.

b) Option b) 27 - 2/3 is equal to 79/3.

a) The expression 9 - 1/2 can be simplified by finding a common denominator for the terms. The common denominator for 9 and 1/2 is 2.

Multiplying 9 by 2/2, we get:

9 * (2/2) = 18/2

So, the expression 9 - 1/2 can be simplified to:

18/2 - 1/2 = 17/2

Therefore, option a) 9 - 1/2 is equal to 17/2.

b) The expression 27 - 2/3 can be simplified in a similar manner by finding a common denominator for the terms. The common denominator for 27 and 2/3 is 3.

Multiplying 27 by 3/3, we get:

27 * (3/3) = 81/3

So, the expression 27 - 2/3 can be simplified to:

81/3 - 2/3 = 79/3

Therefore, option b) 27 - 2/3 is equal to 79/3.

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Answer:

(fg)(x)= (x²+6)(x²-x+9)

multiply the terms:

(fg)(x)= x²(x²-x+9) +6(x²-x+9)

add the like terms:

(fg)(x)= (x⁴-x³+9x²)+(6x²-6x+54)

and you get your final answer:

(fg)(x)= x⁴-x³+15x²-6x+54

Approximately 4.6 pounds of pecans are needed for one pecan pie.You should buy approximately 41.4 pounds of pecans to make 9 pecan pies.

To determine the number of pounds of pecans needed to make 9 pecan pies, we need to consider the amount of pecans required per pie and the number of pies we are making.

The recipe calls for 1 cup of pecans per pie, but we don't have a measuring cup available. However, we do have nutritional information from a bag of pecans, which states that there are 684 calories in 1 cup (99g) of pecans.

To find out how many pecans are in a pound, we can use the information that 1 cup of pecans weighs 99 grams. Since there are 454 grams in a pound, we can set up the following proportion:

1 cup (99g) = x pounds (454g)

Cross-multiplying, we get:

99g * x pounds = 1 cup * 454g

Simplifying, we have:

99x = 454

Dividing both sides by 99, we find:

x ≈ 4.5959 pounds

So, approximately 4.6 pounds of pecans are needed for one pecan pie.

Since we are making 9 pecan pies, we multiply the amount needed for one pie by the number of pies:

4.6 pounds/pie * 9 pies = 41.4 pounds

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Answer:

40, 50, 30 Yes

13, 5, 12 Yes

10, 10, 15 Yes

41, 9, 40 Yes

Step-by-step explanation:

16, 10, 6

10 + 6 = 16

No

40, 50, 30

30 + 40 > 50

Yes

13, 5, 12

5 + 12 > 13

Yes

70, 20, 25

20 + 25 < 70

No

10, 10, 15

10 + 10 >15

Yes

41, 9, 40

9 + 40 > 41

Yes

To show that ABC ≅ AXYZ by SAS, we would need to establish that angle BAC ≅ angle XYZ.

To show that triangles ABC and AXYZ are congruent by the Side-Angle-Side (SAS) criterion, we need to establish that two corresponding sides and the included angle are congruent.

Given AB ≅ XY and BC ≅ YZ, we already have two corresponding sides congruent.

To complete the congruence by the SAS criterion, we need to establish that the included angles are congruent. In this case, the included angle is angle BAC (or angle XYZ).

Therefore, to show that ABC ≅ AXYZ by SAS, we would need to establish that angle BAC ≅ angle XYZ.

None of the answer choices directly addresses the congruence of the angles. So, none of the given options (A, B, C, D) are sufficient to show the congruence of the triangles by SAS.

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Part 1- The lower class boundary for the first class is 100.

Part 2- Approximately 75% of students take exactly two courses.

Part 1:

To find the lower class boundary for the first class, we need to consider the given class intervals. The lower class boundary is the smallest value within each class interval.

Given the class intervals:

100 - 104

105 - 109

110 - 114

115 - 119

120 - 124

125 - 129

130 - 134

The lower class boundary for the first class interval (100 - 104) would be 100.

So, the lower class boundary for the first class is 100.

Part 2:

To determine the percentage of students who take exactly two courses, we need to calculate the relative frequency for that particular category.

Given the data:

of Courses Frequency Relative Frequency Cumulative Frequency

1 23 0.4423 23

2 - 39

3 13 0.25 -

We can see that the cumulative frequency for the second class (2 courses) is 39. To find the relative frequency for this class, we need to divide the frequency by the total number of students surveyed, which is 52.

Relative Frequency = Frequency / Total Number of Students

Relative Frequency for 2 courses = 39 / 52 ≈ 0.75 (rounded to 4 decimal places)

To convert this to a percentage, we multiply the relative frequency by 100.

Percentage of students taking exactly two courses = 0.75 * 100 ≈ 75%

Therefore, approximately 75% of students take exactly two courses.

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Question

Part 1.

Data was collected for 300 fish from the North Atlantic. The length of the fish (in mm) is summarized in the GFDT below.

Lengths (mm) Frequency

100 - 104 1

105 - 109 16

110 - 114 71

115 - 119 108

120 - 124 83

125 - 129 18

130 - 134 3

What is the lower class boundary for the first class?

class boundary =

Part 2

In a student survey, fifty-two part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below:

Please round your answer to 4 decimal places for the Relative Frequency if possible.

# of Courses Frequency Relative Frequency Cumulative Frequency

1 23 0.4423 23

2   39

3 13 0.25

What percent of students take exactly two courses? %

Based on the given transformations, the correct graph that represents function g(x) = tan(x-4) + 2 is option c.

To determine the graph that represents function g(x) = tan(x-4) + 2, we can analyze the transformations applied to the parent tangent function.

The parent tangent function has the form f(x) = tan(x). The given function g(x) = tan(x-4) + 2 introduces two transformations:

1. Horizontal shift: The function is shifted 4 units to the right compared to the parent function f(x) = tan(x). This means the graph of g(x) will be shifted to the right by 4 units.

2. Vertical shift: The function is shifted 2 units up compared to the parent function f(x) = tan(x). This means the graph of g(x) will be shifted upward by 2 units.

Based on the given transformations, the correct graph that represents function g(x) = tan(x-4) + 2 is option c. The graph in option c shows a rightward shift of 4 units (horizontal shift) and an upward shift of 2 units (vertical shift), matching the transformations described for g(x).

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