Determine whether the relation represents y as a function of x.

y = x^2

Answer:

2

Step-by-step explanation:

The slope of a line is defined by the formula:

[tex]m=\dfrac{\Delta \, y}{\Delta \, x} = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]

In this problem, we are given two points in the form [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex].

So, we can define the x's and y's as:

[tex]x_1 = 1[/tex], [tex]y_1 = 2[/tex], [tex]x_2 = 5[/tex], [tex]y_2 = 10[/tex].

Hence, the slope of the line can be solved for.

[tex]m = \dfrac{10-2}{5-1}[/tex]

[tex]m = \dfrac{8}{4}[/tex]

[tex]m = 2[/tex]

The slope of the line with points (1, 2) and (5, 10) is 2.

Through reporting statistical results, we found that  [tex]r^{2}[/tex] is 0.22 and Cohen's [tex]d[/tex] is 0.52 when s = 8, and  [tex]r^{2}[/tex] is 0.04 and Cohen's [tex]d[/tex] is 0.21 when s = 20. It is also observed that when the variability of the scores in the sample, [tex]r^{2}[/tex] decreases, there is simultaneous decrease in the measures of effect size, Cohen's [tex]d[/tex] as well.

It is given to us that -

A random sample of n = 16 scores

Population with a mean of μ = 45

Sample mean is found to be M = 49.2

In Reporting Statistical results, there are two different effect sizes, namely eta-squared [tex]r^{2}[/tex] and Cohen's [tex]d[/tex].

Eta-square, [tex]r^{2} = \frac{t^{2} }{t^{2}+df }[/tex]

where, [tex]df = n-1[/tex]

[tex]SEM = \frac{s}{\sqrt{n} }[/tex]

and, [tex]t=\frac{x_{2} -x_{1} }{SEM}[/tex]

Cohen's [tex]d=\frac{x_{2} -x_{1} }{s}[/tex]

a) Given that the sample standard deviation is s=8

We have n = 16

So, [tex]df = n-1 = 16-1 = 15[/tex]

[tex]SEM = \frac{s}{\sqrt{n} } = \frac{8}{\sqrt{16} } = \frac{8}{4} = 2[/tex]

We also have μ = 45 and M = 49.2. This implies

[tex]t=\frac{x_{2} -x_{1} }{SEM} = \frac{49.2-45}{2} = \frac{4.2}{2} =2.1[/tex]

Now,

[tex]r^{2} = \frac{t^{2} }{t^{2}+df } = \frac{(2.1)^{2} }{(2.1)^{2}+ 15 }= \frac{4.41}{4.41+15} \\= \frac{4.41}{19.41} = 0.22[/tex]

Cohen's [tex]d=\frac{x_{2} -x_{1} }{s} = \frac{49.2-45}{8} = \frac{4.2}{8} = 0.52[/tex]

b) Given that the sample standard deviation is s=20

We have n = 16

So, [tex]df = n-1 = 16-1 = 15[/tex]

[tex]SEM = \frac{s}{\sqrt{n} } = \frac{20}{\sqrt{16} } = \frac{20}{4} = 5[/tex]

We also have μ = 45 and M = 49.2. This implies

[tex]t=\frac{x_{2} -x_{1} }{SEM} = \frac{49.2-45}{5} = \frac{4.2}{5} =0.84[/tex]

Now,

[tex]r^{2} = \frac{t^{2} }{t^{2}+df } = \frac{(0.84)^{2} }{(0.84)^{2}+ 15 }= \frac{0.706}{0.706+15} \\= \frac{0.706}{15.706} = 0.04[/tex]

Cohen's [tex]d=\frac{x_{2} -x_{1} }{s} = \frac{49.2-45}{20} = \frac{4.2}{20} = 0.21[/tex]

c) Comparing a and b, we see that the variability of the scores in the sample, [tex]r^{2}[/tex] is 0.22 when s = 8 and [tex]r^{2}[/tex] is 0.04 when s = 20. Similarly, Cohen's [tex]d[/tex] is 0.52 when s = 8 and [tex]d[/tex] is 0.21 when s = 20.

Thus, we can see that when the variability of the scores in the sample, [tex]r^{2}[/tex] decreases, there is simultaneous decrease in the measures of effect size, Cohen's [tex]d[/tex] as well.

Therefore, through reporting statistical results, we found that  [tex]r^{2}[/tex] is 0.22 and Cohen's [tex]d[/tex] is 0.52 when s = 8, and  [tex]r^{2}[/tex] is 0.04 and Cohen's [tex]d[/tex] is 0.21 when s = 20. It is also observed that when the variability of the scores in the sample, [tex]r^{2}[/tex] decreases, there is simultaneous decrease in the measures of effect size, Cohen's [tex]d[/tex] as well.

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The required measure of ∠P would be 76.825° in the given triangle PQR.

Given that two angles in triangle PQR are congruent, ∠P and ∠Q;

∠R measures 26.35°.

Let the measure of ∠P would be x

∠P = x = ∠Q

We know that the sum of interior angles is always 180 degrees in the triangle.

⇒ ∠P + ∠Q + ∠R = 180°

⇒ x + x  + 26.35° = 180°

⇒ 2x  + 26.35° = 180°

⇒ 2x = 180° - 26.35°

⇒ 2x = 153.65

⇒ x = 153.65/2

⇒ x = 76.825°.

Therefore, the required measure of ∠P would be 76.825° in the given triangle PQR.

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The trigonometric measures for the given angle are as follows:

We are given the measure of the sine for the angle and we need to find the three measures, sine, cosine and tangent for half the angle.

These measures are dependent on the cosine of the function, hence we apply the definition of the tangent as follows:

[tex]\tan{x} = \frac{\sin{x}}{\cos{x}}[/tex]

Hence:

[tex]\sqrt{2} = \frac{\sin{x}}{\cos{x}}[/tex]

[tex]\sin{x} = \sqrt{2}\cos{x}[/tex]

The exact value of the cosine can be found applying the identity as follows:

sin²(x) + cos²(x) = 1.

Then, from the equation of the sine as a function of the cosine from the tangent relation, we have that:

[tex](\sqrt{2}\cos{x})^2 + \cos^2{x} = 1[/tex]

[tex]2^\cos^2{x} + \cos^2{x} = 1[/tex]

[tex]\cos^2{x} = \frac{1}{3}[/tex]

[tex]\cos{x} = \pm \sqrt{\frac{1}{3}}[/tex]

The angle is of the first quadrant, as 0° < x < 90°, hence the cosine is positive, thus:

[tex]\cos{x} = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}[/tex]

The identity that gives the sine for half the angle is:

[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 - \cos{x}}{2}}[/tex]

Hence:

[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 - \frac{\sqrt{3}}{3}}{2}}[/tex]

[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{6}}[/tex]

The identity for the cosine is almost the same, just there is a plus instead of a minus, hence:

[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 + \cos{x}}{2}}[/tex]

[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 + \frac{\sqrt{3}}{3}}{2}}[/tex]

[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 + \sqrt{3}}{6}}[/tex]

The tangent is the sine divided by the cosine, hence the inserting the entire division and the same square root and simplify the common denominator, thus:

[tex]\tan{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{3 + \sqrt{3}}}[/tex]

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The equation of the tangent line of f at the point x = -3 will be y = -16x-48.

According to the question,

We have the following information:

f(x) = [tex]2x^{2} -4x-4[/tex]

Now, we will first find the derivation of this function with respect to x:

Let's take its derivation to be f'(x).

f'(x) = 4x-4

Now, finding the slope of the equation when x = -3:

f'(-3) = 4*(-3)-4

f'(-3) = -12-4

f(-3) = -16

Now, we know that following formula is used to find the equation of a line:

(y-y') = m(x-x')

y-0 = -16{x-(-3)}

y = -16(x+3)

y = -16x-48

Hence, the equation of the tangent line of f at the point x = -3 will be y = -16x-48.

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The correct equation that could be used to solve for side x is x = sin52° × (Hypotenuse)

From the question, we are to determine the correct equation that could be used to solve for side x

From the given diagram, we observe that

Side x is the Opposite

Side z is the Adjacent

Side y is the Hypotenuse

Using SOH CAH TOA, we can write that

sin (angle) = Opposite / Hypotenuse

From the diagram,

Given angle = 52°

Hypotenuse = y = 13

Thus,

sin 52° = x/13

x = sin 52° × 13

OR

x = sin52° × (Hypotenuse)

Hence, the equation is x = sin52° × (Hypotenuse)

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The value of expression a³b will be;

⇒ - 75x⁴ + 100x³i

What is substitution method?

To find the value of any one of the variables from one equation in terms of the other variable is called the substitution method.

Given that;

The expression is,

⇒ a³b

The values are;

a = - 5x

b = 3x - 4i

Now,

Substitute the value of a and b, we get;

The expression is,

⇒ a³b

⇒ (-5x)³ × (3x - 4i)

⇒ - 25x³ (3x - 4i)

⇒ - 75x⁴ + 100x³i

Thus, The value of expression a³b will be;

⇒ - 75x⁴ + 100x³i

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LJ = 46 is the value when  LK  congruent MK .

What do you mean by congruent?

By given figure ,

        MK = MN - KN

               = 9x - 11 - ( x + 3 )

               = 8x - 14

now given LK ≅ MK

           LK = MK

8x - 14 = 7x - 10

8x - 7x = 14 - 10

 x = 4

length of LJ = LK + KJ

                    = 7x - 10 + 28

                    = 7(4) + 18

                    = 28 + 18  ⇒ 46

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The coordinates of point (1,5) after dilation is (3,3). Therefore, 4th option is correct.

It is given to us that -

The coordinates of the original point is (1,5)

=> [tex](x_{o},y_{o} )=(1,5)[/tex] ---- (1)

The dilating factor is 1/3

=> [tex]s=\frac{1}{3}[/tex] (say) ---- (2)

And, the center of dilation is at the point (4,2)

=> [tex](x_{cod},y_{cod})=(4,2)[/tex] ---- (3)

We have to find out the coordinates of point (1,5) after dilation.

Using the formula for dilation coordinates from original to image, we have

[tex][(x_{cod}+s(x_{o}-x_{cod}),y_{cod}+s(y_{o}-y_{cod})]\\=[4+\frac{1}{3}(1-4),2+\frac{1}{3}(5-2)]\\=[(4-1),(2+1)]\\=(3,3)[/tex]

Thus, the coordinates of point (1,5) after dilation is (3,3). Therefore, 4th option is correct.

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The vertex of the Parabola is (2, -3)

The vertex of a parabola is the point where the parabola crosses its axis of symmetry.  If the coefficient of the x2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “ U ”-shape.  If the coefficient of the x2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “ U ”-shape.

From  the function f(x) =[tex]-(x-2)^{2}[/tex] - 3

The equation of parabola in vertex form is

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

where h and k are the coordinates of the vertex

comparing the two equations

h = 2, k = -3

The coordinates of the vertex is (2, -3)

In conclusion, the vertex is (2, -3).

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The measures of the arithmetic sequence are given as follows:

a. [tex]S_{10} = 350[/tex]

b. [tex]A_{n + 1} = 17 + 4n[/tex]

c. [tex]S_n = \frac{n(30 + 4n)}{2}[/tex]

d. [tex]A_n = 17 + 4(n - 1)[/tex]

An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and is called common difference d.

The nth term of an arithmetic sequence is given by the rule shown below:

[tex]a_n = a_1 + (n - 1)d[/tex]

In which [tex]a_1[/tex] is the first term of the sequence.

The sum of the first n terms is given by the rule shown below:

[tex]S_n = \frac{n(a_1 + a_n)}{2}[/tex]

The equation given for this problem is:

[tex]a_{n + 4} = 4n + 17[/tex]

Hence the sequence can be written as follows:

17, 21, 25, 29, 33.

Then the first term and the common ratio are given as follows:

[tex]a_1 = 17, d = 4[/tex]

Then the nth term is of:

[tex]A_n = 17 + 4(n - 1)[/tex]

The (n + 1)th term is of:

[tex]A_{n + 1} = 17 + 4(n + 1 - 1)[/tex]

[tex]A_{n + 1} = 17 + 4n[/tex]

The sum of the first n terms is of:

[tex]S_n = \frac{n(a_1 + a_n)}{2}[/tex]

[tex]S_n = \frac{n(17 + 17 + 4(n - 1))}{2}[/tex]

[tex]S_n = \frac{n(30 + 4n)}{2}[/tex]

The sum of the first ten terms is of:

[tex]S_{10} = \frac{10(30 + 4(10))}{2} = 350[/tex]

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The number of the blueberries is 31.

In this case,  the only way that we can be able to obtain the number of the pints of the blueberries that she sold is by the use of a simultaneous equation.

Let the number of  the blueberries be x and the number of the raspberries be y.

We have

x + y = 57 ------ (1)

1.75x + 2.40y = 116.65 ------- (2)

x = 57 - y ------ (3)

Then we have;

1.75(57 - y) + 2.40y = 116.65

99.75 - 1.75y + 2.40y = 116.65

Collecting like terms;

- 1.75y + 2.40y = 116.65 - 99.75

0.65y = 16.9

y = 16.9/0.65

y = 26

To obtain the number of blueberries

x + 26 = 57

x = 57 - 26

x = 31

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Answer: the one on the top left corner

Answer: the third one (bottom left) but also the second one (top right)

Step-by-step explanation: 2. there are 8 total parts to the figure and four of those parts are shaded. 4/8

3. There are four total parts to the figure but only 2 are shaded. 2/4

2/4 and 4/8 are both EQUIVALENT to 1/2. Therefore, that is the correct answer.

When a quadratic equation is given in the form y = ax^2 + Bx + C the easily seen features are

Equation of a parabola is the equation used to trace the path of a parabola. this equation is a quadratic equation

The formula for the roots of the quadratic equation is derived from the equation of the form y = ax^2 + Bx + C to be

-b ± √{(b² - 4ac)2a)

the y intercepts is gotten by equating x = 0

The axis of symmetry refers to the axis where the parabolic curve can be bisected. this is in the y axis

"a" shows if the curve opens upwards or downwards.

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Any of the four terms of a discrete geometric proportion is called a proportional fourth.

[tex]\bold{5, 2 \: and \: 10.5}[/tex]

A geometric proportion is formed as follows:

[tex]\bold{5 : 2 :: 10 : x}[/tex]

Since the unknown term is an extreme and as we have seen before, one extreme is equal to the means divided by the other extreme, we will have:

[tex]\boxed{\bold{ \: x = \frac{2 \:∗ \: 10}{5} = 4} }[/tex]

Therefore, the geometric proportion is:

[tex]\bold{5 : 2 :: \: :: 10 :4}[/tex]

Benchmark percentage of 1% = 9.6 & 10% = 24.

What is Free Throw Percentage?

Free throw percentage (FT%) puts a player’s successful free throws in perspective to their total attempts.

In basketball, a free throw (or foul shot) is awarded to a player who has been fouled by the other team. The number of free throws depends on where on the court the player was while being fouled.

To find 1% of a number, we can move the decimal point of the number two places to the left.

Therefore, 1% of 120 is 1.2

To find 10% of a number, we can move the decimal point of the number one place to the left.

Therefore, 10% of 120 is 12

Since 10% of 120 is 12 and

20% = 2 x 10%

then 20% of 120 = 2 x 10% of 120

= 2 x 12 = 24

Since 1% of 120 is 1.2 and

8% = 8 x 1%

then 8% of 120 = 8 x 1%  of 120

= 8 x 1.2 = 9.6

Hence the answer is, Benchmark percentage of 1% = 9.6 & 10% = 24.

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The probability of the sets P(F or G) is; P(F or G)  = 0.67

We are given the following;

Sample Space; S = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

Set F = {6, 7, 8, 9, 10}

Set G = {10, 11, 12, 13}

Now, F or G, simple means; F ∪ G

Thus;

F or G = {6, 7, 8, 9, 10, 11, 12, 13}

Number of terms is (F or G) = 8

Number of terms is Sample space = 12

Thus;

P(F or G) = 8/12 = 2/3 = 0.67

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The given phrase on translation to math expression is (6 ÷ 3) + 12 , the correct option is (a) .

In the question ,

an mathematical phrase is given , that is "Twelve more than the quotient of six divided by three" .

we have to translate it into mathematical expression ,

So , the term quotient of six divided by three is written as 6 ÷ 3 .

and phrase " more " is represented by " + " ,

Hence the given phrase,  "Twelve more than the quotient of six divided by three" is  (6 ÷ 3) [tex]+[/tex] 12 .

Therefore , The given phrase on translation to math expression is (6 ÷ 3) + 12 , the correct option is (a) .

The given question is incomplete , the complete question is

Translate the phrase into a math expression , "Twelve more than the quotient of six divided by three" .

(a) (6÷3)+12

(b) 6÷(3+12)

(c) (3÷6)+12

(d) 3÷(6+12)

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The difference between their ages one year ago was 2 years.

Difference in maths, the result of one of the important mathematical operations, which is obtained by subtracting two numbers.

Given that, Jack is 2 years older than Bob,

Let Bob's age be x then, Jack's age will be (x+2)

Their ages before 1 year was =

Bob's = x-1

Jack's = (x+1)

Difference = x + 1 - x + 1 = 2

Hence, The difference between their ages one year ago was 2 years.

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The distance is d = √(-3 - a)² + (3 - b)²

What is Distance between two points ?

Distance between two points is the length of the line segment that connects the two points in a plane.

The formula is d =√(x₁ - x₂)² + (y₁ - y₂)²

Points are (-3, 3) and (a, b)

We know, the distance formula is

d =√(x₁ - x₂)² + (y₁ - y₂)²

Let, (x₁, y₁) = (-3, 3)

(x₂,  y₂) = (a, b)

Now, plug in the values in given formula

d = √(-3 - a)² + (3 - b)²

Hence, the distance is d = √(-3 - a)² + (3 - b)²

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Answer: x - 5 is less than or equal to -25

Step-by-step explanation:

Step-by-step explanation:

If x = 3 :

x³ ÷ 3 = y

3³ ÷ 3 = y

27 ÷ 3 = y

y = 27 ÷ 3 = 9

The point S of line segment RT such that RS / ST = 2 / 5 is equal to (- 3, 0).

Herein we find a line segment whose endpoints are known (R(x, y) = (- 5, - 2), T(x, y) = (2, 5)) and in which we must determine the coordinates of a point S within line segment RT such that the partition ratio is observed:

RS / ST = 2 / 5

[tex]\overrightarrow{RS} = \frac{2}{5} \cdot \overrightarrow {ST}[/tex]

S(x, y) - R(x, y) = (2 / 5) · [T(x, y) - S(x, y)]

(7 / 5) · S(x, y) = (2 / 5) · T(x, y) + R(x, y)

S(x, y) = (2 / 7) · T(x, y) + (5 / 7) · R(x, y)

Now we determine the location of point S:

S(x, y) = (2 / 7) · (2, 5) + (5 / 7) · (- 5, - 2)

S(x, y) = (4 / 7, 10 / 7) + (- 25 / 7, - 10 / 7)

S(x, y) = (- 21 / 7, 0)

S(x, y) = (- 3, 0)

The location of point S is (- 3, 0). A representation of the geometric system is shown in the image attached below.

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Answer:

d /c

Step-by-step explanation:

if its rotated at an 270 angle then its d or c cause they are the same

Step-by-step explanation:

if I understand your typing correctly, and there is nothing missing, we have

f(x) = 0.6x⁵ - 2x⁴ + 8x

the "end behavior" means the general tendency of the result values for very large or very low values of x (going to +infinity and -infinity).

the higher (or lower in the negative direction) x gets, the more the highest exponent will dominate the result values.

it does not matter, that it has a diminishing factor (or coefficient) like 0.6.

the much stronger progression of x⁵ vs. smaller exponents like x⁴ or x will easily compensate for that with sufficiently large x.

so, ultimately, the term with the highest exponent (in our case 0.6x⁵) defines the end behavior.

with x going to +infinity, so does the function result (+infinity).

with x going to -infinity, so does the function result (-infinity, because an odd exponent number like 5 will maintain the sign of the argument).

1. Two constants of proportionality representing the proportional relationship between distance in kilometers and number of hours are 400 km and 8 hours.

2. The relationship (ratio) between the two values is 50 km per hour or the equation, d = 50t.

3. Another equation that relates distance and time in this context is t = d/50 or 400/50.

The constant of proportionality is the ratio relating two given values in a proportional relationship.

Other names for the constant of proportionality include:

Distance, d = 400 km

Constant speed, t = 8 hours

d = 400/8

d = 50t

Constant of proportionality = 50.

t = d/s

Where d, distance = 400 km and s, speed = 50 km/h

= 400/50

= 8 hours

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An arrangement of the order the numbers would appear on a number line from least to greatest is: -31/3, -10, -9/8, 0, 1, 1/4, 9/8, 8.

In Mathematics, a number line can be defined as a type of graph with a graduated straight line which is composed of both positive and negative numbers that are placed at equal intervals along its length.

Generally speaking, a number line typically increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).

In order to arrange the given numbers in order from least to greatest, we would convert them into a decimal number as follows:

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Answer:

The y-intercept represents the number of mammals when they were originally counted.

The line y = 0 is an asymptote of the graph.

The y-intercept is 120.

The asymptote indicates that as years pass, the number of mammals will approach 0.

The point P is on the perpendicular line to AB that passes through its midpoint.

We know perpendicular lines have opposite-reciprocal slopes.

So the line we are looking for has a slope of 7/5.

Use the point-slope equation to find the line:

Choose any form above of the same line.

Answer:

[tex]\textsf{Slope-intercept form}: \quad y=\dfrac{7}{5}x-\dfrac{19}{5}[/tex]

[tex]\textsf{Standard form}: \quad 7x-5y=19[/tex]

Step-by-step explanation:

A perpendicular bisector is a line that intersects another line segment at 90°, dividing it into two equal parts.

To find the perpendicular bisector of segment AB, find the slope of AB and the midpoint of AB.

Define the points:

Slope of AB

[tex]\textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-6-4}{9-(-5)}=\dfrac{-10}{14}=-\dfrac{5}{7}[/tex]

Midpoint of AB

[tex]\textsf{Midpoint}=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)=\left(\dfrac{9+(-5)}{2},\dfrac{-6+4}{2}\right)=(2,-1)[/tex]

If two lines are perpendicular to each other, their slopes are negative reciprocals.

Therefore, the slope of the line that is perpendicular to line segment AB is ⁷/₅.

Substitute the found perpendicular slope and the midpoint of AB into the point-slope formula to create an equation for the line that is the perpendicular bisector of line segment AB:

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

[tex]\implies y-(-1)=\dfrac{7}{5}(x-2)[/tex]

[tex]\implies y+1=\dfrac{7}{5}x-\dfrac{14}{5}[/tex]

[tex]\implies y=\dfrac{7}{5}x-\dfrac{14}{5}-1[/tex]

[tex]\implies y=\dfrac{7}{5}x-\dfrac{19}{5}[/tex]

Therefore, the formula that expresses the fact that an arbitrary point P(x, y) is on the perpendicular bisector of segment AB is:

[tex]\textsf{Slope-intercept form}: \quad y=\dfrac{7}{5}x-\dfrac{19}{5}[/tex]

[tex]\textsf{Standard form}: \quad 7x-5y=19[/tex]