you want to save $12,000 for a downpayment on a house. how much should be put into an account that pays 3.95% interest compounded continuously for 5 years

k is equal to 2, and the function g(x) is obtained by shifting the graph of f(x) two units to the left.

A function in mathematics is a relationship between a set of inputs (also known as the domain) and a set of outputs (also known as the range), where each input is connected to each output inexactly.

It can be compared to a machine that processes inputs and outputs in accordance with predetermined rules or algorithms.

In order to model real-world scenarios, carry out calculations, and examine mathematical relationships, functions are used.

They are frequently depicted through graphs or algebraic equations.

Since g(x) = f(x + k), we need to find the value of k that will shift the graph of f(x) to the graph of g(x). To do this, we can use the fact that g(x) = f(x + k) and the values in the table to determine the value of k.

When x = -1, g(x) = 5 = f(-1 + k). Since f has a point at (-1, 3), we know that f(-1) = 3.

Therefore, we have:

5 = f(-1 + k) = f(-1) + k = 3 + k

Solving for k, we get:

k = 5 - 3 = 2

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Answer:

There are 46 numbers in this sequence.

Hence inverse of the function h is

h⁻¹ :x→  2/(x-1)  -1

An inverse function is a function that "undoes" the action of another function. In other words, if a function f(x) takes an input x and produces an output y, then its inverse function, denoted as f⁻¹(y) or sometimes as g(y), takes the output y and produces the input x such that f(x) = y.

 A function is a rule that associates each element of a set (called the domain) with a unique element of another set (called the range or codomain).

Given function

f(x)=1+1/x for x>0

g(x)=(x+1)/2  for x>0

Given: h=fg

h=f(g(x))

h=1+1/(x+1)/2

Now, taking the inverse of the function h,

h=1+1/(x+1)/2

h-1=1/(x+1)/2

1/(h-1)=(x+1)/2

2/(h-1)=x+1

x=2/(h-1) -1

Hence inverse of the function h is

h⁻¹:x→ 2/(x-1) -1

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Answer:

Kate is 3 years old, Jane is 9 years old

Step-by-step explanation:

1.) First, assign variables to all of their ages. If we say Kate's age is x, Jane's age is 3 times this, which can be written as j, is 3x.

2.) Jane's age is also 2 less than twice of Kate's in 5 years. This means that her age is also 2(x+5) - 2 = j + 5. With a little simplification, you get that 2x + 8 = j + 5.

3.) Since j, Jane's age, is also 3x, we can substitute 3x in for j in the second equation. If you do this, you get 2x + 8 = 3x + 5.

4.) By moving the x onto one side and the numbers onto another, you get x = 3. X was Kate's age, meaning that Kate is 3 years old.

5.) Finally, since Jane's age is 3 times Kate's age, Jane's age is 3 * 3, which is 9. Jane is 9 years old.

Jane is 9 years old and Kate is 3 years old.To solve the problem, let's first establish variables for Jane and Kate's ages.

Let J represent Jane's age and K represent Kate's age.
According to the student question, Jane is 3 times as old as Kate, which can be represented as:
J = 3K
In 5 years, Jane's age will be 2 less than twice Kate's age, which can be represented as:
J + 5 = 2(K + 5) - 2
Now we can solve the equations step by step:

Substitute the first equation into the second equation to eliminate one of the variables:
3K + 5 = 2(K + 5) - 2
Distribute the 2 on the right side of the equation:
3K + 5 = 2K + 10 - 2
Simplify the equation by combining like terms:
3K + 5 = 2K + 8
Move the 2K term to the left side of the equation:
K = 3
now we know that Kate is currently 3 years old.
Substitute K's value back into the first equation to find Jane's age:
J = 3K
J = 3(3)
Simplify to find Jane's age:
J = 9
So, Jane is currently 9 years old.
Jane is 9 years old and Kate is 3 years old.

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Answer:

A

Step-by-step explanation:

given the x- intercepts x = a and x = b then the corresponding factors are

(x - a) and (x - b)

the equation of the quadratic is then the product of the factors , that is

y = a(x - a)(x - b) ← a is a multiplier

here the x- intercepts are x = 4 and x = 8 , then factors are

(x - 4) and (x - 8 ) , so

y = a(x - 4)(x - 8)

to find a substitute any other point on the graph into the equation

given vertex = (6, - 4 ) , then

- 4 = a(6 - 4)(6 - 8)

- 4 = a(2)(- 2) = - 4a ( divide both sides by - 4 )

1 = a , then

y = (x - 4)(x - 8) ← expand using FOIL

y = x² - 12x + 32

The vector FC is (14, 0, 3/2, -7, -8, 6).

If you are given two vectors FC and F and C, with F represented as F(-2, 0, 0, 8, 2, 1) and C represented as C(12, 0, 3/2,

1, -6, 7), you can find the vector FC by subtracting the F vector from the C vector component-wise.

Step 1: Write down the F and C vectors.

F = (-2, 0, 0, 8, 2, 1)

C = (12, 0, 3/2, 1, -6, 7)

Step 2: Subtract the F vector from the C vector component-wise.

FC = C - F

FC = (12 - (-2), 0 - 0, 3/2 - 0, 1 - 8, -6 - 2, 7 - 1)

Step 3: Simplify the subtraction.

FC = (14, 0, 3/2, -7, -8, 6)

So, the vector FC is (14, 0, 3/2, -7, -8, 6).

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Radius of cylindrical container is 10 cm, if cylindrical container holds 4.4 litres of water when full and If the height of the container is 14 cm.

How to measure radius of the cylinder?

To solve this problem, we can use the cylinder volume formula:

Volume of cylinder =

π × radius^2 × height

We know that the volume of water that the cylinder can hold is 4.4 litres. We can convert this to cubic centimeters (cm^3) by multiplying by 1000, since there are 1000 cm^3 in a liter:

4.4 litres =

4.4 × 1000 = 4400 cm^3

We also know the height of the cylinder is 14 cm. So we can plug in these values and solve for the radius:

4400 = π × radius^2 × 14

Dividing both sides by 14π, we get:

radius^2 = 4400 / (14 × π) = 100

Taking the square root of the both sides yields:

radius = √100 = 10 cm

As a result, the cylindrical container's radius is 10 cm.

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As per the given scenario, a cereal comes in three different package sizes.

The ratio of the cost of an 8-ounce box to a 16-ounce box of cereal can be calculated as follows :Ratio of 8-ounce box to 16-ounce box= 8 : 16Here, the ratio of 8 and 16 can be simplified by dividing both the terms by [tex]8.8 : 16 = 1 : 2[/tex]

Therefore, the ratio of the cost of an 8-ounce box to a 16-ounce box of cereal is 1 : 2.

Note: The answer should be represented in the simplest form possible.

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Answer:

1 2 3 6 9 and 18

Step-by-step explanation:

the 111 factors of 18 is 1 2 3 6 9 ang 18

Use circumcenter to draw a circle through three non-collinear points.

a) Draw diameter, construct equilateral triangle with one vertex at center and other two on circle.

b) Draw diameter, construct regular hexagon with one vertex at center and other vertices on circle.

Step 1: To construct a circle through three points not on a line, we can use the circumcenter of a triangle. We first construct the perpendicular bisectors of any two sides of the triangle. The point where these two lines intersect is the circumcenter of the triangle. We can then draw a circle with this point as the center, which passes through all three vertices of the triangle.

Step 2:

a) To construct an equilateral triangle inscribed in a circle, we can draw a diameter of the circle and then construct an equilateral triangle with one of the endpoints at the center of the circle and the other two endpoints on the circle itself.

b) To construct a regular hexagon inscribed in a circle, we can draw a diameter of the circle and then construct a regular hexagon with one of the vertices at the center of the circle and the other vertices on the circle itself.

c) To construct a square inscribed in a circle, we can draw a diameter of the circle and then construct a square with one of the vertices at the center of the circle and the other vertices on the circle itself.

Step 3:

a) To construct a tangent to a circle through a point on the circle, we can draw a radius of the circle to the point of tangency, and then construct the perpendicular bisector of this radius. The line where the perpendicular bisector intersects the circle is the tangent line.

b) To construct a tangent to a circle through a point outside the circle, we can draw a line from the point to the center of the circle, and then construct the perpendicular to this line at the point of tangency. The line where the perpendicular intersects the circle is the tangent line.

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For the given linear equations, x = 63/4 i.e. None of the above options.

What is a linear equation, exactly?

Each term in a linear equation is either a constant or the product of a constant and a single variable raised to the first power. A linear equation in one variable has the general form: axe + b = 0, where a and b are constants and x is the variable. This equation's answer is a single value of x that solves the problem.

Now,

To use Cramer's Rule to solve this system of linear equations, we need to set up the following matrices:

A = 2/5 1/4

2/3 5/12

B = 9/20

3/4

The determinant of A is given by:

|A| = (2/5)(5/12) - (1/4)(2/3) = 1/30 - 1/18 = -1/90

The determinant of the matrix obtained by replacing the first column of A with B is given by:

|A1| = (9/20)(5/12) - (3/4)(2/3) = 9/80 - 1/4 = -7/40

The determinant of the matrix obtained by replacing the second column of A with B is given by:

|A2| = (2/5)(3/4) - (1/4)(9/20) = 3/10 - 9/80 = 3/40

Now, we can use Cramer's Rule to find the value of x:

x = |A1|/|A| = (-7/40)/(-1/90) = 63/4

Therefore, the value of x

x = 63/4

So the answer is not any of the given options.

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The angle of elevation of the ramp to the nearest degree is 34°.

we can use trigonometry to determine the angle of elevation of the ramp.To begin, we need to draw a diagram to visualize the problem.

Let's assume that the height of the end of the ramp is h, and the horizontal distance from the bottom of the ramp to the building is d.

From the diagram, we can see that we have a right-angled triangle where the height of the triangle is h, the base of the triangle is d, and the hypotenuse of the triangle is the length of the ramp.

Using the tangent ratio, we can write:

tanθ = opposite/adjacent

where opposite is the height of the triangle, and adjacent is the base of the triangle. Substituting the values we have, we get:

tanθ = h/d

Rearranging this formula, we can write:

θ = tan⁻¹(h/d)

Substituting the values we have, we get:θ = tan⁻¹(10/15)θ = 33.69°

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Answer:

104

Step-by-step explanation:

You add 65+ 39 and that gives you 104

Note that the sum of all the regions equals the size of the universal set U, as it should be:

5 + 12 + 4 + 24 + 19 + 9 + 5 + 6 = 70.

To fill in the Venn diagram, we start with the given information:

n(A) = 33

n(B) = 36

n (B ∩ C) = 14

n (A ∩ C) = 9

n (A ∩ B ∩ C) = 5

n(U) = 70

n(C) = 24

n (A ∩ B) = 17

We can use the formula for the size of a set union to find n (B ∪ C):

n (B ∪ C) = n(B) + n(C) - n (B ∩ C)

n (B ∪ C) = 36 + 24 - 14

n (B ∪ C) = 46

We can also use the formula for the size of a set intersection to find n(A ∩ B):

n (A ∩ B) = n(A) + n(B) - n (A ∪ B)

n (A ∪ B) = n(A) + n(B) - n (A ∩ B)

n (A ∪ B) = 33 + 36 - 17

n (A ∪ B) = 52

n (A ∩ B) = 33 + 36 - 52

n (A ∩ B) = 17

Now we can start filling in the Venn diagram:

I = A ∩ B ∩ C = 5

II = A ∩ B - C = 12 (since n(A ∩ B) = 17 and n(A ∩ B ∩ C) = 5)

III = A ∩ C - B = 4 (since n(A ∩ C) = 9 and n(A ∩ B ∩ C) = 5)

IV = A - B - C = 24 (since n(A) = 33 and n(A ∩ B) = 17 and n(A ∩ C) = 9 and n (A ∩ B ∩ C) = 5)

V = B - A - C = 19 (since n(B) = 36 and n(A ∩ B) = 17 and n(B ∩ C) = 14 and n (A ∩ B ∩ C) = 5)

VI = B ∩ C - A = 9 (since n(B ∩ C) = 14 and n(A ∩ B ∩ C) = 5)

VII = C - A - B = 5 (since n(C) = 24 and n(A ∩ C) = 9 and n(B ∩ C) = 14 and n (A ∩ B ∩ C) = 5)

VIII = U - (A ∪ B ∪ C) = 6 (since n(U) = 70 and n(A) = 33 and n(B) = 36 and n(C) = 24)

Therefore, the completed Venn diagram would have:

I = 5

II = 12

III = 4

IV = 24

V = 19

VI = 9

VII = 5

VIII = 6

Note that the sum of all the regions equals the size of the universal set U, as it should be:

5 + 12 + 4 + 24 + 19 + 9 + 5 + 6 = 70

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Answer:

  B. 1/4(x-5) = 13

Step-by-step explanation:

You want to know which number sentence has one solution.

Typically, an inequality has an infinite set of solutions. The inequalities here are linear inequalities in one variable, so will have an infinite solution set consisting of numbers greater than, or less than, some boundary value.

If the equation can be arranged to be in general form, ax +b = 0, where a ≠ 0, then it will have exactly one solution: x = -b/a.

The set of number sentences contains exactly one equation. That is the number sentence that has exactly one solution:

  B. 1/4(x-5) = 13

The distribution will exhibit symmetry. The solution has been obtained by using the histogram.

What is a histogram?

A graph known as a histogram uses rectangles to show the frequency of numerical values. The vertical axis of a rectangle's height (which represents the distribution frequency of a variable) (the amount, or how often that variable appears).

The given histogram's form demonstrates that the distribution's shape exhibits symmetry (i.e. the shorter bars are to the left and to the right while the longer bars are in the middle).

Sales of vehicles under $5,000 and those between $45,000 and $50,000, with expected sales of 200 vehicles each, will be added, resulting in bars at both ends of the histogram that are the same size as the shortest bar.

Due to the distribution's continued symmetry, this won't change the histogram's initial shape of the distribution.

Hence, the first option is the correct answer.

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The complete question and the figure is attached below.

Question: A salesman sells cars with prices ranging from $5,000 to $45,000. the histogram shows the distribution of the numbers of cars he expects to sell over the next 10 years.

How would the shape of the distribution change if the salesman decides to also deal in cars priced under $5,000 and in cars priced from $45,000 to $50,000 and projects sales of 200 cars in each category?

a. the distribution will exhibit symmetry.

b. the distribution will exhibit a positive skew.

c. the distribution will exhibit a negative skew.

d. the distribution will uniform throughout.

Answer:

No.

Step-by-step explanation:

To determine whether eight-digit numbers with no digits 9 in their decimal representations constitute more than half of all eight-digit numbers, we need to calculate the total number of eight-digit numbers and the number of eight-digit numbers with no digit 9.There are 10 possible digits for each position in an eight-digit number, so there are 10^8 (or 100,000,000) total eight-digit numbers.

To count the number of eight-digit numbers with no digit 9, we can note that each digit in the number has 9 possible choices (0, 1, 2, 3, 4, 5, 6, 7, or 8), and since there are eight digits in the number, there are 9^8 (or 43,046,721) eight-digit numbers with no digit 9.Therefore, the proportion of eight-digit numbers with no digit 9 is:

9^8 / 10^8 = 0.43046721

This is less than half, so we can conclude that eight-digit numbers with no digits 9 in their decimal representations do not constitute more than half of all eight-digit numbers.

No, eight-digit numbers with no digit 9 in their decimal representations do not constitute more than half of all eight-digit numbers.

There are a total of 9 digits (0-9) in the decimal system, and an eight-digit number can have any of these digits in each of its eight places.

So, the total number of eight-digit numbers that can be formed is:

9 × 9 × 9 × 9 × 9 × 9 × 9 × 9 = 43,046,721

Out of these, there are 8 × 8 × 8 × 8 × 8 × 8 × 8 × 8 = 16,777,216 numbers that can have digits other than 9 in each of the eight places.

Therefore, the remaining numbers (43,046,721 - 16,777,216 = 26,269,505) will have at least one 9 in their decimal representations.

So, the eight-digit numbers with no digit 9 in their decimal representations constitute:16,777,216 / 43,046,721 = 0.3906 or approximately 39.06%of all eight-digit numbers. Therefore, they do not constitute more than half of all eight-digit numbers.
No, eight-digit numbers with no digit 9 in their decimal representations do not constitute more than half of all eight-digit numbers.

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With these steps, you can successfully find the mean, mode, and range of the data represented in the dot plot.

To find the mean, mode, and range of the dot plot, follow these steps:
Step 1: Count the total number of data points (dots) in the dot plot.
Step 2: Add up the values represented by the data points.
Step 3: Divide the sum from Step 2 by the total number of data points from Step 1 to calculate the mean.
Step 4: Identify the mode by finding the value(s) that appears most frequently in the dot plot (i.e., the value with the highest number of dots).
Step 5: Determine the range by finding the difference between the highest and lowest values in the dot plot.

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Step-by-step explanation:

This is kinda like the Pythagorean Theorem with THREE dimensions instead of two

diagonal^2 = 3^2 + 4^2  +  2^2

diagonal = sqrt ( 29) = 5.39 ft

Sοlving the equatiοn we get x=7

In mathematics an equatiοn is a mathematical statement which is built by twο expressiοns cοnnected by an equal ('=')sign. Fοr example, 3x – 8 = 16 is an equatiοn. Sοlving this equatiοn, we get the  x = 8.

Given that,

3/x²= x-4/3x² +2/x²

Tο sοlve the given equation  at first we will take LCM of the denominator of RHS

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

⇒[tex]\frac{3}{x^{2} }[/tex] = [tex]\frac{x+2}{3x^{2} }[/tex]

By crο ss multiplicatiοn we get,

  x+2= 9

Subtracting 2 frοm both sides of equation we get

  x= 9-2

 x=7

Hence sοlving the equation we get x=7

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we have expressed log146 - log14 3 2 as a single logarithm in terms of natural logarithms, which is ln (2/3) / ln 14.

Using the properties of logarithms, we can simplify the given expression as follows:

[tex]log_{14}[/tex](6) = [tex]log_{14}[/tex](4) - [tex]log_{14}[/tex]([tex]3^{2}[/tex])

= 1 - 2[tex]log_{14}[/tex](3)

So the expression log146 − log1432 can be written as [tex]log_{14}[/tex](6) = 1 - 2[tex]log_{14}[/tex](3).

The first step used the property [tex]log_{a}[/tex] (b/c) =  [tex]log_{a}[/tex](b) -  [tex]log_{a}[/tex](c), and the second step used the property  [tex]log_{a}[/tex](b^c) = c [tex]log_{a}[/tex](b).

In simpler terms, we can say that the given expression is equivalent to subtracting the logarithm of [tex]3^{2}[/tex] from the logarithm of 4, both to the base 14. We can then use the rules of logarithms to simplify the expression into a single logarithm of 6 to the base 14, with a coefficient of -2 in front of the logarithm of 3 to the base 14.

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The probability of drawing a red marble on the first draw is 3/8 since there are 3 red marbles out of 8 total marbles.

After the first marble is drawn, 7 marbles are remaining, so the probability of drawing a white marble on the second draw is 4/7 since 4 white marbles are remaining out of 7 total marbles.

To find the probability of drawing a red marble followed by a white marble, we multiply the probabilities of each event:

P(red, then white) = (3/8) x (4/7) = 12/56

Simplifying this fraction by dividing both the numerator and denominator by their greatest common factor of 4, we get:

P(red, then white) = 3/14

Therefore, the probability of drawing a red marble followed by a white marble is 3/14.

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The area of the given triangle in the question is 48cm².

Area = (1/2) x base x height

where the base is 8cm and the height is 12cm, as given in the question.

Substituting these values, we get:

Area = (1/2) x 8cm x 12cm

Area = 48cm²

Therefore, the area of the triangle with a base of 8cm, a height of 12cm, and an angle of 59° is 48cm². A triangle is a 2-dimensional geometric shape that consists of three line segments or sides and three angles. The sum of the interior angles of a triangle is always 180 degrees. The area of a triangle is the amount of space inside the triangle and is calculated using the formula A = (1/2) x b x h, where A is the area, b is the base of the triangle, and h is the height of the triangle. The base of a triangle is any one of its sides and the height is the perpendicular distance from the base to the opposite vertex. The area of a triangle can be used to solve various real-world problems, such as calculating the amount of material needed to construct a roof or a sail.

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The area οf the trapezium is 351 cm². Given that 2x² + x-351=0, sο the value οf x is 13 cm.

The sοlutiοns tο the quadratic equatiοn are the values οf the unknοwn variable x, which satisfy the equatiοn. These sοlutiοns are called rοοts οr zerοs οf quadratic equatiοns. The rοοts οf any pοlynοmial are the sοlutiοns fοr the given equatiοn.

We have the equatiοn 2x² + x - 351 = 0.

We can sοlve fοr x by factοring οr using the quadratic fοrmula. Since the cοefficient οf x² is 2, it's easier tο use the quadratic fοrmula:

x = (-b ± √(b² - 4ac)) / 2a

Here, a = 2, b = 1, and c = -351. Plugging in these values, we get:

x = (-1 ± √(1² - 4(2)(-351))) / 2(2)

x = (-1 ± √(1 + 2808)) / 4

x = (-1 ± √(2809)) / 4

We can simplify [tex]\sqrt{(2809)[/tex] tο 53, since 53² = 2809. Sο we have:

x = (-1 ± 53) / 4

Taking the pοsitive sοlutiοn, we get:

x = (53 - 1) / 4

x = 52 / 4

x = 13

Therefοre, the value οf x is 13 cm.

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Answer:

[tex]\begin{array}{|c|c|}\cline{1-2}\vphantom{\dfrac12}x&y\\\cline{1-2}\vphantom{\dfrac12}-2&\rm DNE\\\cline{1-2}\vphantom{\dfrac12}-1&\rm DNE\\\cline{1-2}\vphantom{\dfrac12}0&\rm DNE\\\cline{1-2}\vphantom{\dfrac12}1&-1\\\cline{1-2}\vphantom{\dfrac12}2&0\\\cline{1-2}\end{array}[/tex]

Step-by-step explanation:

Given equation:

[tex]y=\log_2x-1[/tex]

To fill out the given x-y chart, substitute each given value of x into the given log equation.

The argument of a log function can only take positive arguments, so when x = -2, x = -1 and x = 0, the y-values are undefined.

The values of y for x = 1 and x = 2 are:

[tex]\begin{aligned}x=1 \implies y&=\log_21-1\\&=0-1\\&=-1\end{aligned}[/tex]

[tex]\begin{aligned}x=2 \implies y&=\log_22-1\\&=1-1\\&=0\end{aligned}[/tex]

Therefore, the completed x-y chart for the equation y = log₂x - 1 is:

[tex]\begin{array}{|c|c|}\cline{1-2}\vphantom{\dfrac12}x&y\\\cline{1-2}\vphantom{\dfrac12}-2&\rm DNE\\\cline{1-2}\vphantom{\dfrac12}-1&\rm DNE\\\cline{1-2}\vphantom{\dfrac12}0&\rm DNE\\\cline{1-2}\vphantom{\dfrac12}1&-1\\\cline{1-2}\vphantom{\dfrac12}2&0\\\cline{1-2}\end{array}[/tex]

Note: I have used DNE for does not exist.

The patient is recieving nearly 35.5 miligrams of the drugs for  per kilogram per 24 hours.

First, we need to convert the child's weight from pounds to kilograms:

99 lbs / 2.205 = 44.9 kg

Next, we calculate the recommended dose for this weight:

40 mg/kg/24h x 44.9 kg = 1796 mg/24h

Since the recommended dose is divided into four equal doses, each dose should be:

1796 mg/24h ÷ 4 doses = 449 mg/dose

However, the physician ordered 400 mg every 6 hours, which is not the same as 449 mg every 6 hours. To calculate the actual dose per kilogram per 24 hours, we need to convert the ordered dose to the recommended dose:

400 mg/dose x 4 doses = 1600 mg/24h

Then, we divide the actual dose by the child's weight in kilograms:

1600 mg/24h ÷ 44.9 kg = 35.6 mg/kg/24h

Therefore, the patient is receiving 35.6 mg/kg/24h, which is slightly lower than the recommended dose of 40 mg/kg/24h.

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Vertex of h(x) is (-2, 0), and the axis of symmetry is the line x = -2, vertex of g(x) is (3, 0), and the axis of symmetry is the line x = 3, vertex of j(x) is (3, 2), and the axis of symmetry is the line x = 3.

What is axis of symmetry?

In mathematics, the axis of symmetry is a line that divides a given shape into two symmetrical parts, such that each part is a mirror image of the other.

We can use the standard form of a quadratic function, which is given by:

[tex]F(x) = a(x - h)^2 + k[/tex]

where (h, k) is the vertex of the parabola and the axis of symmetry is the line x = h.

For [tex]h(x) = (x + 2)^2[/tex], we have:

h = -2 (since x + 2 = 0 when x = -2)

k = 0 (since the square of any real number is non-negative)

Therefore, the vertex of h(x) is (-2, 0), and the axis of symmetry is the line x = -2.

For [tex]g(x) = (x - 3)^2[/tex], we have:

h = 3

k = 0

Therefore, the vertex of g(x) is (3, 0), and the axis of symmetry is the line x = 3.

For [tex]j(x) = (x - 3)^2 + 2[/tex], we have:

h = 3

k = 2

Therefore, the vertex of j(x) is (3, 2), and the axis of symmetry is the line x = 3.

To find the y-intercept of each function, we can simply plug in x = 0:

For h(x):

[tex]h(0) = (0 + 2)^2 = 4[/tex]

Therefore, the y-intercept of h(x) is 4.

For g(x):

[tex]g(0) = (0 - 3)^2 = 9[/tex]

Therefore, the y-intercept of g(x) is 9.

For j(x):

[tex]j(0) = (0 - 3)^2 + 2 = 11[/tex]

Therefore, the y-intercept of j(x) is 11.

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Answer:

  0 < m < 4-√12 ≈ 0.535898

Step-by-step explanation:

You want to know the range of values of m that will give |(x-1)(x-3)| = mx four distinct solutions.

The quadratic function f(x) = (x -1)(x -3) will be negative for values of x between the zeros: 1 < x < 3. Hence the absolute value function will invert the graph in that interval, as shown by the red curve in the attachment.

The line y = mx can only intersect that graph in 4 places in the first quadrant. The value of m must be greater than 0 and less than 1.

The upper limit of the slope will be defined by the value of m that makes the line intersect the inverted quadratic exactly once. That is, the discriminant of mx -(-f(x)) = 0 will be zero.

  mx +(x -1)(x -3) = x² +(m -4)x +3 = 0

  D = (m -4)² -4(1)(3) = (m -4)² -12 = 0

Solving for m gives ...

  (m -4)² = 12

  m -4 = ±√12

  m = 4 ±√12 ≈ 0.54 or 7.46

We can see from the attached graph that m ≈ 7.46 is an extraneous solution. This means the range of m will be ...

  0 < m < 4-√12