16. The dimensions of a window are 3x + 10 and 2x + 6. What is the area of the window?

Answer:

x.

Step-by-step explanation:

If h(x) is the inverse of f(x) them h(f(x)) = x.

If h(x) is the inverse of f(x) then the  the value of h(f(x)) is x.

A relation is a function if it has only One y-value for each x-value.

The inverse function of a function f is a function that undoes the operation of f.

If h(x) is the inverse of f(x), then by definition we have:

h(f(x)) = x

This is because the composition of h and f is the identity function, which means that applying h to f(x) gives us back x.

Therefore, the value of h(f(x)) is x.

Hence, If h(x) is the inverse of f(x) then the  the value of h(f(x)) is x.

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Answer:

x=8 is a true solution of the radical equation

Step-by-step explanation:

we have

[tex]x-3=\sqrt{4x-7}[/tex]

Solve for x

squared both sides

[tex](x-3)^{2}=4x-7\\\\x^{2}-6x+9=4x-7\\\\ x^{2}-10x+16=0[/tex]

Convert to factored form

[tex]x^{2}-10x+16=(x-2)(x-8)[/tex]

The solutions are x=2 and x=8

Verify the solutions

For x=2

Substitute in the original equation

[tex]2-3=\sqrt{4(2)-7}[/tex]

[tex]-1=1[/tex] ----> is not true

therefore

x=2 is not a true solution of the radical equation

For x=8

Substitute in the original equation

[tex]8-3=\sqrt{4(8)-7}[/tex]

[tex]5=5[/tex] ----> is true

therefore

x=8 is a true solution of the radical equation

The true solution to the radical equation is x = 8, after checking the potential solutions by substituting them back into the original equation.

The student is asked to find the true solution(s) to the radical equation x – 3 = square root of 4x-7. The student has solved the equation, but it is essential to check the solutions by substituting them back into the original equation to ensure they are not extraneous. The student got x = 2 and x = 8 as potential solutions. We must substitute these values back into the original equation to determine their validity:

Hence, the only true solution to the radical equation x – 3 = square root of 4x-7 is x = 8.

Answer: 90 degrees.

Step-by-step explanation: since 360 degrees is a whole circle, subtract 360 from 270 to get 90 degrees.

Answer:

The concentric circles are

[tex]3x^{2}+3y^{2}+12x-6y-21=0[/tex]  and [tex]4x^{2}+4y^{2}+16x-8y-308=0[/tex]

[tex]5x^{2}+5y^{2}-30x+20y-10=0[/tex]  and [tex]3x^{2}+3y^{2}-18x+12y-81=0[/tex]

[tex]4x^{2}+4y^{2}-16x+24y-28=0[/tex]  and [tex]2x^{2}+2y^{2}-8x+12y-40=0[/tex]

[tex]x^{2}+y^{2}-2x+8y-13=0[/tex]  and  [tex]5x^{2}+5y^{2}-10x+40y-75=0[/tex]

Step-by-step explanation:

we know that

The equation of the circle in standard form is equal to

[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex]

where

(h,k) is the center and r is the radius

Remember that

Concentric circles, are circles that have the same center

so

Convert each equation in standard form and then compare the centers

The complete answer in the attached document

Part 1) we have

[tex]3x^{2}+3y^{2}+12x-6y-21=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](3x^{2}+12x)+(3y^{2}-6y)=21[/tex]

Factor the leading coefficient of each expression

[tex]3(x^{2}+4x)+3(y^{2}-2y)=21[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]3(x^{2}+4x+4)+3(y^{2}-2y+1)=21+12+3[/tex]

[tex]3(x^{2}+4x+4)+3(y^{2}-2y+1)=36[/tex]

Rewrite as perfect squares

[tex]3(x+2)^{2}+3(y-1)^{2}=36[/tex]

[tex](x+2)^{2}+(y-1)^{2}=12[/tex]

therefore

The center is the point (-2,1)                                  

Part 2) we have

[tex]5x^{2}+5y^{2}-30x+20y-10=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](5x^{2}-30x)+(5y^{2}+20y)=10[/tex]

Factor the leading coefficient of each expression

[tex]5(x^{2}-6x)+5(y^{2}+4y)=10[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]5(x^{2}-6x+9)+5(y^{2}+4y+4)=10+45+20[/tex]

[tex]5(x^{2}-6x+9)+5(y^{2}+4y+4)=75[/tex]

Rewrite as perfect squares

[tex]5(x-3)^{2}+5(y+2)^{2}=75[/tex]

[tex](x-3)^{2}+(y+2)^{2}=15[/tex]

therefore

The center is the point (3,-2)      

Part 3) we have

[tex]x^{2}+y^{2}-12x-8y-100=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](x^{2}-12x)+(y^{2}-8y)=100[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex](x^{2}-12x+36)+(y^{2}-8y+16)=100+36+16[/tex]

[tex](x^{2}-12x+36)+(y^{2}-8y+16)=152[/tex]

Rewrite as perfect squares

[tex](x-6)^{2}+(y-4)^{2}=152[/tex]

therefore

The center is the point (6,4)      

Part 4) we have

[tex]4x^{2}+4y^{2}-16x+24y-28=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](4x^{2}-16x)+(4y^{2}+24y)=28[/tex]

Factor the leading coefficient of each expression

[tex]4(x^{2}-4x)+4(y^{2}+6y)=28[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]4(x^{2}-4x+4)+4(y^{2}+6y+9)=28+16+36[/tex]

[tex]4(x^{2}-4x+4)+4(y^{2}+6y+9)=80[/tex]

Rewrite as perfect squares

[tex]4(x-2)^{2}+4(y+3)^{2}=80[/tex]

[tex](x-2)^{2}+(y+3)^{2}=20[/tex]

therefore

The center is the point (2,-3)  

Part 5) we have

[tex]x^{2}+y^{2}-2x+8y-13=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](x^{2}-2x)+(y^{2}+8y)=13[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex](x^{2}-2x+1)+(y^{2}+8y+16)=13+1+16[/tex]

[tex](x^{2}-2x+1)+(y^{2}+8y+16)=30[/tex]

Rewrite as perfect squares

[tex](x-1)^{2}+(y+4)^{2}=30[/tex]

therefore

The center is the point (1,-4)  

Part 6) we have

[tex]5x^{2}+5y^{2}-10x+40y-75=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](5x^{2}-10x)+(5y^{2}+40y)=75[/tex]

Factor the leading coefficient of each expression

[tex]5(x^{2}-2x)+5(y^{2}+8y)=75[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]5(x^{2}-2x+1)+5(y^{2}+8y+16)=75+5+80[/tex]

[tex]5(x^{2}-2x+1)+5(y^{2}+8y+16)=160[/tex]

Rewrite as perfect squares

[tex]5(x-1)^{2}+5(y+4)^{2}=160[/tex]

[tex](x-1)^{2}+(y+4)^{2}=32[/tex]

therefore

The center is the point (1,-4)  

Part 7) we have

[tex]4x^{2}+4y^{2}+16x-8y-308=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](4x^{2}+16x)+(4y^{2}-8y)=308[/tex]

Factor the leading coefficient of each expression

[tex]4(x^{2}+4x)+4(y^{2}-2y)=308[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]4(x^{2}+4x+4)+4(y^{2}-2y+1)=308+16+4[/tex]

[tex]4(x^{2}+4x+4)+4(y^{2}-2y+1)=328[/tex]

Rewrite as perfect squares

[tex]4(x+2)^{2}+4(y-1)^{2}=328[/tex]

[tex](x+2)^{2}+(y-1)^{2}=82[/tex]

therefore

The center is the point (-2,1)  

Part 8) Part 9) and Part 10)  in the attached document

Answer:

see explanation

Step-by-step explanation:

Under a reflection in the x- axis

a point (x, y ) → (x, - y )

Hence

A(1, 1 ) →A'(1, - 1 )

B(1, 4 ) → B'(1, - 4 )

P(3, 1 ) → P'(3, - 1 )

Q(3, 4 ) → Q'(3, - 4 )

The number 18 expressed as a percentage of 24 is; 75%

To determine what percentage of 24 is 18;

We must evaluate the percentage as follows;

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Answer:

Step-by-step explanation:

The point-slope form of an equation of a line:

[tex]y-y_1=m(x-x_1)[/tex]

m - slope

The formula of a slope:

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

============================================

We have the points (-4, -3) and (12, 1).

Substitute:

[tex]m=d\frac{1-(-3)}{12-(-4)}=\dfrac{4}{16}=\dfrac{1}{4}[/tex]

Put the volume of a slope and the coordinates of the point (12, 1) to the equation of a line:

[tex]y-1=\dfrac{1}{4}(x-12)[/tex]

The standard formula of an equation of a line:

[tex]Ax+By=C[/tex]

Convert:

[tex]y-1=\dfrac{1}{4}(x-12)[/tex]           multiply both sides by 4

[tex]4y-4=x-12[/tex]             add 4 to both sides

[tex]4y=x-8[/tex]             subtract x from both sides

[tex]-x+4y=-8[/tex]         change the signs

[tex]x-4y=8[/tex]

Step-by-step explanation:

[tex]4x^3+x^2-8x-2\qquad\text{distributive}\\\\=x^2(4x+1)-2(4x+1)\\\\=(4x+1)(x^2-2)[/tex]

Answer:

(-5, -6)

Step-by-step explanation:

The general form of absolute function is,  and

its vertex form is given by:

         .....[1]

where, (h, k) is the vertex

As per the statement:

we have to find the  vertex of the graph of f(x).

On comparing given equation with [1] we have;

we have;

h =-5 and k = -6

⇒Vertex = (-5, -6)

Therefore, the vertex of the graph of f(x) = |x + 5| – 6 is,  (-5, -6)

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Answer:

The answer is (-5, -6)

Answer:

1. W(5,1),X(1,7),Y(9,9) and Z(11,7).

2.A(-8,-4),B(-4,-10),C(-12,-12) and D(-14,-10).

3. E(5,6) ,F(1,12),G(9,14) and H(11,12).

4.O(10,1),P(6,7),Q(14,9) and R(16,7).

Step-by-step explanation:

We are given that a quadrilateral JKLM with vertices J(8,4),K(4,10),L(12,12) and M(14,10)

We have to match a quadrilateral with its correct transformation of given quadrilateral JKLM

1.a transformation 3 units down and 3 units left

By using transformation rule [tex](x,y)\rightarrow (x-3,y-3)[/tex]

The new  vertices  of quadrilateral is (5,1), (1,7),(9,9) and (11,7).

Hence, the quadrilateral WXYZ with vertices W(5,1),X(1,7),Y(9,9) and Z(11,7).

2.A sequence of reflection across x- axis and y-axis in order

Reflection across x- axis

The transformation  rule [tex](x,y)\rightarrow (x,-y)[/tex]

By using this rule

The vertices of quadrilateral are  (8,-4),(4,-10),(12,-12) and (14,-10).

After the reflection across y- axis

The transformations rule

[tex](x,y)\rightarrow (-x,y)[/tex]

By using this rule

We get the new vertices of quadrilateral are (-8,-4),(-4,-10),(-12,-12) and (-14,-10).

Hence, the quadrilateral ABCD with vertices A(-8,-4),B(-4,-10),C(-12,-12) and D(-14,-10).

3.a translation 3 unit left and 2 units up

The transformation rule [tex](x,y)\rightarrow (x-3,y+2)[/tex]

By using this rule

The new vertices are (5,6),(1,12),(9,14) and (11,12).

Hence, the quadrilateral EFGH with vertices E(5,6) ,F(1,12),G(9,14) and H(11,12).

4.a translation 2 units right and 3 units down

The transformation rule

[tex](x,y)\rightarrow (x+2,y-3)[/tex]

By using this rule

The new vertices are (10,1),(6,7),(14,9) and (16,7)

Hence, the quadrilateral OPQR with vertices O(10,1),P(6,7),Q(14,9) and R(16,7).

Answer:

C.

Step-by-step explanation:

Your fractions are missing there fraction bars:

3/4 (20y − 8) + 5 = 1/2 y + 1/4 (20y + 8)  

15y − 6 + 5 = 1/2 y + 5y + 2  

15y − 1 = 11/2 y + 2

A. 11/2y and 2 aren't like terms because one contains the variable y and the other contains no variable

B. The distribute property can't be used there because you don't have 15(y-1) you have 15y-1

C. Subtracting 11/2y sounds like a good step because there is a y term on the opposing side.

15y-1=11/2y+2

Subtracing 11/2y on both sides

9.5y-1=2

That looks pretty good because then you would add 1 on both sides giving:

9.5y =3

Last step would get the y by itself which is dividing both sides by 9.5 giving you 6/19.

D. You could actually do this but it doesn't help you get x by itself.  The equation would look like this: 15/2 y-1/2=11/4 y+1

Answer:

C. Subtract 11/2y from each side of the equation.

Step-by-step explanation:

The methods that we can use to calculate the y-coordinate of the midpoint of a vertical line segment with endpoints at (0,0) and (0,15) are: Option B: Count by hand, Option D: Dividie 15 by 2

Midpoint of a line segment that lies in the mid of that line segment, as the name 'midpoint' suggests.

If the endpoints of the considered line segments are (a,b), and (c,d), then the coordinates of the midpoint would be:

[tex](x,y) = \left(\dfrac{c-a}{2}, \dfrac{d-b}{2}\right)[/tex]

We're specified here that:

Since the line is vertical, we can easily find its midpoint by going up by half of the length of the line segment.

The y-coordinate starts from 0 and goes to 15 and x-coordinate is still all along the line as the line is vertical, so the length's half is (15-0)/2 =15/2 = 7.5 units. This gives the y-coordinate of the midpoint as visible in the formula specified above.

If we go this units up, we will reach the midpoint. Since x-coordinates of the points in the line segment are fixed to 0, so the midpoint's coordinates are (0, 7.5)

We can also count by hand as there is motion only in y-coordinates, so move half of the total motion upwards from (0,0) or half of the total length downwards from (0,15).

So we see, that the second method and the fourth method listed in the option can be used.

Thus, the methods that we can use to calculate the y-coordinate of the midpoint of a vertical line segment with endpoints at (0,0) and (0,15) are: Option B: Count by hand, Option D: Dividie 15 by 2

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Final answer:

To find the y-coordinate of a vertical line segment midpoint between (0,0) and (0,15), you can count by hand to get an approximate location, add the y-coordinates of the endpoints and then divide by 2, or simply divide 15 by 2 to get the correct value of 7.5.

Explanation:

To calculate the y-coordinate of the midpoint of a vertical line segment with endpoints at (0,0) and (0,15), you can use the following methods:

Option d is the most direct mathematical approach to finding the midpoint's y-coordinate. Option b is a valid but less precise method that relies on visual estimation. Option c is essentially part of the formula used in option d.

Answer:

-1

Step-by-step explanation:

So you are asked to find 3f(1)-4g(-2).

In order to find this you must first find f(1) and g(-2).

So f(1) means what is the value of the expression 3x-2 for when x=1.

f(1)=3(1)-2

f(1)=3-2

f(1)=1

And g(-2) mean what is the value of the expression 5-x^2 for when x=-2.

g(-2)=5-(-2)^2

g(-2)=5-(4)

g(-2)=5-4

g(-2)=1

So both of those function values were 1.

Our problem of 3f(1)-4g(-2) becomes 3(1)-4(1)=3-4=-1.

Answer:

c. 34.49

Step-by-step explanation:

Note that the nearest tenth place value is directly next to the decimal point (located on the right). Round each number to the nearest tenth. Look at the number in the hundredth place value. If it is 5 or greater round up, if 4 and less, round down.

a. 34.62 rounded to the nearest tenth is 34.6

c. 34.49 rounded to the nearest tenth is 34.5

b. 34.59 rounded to the nearest tenth is 34.6

d. 34.56 rounded to the nearest tenth is 34.6

This makes it so that C. 34.49 is your answer.

~

Answer:

y = 1/16

Step-by-step explanation:

y = 4 • 2^x

Let x = -6

y = 4 • 2^(-6)

The negative exponent puts it in the denominator

y = 4 • 1/ 2^6

y = 4 * 1/64

y = 4/64

y = 1/16  

Answer:

The measure of angle ABC is 34°

Step-by-step explanation:

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

so

∠ABC=(1/2)[major arc AC-minor arc AC]

∠ABC=(1/2)[major arc AC-146°]

Find the measure of major arc AC

major arc AC=360°-146°=214°

substitute

∠ABC=(1/2)[214°-146°]=34°

Answer:

∠F = 106°

Step-by-step explanation:

The opposite angles of a parallelogram are congruent, hence

∠F = ∠J = 106°

Answer: rational number answer is desired

Step-by-step explanation:

Answer:

7 students.

Step-by-step explanation:

To pass science, a student requires at least 70 grades so students who scored below 70 will be failed.

From the given bar chart we can calculate the number of students who scored below 70 grades

Students who earned 50 - 59 = 2

Students who earned 60 - 69 = 5

Therefore, number of students who failed were

2 + 5 = 7 students.

Answer:7 students

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

The GCF of 52 and 84 is 4.

52 = 2 × 2 × 13

84 = 2 × 2 × 3 × 7

Therefore, GCF = 2 × 2

GCF = 4

The GCF of 52 and 84 is found by finding the prime factors of each number and identifying the common factors. The GCF is 4.

The GCF (Greatest Common Factor) of two numbers is the largest number that can evenly divide both numbers. To find the GCF of 52 and 84, we need to find the prime factors of each number and then find the highest common factor of these sets of prime factors.

The common factors are 2 and 2, so the GCF of 52 and 84 is 2 x 2, which equals 4.

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Remember that the slope intercept formula is:

y = mx + b

m is the slope

b is the y-intercept

It tells us that the slope (m) is -3 and the y-intercept (b) is 5. Plug this into the formula given above:

y = -3x + 5

Hope this helped!

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[tex]y=x^2-6x+6\\y=x^2-6x+9-3\\y=(x-3)^2-3[/tex]

The vertex form of the quadratic equation is y = (x-3)^2 + 3.

The vertex form of a quadratic function is given by y = a(x-h)^2 + k, where (h,k) represents the coordinates of the vertex. To convert the quadratic equation y = x^2-6x+6 into vertex form, we need to complete the square.

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Answer:

[tex]f(x)=2(x-2)^{3}(x+2)^{2}[/tex]

Step-by-step explanation:

we know that

2 is a zero of multiplicity 3 of the polynomial

so

we have that

x=2  is a solution of the polynomial

A factor of the polynomial is

[tex](x-2)^{3}[/tex] ----> is elevated to the cube because is a multiplicity 3

and the other solution is x=-2

since the polynomial  is fifth degree, x=-2 must have a multiplicity 2

so

the other factor of the polynomial is  

[tex](x+2)^{2}[/tex] ----> is squared because is a multiplicity 2

therefore

The polynomial is equal to multiply the factors by the leading coefficient

so

[tex]f(x)=2(x-2)^{3}(x+2)^{2}[/tex]

Answer:

14h <56

Step-by-step explanation:

The area of a rectangle is given by

A =bh

We know the area is less the 56

bh <56

The base length is 14

14h <56

Answer:

D

Step-by-step explanation:

i got this question right on edgeinuity

Answer:

y=(3/2)x+6

If your equation is in a different form, let me know.

Step-by-step explanation:

So the slope-intercept form of a line is y=mx+b where m is the slope and b is the y-intercept.

Parallel lines have the same slope, m (different y-intercept (b) though).

So we need to find the slope going through (1,6) and (-7,-6).

To do this you could use [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex].

Or, what I like to do is line the points up vertically and subtract vertically then put 2nd difference over first difference. Like so:

(  1  ,  6)

-( -7,  -6)

---------------

  8     12

So the slope of our line is 12/8.

Let's reduce it! Both numerator and denominator are divisible by 4 so divide top and bottom by 4 giving 3/2.

Again parallel lines have the same slope.  

So we know the line we are looking for is in the form y=(3/2)x+b where we don't know the y-intercept (b) yet.

But we do know a point (x,y)=(2,9) that should be on our line.

So let's plug it in to find b.

y=(3/2)x+b  with (x,y)=(2,9)

9=(3/2)2+b

9=3       +b

Subtract 3 on both sides:

9-3=b

6=b

So the equation in slope intercept form is y=(3/2)x+6

Answer:

In short, you are looking a diagonal line passing through the origin. And yes diagonal line is a straight-edged line that is running diagonally.

Step-by-step explanation:

If you want a better answer you can post the graphs.

I will tell you the trick, direct variations is another words for the relation is proportional.

The graph will have equation y=kx where k (k can't be zero) is the slope and your y-intercept is 0. The graph should be a diagonal line.

In short, you are looking a diagonal line passing through the origin. And yes diagonal line is a straight-edged line that is running diagonally.

Answer:

(9,9)

Step-by-step explanation:

I like to use similar triangles.

To take some confusion out of of this let's translate the line down 3 units and left 3 units.

Alright from the drawing we get:

a/8  = b/8                 and  a/8  =  3k/4k    (or    b/8=3k/4k  )

We don't need the k's, they cancel.

a/8   = b/8                  and             a/8=3/4

So the first equation means a=b.

Now let's see what a and b are by solving a/8 = 3/4.

Cross multiply:

[tex]\frac{a}{8}=\frac{3}{4}[/tex]

[tex]a(4)=8(3)[/tex]

[tex]4a=24[/tex]

a=6

So if a=b and a=6, then b=6.

The ordered pair is (6,6).

Now let's move the line back.

We have to move it up 3 units and right 3 units which gives us the point (9,9).

Answer:

It is B.

Step-by-step explanation:

x^2+x-6 /  x^2-x-6 = 0

The numerator equates to zero:

x^2 + x - 6 = 0

(x + 3)(x - 2) = 0

the zeroes are {-3, 2).

The zeroes of the function f(x) = (x² + x - 6)/(x² - x - 6) are -3, 2.

Hence, option B. is the right choice.

A function (say f(x)) is defined over x, when an expression in x, gives only one value f(x) for all the x in its domain.

A zero of a function (say f(x)) is the value x, where f(x) = 0.

We are asked to find the zeroes of the function

f(x) = (x² + x - 6)/(x² - x - 6).

To find the zeroes, we put the value of f(x) = 0, in the above equation to get:

(x² + x - 6)/(x² - x - 6) = 0

or, (x² + 3x - 2x - 6)/(x² + 2x - 3x - 6) = 0

or, {x(x + 3) -2(x + 3)}/{x(x + 2) -3(x + 2)} = 0

or, {(x - 2)(x + 3)}/{(x - 3)(x + 2)} = 0

Zeroes are where the numerator = 0, as the denominator can not be 0.

∴ (x + 3)(x - 2) = 0

∴ Zeroes of the function x + 3 = 0 ⇒ x = -3, and x - 2 =0 ⇒ x = 2, that is

-3, 2. Hence, option B. is the right choice.

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Answer:

9/8 = n

Step-by-step explanation:

9=8n

Divide each side by 8

9/8 = 8n/8

9/8 = n