Which figure shows a reflection of pre-image QRS over the line t?
A.
90° clockwise rotation around point B

B.
180° clockwise rotation around point B

C.
reflection over line a

D.
translation 4 units up

Comparing the graphs of F(x) = [tex]x^2[/tex] and G(x) = [tex]2x^2[/tex] - 5 shows that G(x)'s graph is F(x)'s graph stretched vertically by a factor of 2 and then shifted 5 units down.

There is no horizontal shift involved.

Comparing the functions F(x) = [tex]x^2[/tex] and G(x) = [tex]2x^2[/tex] - 5, we observe two main transformations applied to F(x) to obtain G(x).

First, the coefficient 2 in front of[tex]x^2[/tex] in G(x) indicates that the graph of F(x) is stretched vertically by a factor of 2.

This stretching makes the graph of G(x) stretch away from the x-axis, becoming narrower compared to F(x).

Second, the term -5 added to [tex]2x^2[/tex] suggests that the entire graph of F(x) after being stretched is then shifted 5 units down.

It's important to note that this vertical shift is down because of the negative sign in front of 5; there is no horizontal shift involved.

Therefore, the statement that best compares the graph of G(x) to the graph of F(x) is:

Based on the analysis, the correct statement is:
D. The graph of G(x) is the graph of F(x) stretched vertically and shifted 5 units down.

1. Identify the base function:

Both F(x) = x^2 and G(x) = 2x^2-5 share the same base function, which is x^2. This means their graphs have the same basic shape, a parabola.

2. Analyze the transformations:

Vertical stretch: The coefficient of x^2 in G(x) is 2, which is a vertical stretch factor of 2 compared to F(x). This stretches the graph of G(x) vertically by a factor of 2, making it narrower.

Vertical shift: The constant term in G(x) is -5, which corresponds to a downward shift of 5 units compared to F(x). This moves the entire graph of G(x) 5 units down.

3. Combine the transformations:

The graph of G(x) is obtained by taking the graph of F(x), stretching it vertically by a factor of 2, and then shifting it down by 5 units.

For graph refer to image:

The digit in the tenths place of the number 123.456 is 4. In the general context of decimals and rounding, if the following digit (hundredths place) is 5 or higher, the tenths place is rounded up when dropped.

The number in the tenths place of 123.456 is 4. When looking at decimal numbers, the first digit to the right of the decimal point represents the tenths place. To illustrate, the number 123.456 can be broken down as (1  imes 10^2) + (2  imes 10^1) + (3  imes 10^0) + (4  imes 10^-1) + (5  imes 10^-2) + (6  imes 10^-3), where the digit 4 is in the tenths place and holds the value of four-tenths or 0.4.

Regarding rounding to the tenths place, if you had a number like 1,459.08 and need to round it, you would look at the digit in the hundredths place which is 8. Since the first dropped digit is 5 or higher, you round up, resulting in 1,459.1.

The volume of the solid formed by rotating the region enclosed by y = x^3 and y = 9x about the x-axis in the first quadrant can be found by integrating from 0 to 3, the square of the outer and inner radius, multiplied by π. Solve the integral to get the volume.

To find the volume of the solid formed by rotating the region enclosed by y = x^3 and y = 9x about the x-axis, we need to use the method of discs/washers. The volume V is given by the following integral:

 ∫[a,b] π(r(x)^2 - R(x)^2) dx

For our given curves, r(x) = 9x and R(x) = x^3 since 9x ≥ x^3 for 0 ≤ x ≤ 3. Therefore, we get:

 V = ∫[0,3] π[(9x)^2 - (x^3)^2] dx

Solving this integral will yield the volume of the solid:

 V = π ∫[0,3] (81x^2 - x^6) dx

Calculating this integral will give the volume of the solid.

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The student is required to combine three algebraic fractions with different denominators using factoring to find a common denominator and then simplify the expression.

The question entails a topic in algebra, specifically with respect to combining expressions with different denominators, which requires finding a common denominator, and working with signs and exponents. The problem presents three fractions that should be combined: (x+6)/(x^2+8x+15), (3x)/(x+5), and (x-3)/(x+3).

Firstly, note that the denominator x^2+8x+15 can be factored into (x+3)(x+5). This will allow us to identify a common denominator for all three fractions, which is (x+3)(x+5). We rewrite the fractions so that each has the common denominator:

Now we simply combine these three fractions over the common denominator:

After the combination, the terms must be simplified and ordered in descending powers of x, which is the final answer.

Descending powers of x is [tex]\[ \frac{2x^2 + 8x + 15}{(x + 3)(x + 5)} \][/tex].

To combine the given expressions, we need to find a common denominator and then combine the numerators. Here are the expressions given:

[tex]\[ \frac{x}{x^2 + 8x + 15} + \frac{3x}{x + 5} - \frac{x - 3}{x + 3} \][/tex]

First, let's factor the quadratic denominator in the first term if possible and identify the common denominator:

The quadratic [tex]\( x^2 + 8x + 15 \)[/tex] can be factored into [tex]\( (x + 3)(x + 5) \)[/tex], since 3 and 5 are factors of 15 that add up to 8.

Now we have:

[tex]\[ \frac{x}{(x + 3)(x + 5)} + \frac{3x}{x + 5} - \frac{x - 3}{x + 3} \][/tex]

The common denominator will be [tex]\( (x + 3)(x + 5) \)[/tex].

Now, let's rewrite each fraction with the common denominator:

The second term already has [tex]\( x + 5 \)[/tex] in the denominator, so we multiply the numerator and denominator by [tex]\( x + 3 \)[/tex]to have the common denominator.

[tex]\[ \frac{3x}{x + 5} \rightarrow \frac{3x(x + 3)}{(x + 5)(x + 3)} \][/tex]

The third term has [tex]\( x + 3 \)[/tex] in the denominator, so we multiply the numerator and denominator by \( x + 5 \) to have the common denominator.

[tex]\[ \frac{x - 3}{x + 3} \rightarrow \frac{(x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]

Now all terms have a common denominator, and we can combine them as follows:

[tex]\[ \frac{x}{(x + 3)(x + 5)} + \frac{3x(x + 3)}{(x + 3)(x + 5)} - \frac{(x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]

Combine the numerators while keeping the denominator the same:

[tex]\[ \frac{x + 3x(x + 3) - (x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]

Now, let's expand and simplify the numerator:

[tex]\[ x + 3x^2 + 9x - (x^2 + 2x - 15) \][/tex]

[tex]\[ x + 3x^2 + 9x - x^2 - 2x + 15 \][/tex]

Combine like terms:

[tex]\[ 3x^2 - x^2 + x + 9x - 2x + 15 \][/tex]

[tex]\[ 2x^2 + 8x + 15 \][/tex]

Now, let's put it all over the common denominator:

[tex]\[ \frac{2x^2 + 8x + 15}{(x + 3)(x + 5)} \][/tex]

This is the simplified expression in descending powers of \( x \). Since the numerator is already in descending powers of \( x \), this is the final answer. There is no further simplification possible because the numerator and the denominator do not have common factors other than 1.

Answer:

therers no box

Step-by-step explanation:

When [tex]x =  - 7, 2{x^2} + 10x - 20 = 8[/tex] and If [tex]2{x^2} + 10x -20[/tex] is divided by [tex]\left( {x + 7} \right)[/tex], the remainder is 12. Option (D) is correct and option (E) is correct.

Further Explanation:

Explanation:

If division of a polynomial by a binomial result in a remainder of zero means that the binomial is a factor of polynomial.

The synthetic division can be expressed as follows,

[tex]\begin{aligned}- 7\left| \!{\nderline {\,{2\,\,\,\,\,\,\,\,\,\,10\,\,\,\,\,\,\,\,\,\, - 20} \,}} \right.\hfill\\\,\,\,\,\,\,\underline {\,\,\,\,\,\,\,\,\,\, - 14\,\,\,\,\,\,\,\,\,\,\,\,\,\,28}  \hfill\\ \,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\, - 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8 \hfill\\\end{aligned}[/tex]

The last entry of the synthetic division tells us about remainder and the last entry of the synthetic division is [tex]8[/tex]. Therefore, the remainder of the synthetic division is [tex]8[/tex].

When [tex]x =  - 7, 2{x^2} + 10x - 20 = 8[/tex] and If [tex]2{x^2} + 10x -20[/tex] is divided by [tex]\left( {x + 7} \right)[/tex], the remainder is 12. Option (D) is correct and option (E) is correct.

Option (A) is not correct.

Option (B) is not correct.

Option (C) is not correct.

Option (F) is not correct.

Learn more:

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Synthetic Division

Keywords: division, factor (x+7), remainder 8, statements, true, apply, divided, binomial synthetic division, long division method, coefficients, quotients, remainders, numerator, denominator, polynomial, zeros, degree.

The average mileage covered per day is (x + y + z) / 3. It provides a balanced representation of the man's daily driving performance throughout the three-day period.

To find the average mileage covered per day, you need to calculate the total mileage covered over the three days and then divide it by the number of days (which is 3 in this case).

The total mileage covered over the three days is: x + y + z

The average mileage covered per day is: (x + y + z) / 3

This formula finds the mean distance covered each day.

By dividing the total distance by the number of days, the average mileage smooths out any fluctuations in daily distances and gives a more comprehensive view of his overall performance.

Learn more about the distance here:

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Final answer:

To find the tangent lines to the curve y = (x - 1)/(x + 1) that are parallel to the given line, convert the given line to slope-intercept form to find the slope, take the derivative of the curve to find where its slope matches the line's slope, and utilize these points to write the equations of the tangent lines.

Explanation:

To find the equations of the tangent lines to the curve y = (x − 1)/(x + 1) that are parallel to the line x − 2y = 3, we first need to find the slope of the given line by rewriting it in slope-intercept form (y = mx + b), where m is the slope. Rewriting x − 2y = 3 gives us y = ⅓x - ⅓; thus, the slope (m) is ⅓.

Next, we find the derivative of the curve, y' = dy/dx, which will give us the slope of the tangent at any point x. Taking the derivative of y = (x − 1)/(x + 1) using the quotient rule or another differentiation method, we find a general expression for y'. We then set y' equal to ⅓ to find the points where the slope of the tangent is equal to the slope of the given line.

After determining the x-values where the tangent has the correct slope, we calculate the corresponding y-values on the curve and use these points to write the equations of the tangent lines in the form y = mx + b, substituting the slope (⅓) and our found points (x, y).

To find the equations of the tangent lines to the given curve that are parallel to the given line, we differentiate the curve's equation to find its slope, equate it to the slope of the given line, solve for x, substitute the values back into the curve's equation to find the corresponding y-values, and use the point-slope form of the equation of a line to find the equations of the tangent lines.

To find the equations of the tangent lines to the curve y = (x − 1)/(x + 1) that are parallel to the line x − 2y = 3, we can use the slope of the given line as the slope of the tangent lines. The slope of the given line is 1/2, so the slope of the tangent lines is also 1/2.

Next, we can differentiate the equation of the curve y = (x − 1)/(x + 1) with respect to x to find the slope of the curve at any point. Taking the derivative, we get dy/dx = 2/(x + 1)².

Since the tangent lines are parallel to the given line, their slopes are equal. Therefore, we can equate the slope of the curve to the slope of the tangent lines and solve for x:

2/(x + 1)² = 1/2

Solving this equation, we get x = -1 or x = 1.

Substituting these values of x back into the equation of the curve, we can find the corresponding y-values. The coordinates of the points where the tangent lines intersect the curve are (-1, -2) and (1, 2).

Finally, we can use the point-slope form of the equation of a line to find the equations of the tangent lines:

Tangent line at (-1, -2): y + 2 = (1/2)(x + 1)

Tangent line at (1, 2): y - 2 = (1/2)(x - 1)

Final answer:

Upon reviewing the proof and the assumed relationship between the angles, the given information leads to an incorrect conclusion of angle BAC being 90°. The mistake indicates a reassessment of angle relationships is required to determine the true measure of angle BAC in this question.

Explanation:

To prove that angle BAC is 72° in a triangle ABC where angle B is twice angle C and where AD bisects angle BAC with AB equal to CD, we proceed as follows:

Let angle BAC be represented as 2x. Therefore, since AD bisects angle BAC, each angle BAD and DAC is x.

Since angle B is twice angle C, let angle C be x and angle B then is 2x. It is given that AB is equal to CD, meaning triangle ABD is isosceles with angles BAD = DAC.

In isosceles triangle ABD, the angles at base AD are equal, which means each of these angles is x. Thus, the sum of angles in triangle ABD is x (at A) + 2x (at B) + x (at D) =  180°.

Combining these angles, we get 4x = 180°. Dividing both sides by 4, we obtain x = 45°.

Since angle BAC is 2x and x is 45°, angle BAC is therefore 90°.

This leads to a contradiction to the original assumption and upon review reveals the mistake in the assumption about the relationship of the angles given as twice. The correct relationship should be considered to find the accurate measure of angle BAC.

The correct answer is 8

Answer:

8 inches

Step-by-step explanation:

Given,

In two quadrilateral GAME and G'A'M'E',

ME = 35 inches, AM = GE = 56 inches,

M'E' = 14 inches,

Also, the perimeter of quadrilateral GAME = 167 inches,

⇒ GA + AM + ME + GE = 167

⇒ GA + 56 + 35 + 56 = 167

⇒ GA + 147 = 167

⇒ GA = 20 inches.

Now, GAME is similar to G'A'M'E' are similar,

By the property of similar figures,

[tex]\frac{ME}{M'E'}=\frac{GA}{G'A'}[/tex]

[tex]\implies G'A'=\frac{M'E'\times GA}{ME}=\frac{14\times 20}{35}=\frac{280}{35}=8\text{ in}[/tex]

Hence, the length of Segment line G'A' is 8 inches.

Answer:

C.[tex]8.22 h\geq[/tex]$ 623

Step-by-step explanation:

We are given that Sophia is saving money for a new bicycle.

The bicycle will  cost atleast $623.

Sophia makes $8.22 per hour.

We have to find the inequality that could be used to find the number of hours Sophia needs to work to make enough money to buy a new bicycle.

Let Sophia works h hours to make enough money to buy a new bicycle.

Sophia makes money per hour =$8.22

Total money made by Sophia in h hours =[tex]8.22 h[/tex]

According to question

[tex]8.22 h\geq [/tex]$623

Hence, option C is true.

Answer with explanation:

Slope of Line= -2

The line passes through the point , (6,8).

⇒Equation of line passing through point , (a,b) having slope ,m is

y -b = m (x -a)

⇒≡Equation of line passing through point , (6,8) having slope ,-2 is

→y -8 = -2× (x -6)

→y -8 = -2 x + 12⇒⇒ Using Distributive property of multiplication with respect to Subtraction

→2 x + y= 12 + 8

→2 x + y=20

Required Equation of line.

Answer:

see the explanation

Step-by-step explanation:

we know that

The Side Angle Side postulate (SAS) states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.

In this problem

Triangles ABC and DEC are congruent by SAS postulate

Because

BC≅EC

AC≅DC

and the include angle

m∠BCA≅m∠ECD ----> by vertical angles

Remember that

If two triangles are congruent, its corresponding sides and corresponding angles are congruent

therefore

BC≅EC  

AC≅DC

and

BA≅ED

Answer:

Partial and negative tickets cannot be sold, so the minimum number values of e and s are 0. If s = 0, then e = 16, and if e = 0, then s = 20. Therefore, the values of s are whole numbers from 0 to 20 and the values of e are whole numbers between 0 and 16. The greatest number of student tickets sold was 20 and the greatest number of nonstudent tickets sold was 16.

Step-by-step explanation:

Answer:

Option (b) is correct.

The general term of the sequence is 2n

Step-by-step explanation:

Given : The  sequence 2, 4, 6, 8, 10, . . .

We have to find the representation of the general term of the given sequence 2, 4, 6, 8, 10, . . .

Consider the given sequence 2, 4, 6, .....

The general term of an arithmetic sequence is given by [tex]a_n=a+(n-1)d[/tex]

where, a = first term

d is common difference

For the given sequence a = 2

and d = 2

Then [tex]a_n=2+(n-1)2=2+2n-2=2n[/tex]

Thus, The general term of the sequence is 2n