ANSWER
The slope of the line perpendicular line of the equation is 3
STEP-BY-STEP EXPLANATION:
What to find? The slope of a line perpendicular to the line whose equation is x + 3y = -15
Given the equation
x + 3y = -15
The slope-intercept form of an equation is given as
[tex]y\text{ = mx + b}[/tex]Where m = slope of the line
y = the intercept of the y-axis
The next step is to re-arrange the above equation in the slope-intercept format
[tex]\begin{gathered} \text{Given the equation of a straight line as} \\ x\text{ + 3y = -15} \\ \text{Isolate 3y by substracting x from both sides} \\ x\text{ - x + 3y = -15 - x} \\ 3y\text{ = -x - 15} \\ \text{Divide through by 3} \\ \frac{3y}{3}\text{ = }\frac{-1}{3}x\text{ -}\frac{15}{3} \\ y\text{ = }\frac{-1}{3}x\text{ - 5} \\ \text{Hence, the slope}-\text{intercept form of the above equation is given as} \\ y\text{ = }\frac{-1}{3}x\text{ - 5} \end{gathered}[/tex]NB: That the two lines are perpendicular to each other
From y = mx + b
m = -1/3
The slope of the equation
[tex]\begin{gathered} \text{ For two perpendicular lines, we can calculate the slope as follows} \\ m_1\cdot m_2\text{ =- 1} \\ \text{where m}_1\text{ = }\frac{-1}{3} \\ \frac{-1}{3}\cdot m_2\text{ = -1} \\ \frac{-1\cdot m_2}{3}=\text{ -1} \\ \text{Cross multiply} \\ -m_2\text{ = -1 }\cdot\text{ 3} \\ -m_2\text{ = -3} \\ \text{Divide through by -1} \\ \frac{-m_2}{-1}\text{ = }\frac{-3}{-1} \\ m_2\text{ = }3 \\ \text{Hence, the slope of the perpendicular line to the equation is 3} \end{gathered}[/tex]Choose the best answer. The diagonals of a rhombus:A. bisect each other and intersect at different anglesB. are the same length and intersect at a right angleC. are the same length and intersect at different anglesD. bisect each other at right angles
The property of a rhombus regarding the two diagonals states that they bisect each other at right angles.
Hence, option D is the correct answer.
What is the average rate of change please write your answer as an integer or simplify fraction
Given:
f(x)=6x+3
Required:
To calculate the average rate
Explanation:
[tex]\begin{gathered} Average\text{ rate of change} \\ \\ y=6x+3\text{ at \lparen x=5\rparen} \\ \\ y=6(5)+3=33 \end{gathered}[/tex][tex]\begin{gathered} y=6x+3\text{ at\lparen x=10\rparen} \\ \\ y=6(10)+3 \\ \\ y=63 \end{gathered}[/tex][tex]\begin{gathered} average\text{ }rate\text{ }of\text{ }change \\ \\ y=63-33 \\ \\ y=30 \end{gathered}[/tex]Required answer:
30
Under which transformation is size not preserved?A. reflectionB. dilationC. rotationD. translation
The philosophy of dilation is to resize uniformly the figure in question. Under this intuitive idea, dilation is the answer. Now, what does uniformly mean here?
It means that, as can be seen in the figure, the length of every side of the right triangle can be calculated by multiplying its corresponding side in the left ("small") triangle by a constant. However, this can be done from the right triangle to the left triangle; that's why I put a bidirectional arrow.
Finally, I want to give you some motivation for this concept: Every time you are resizing an image on your phone or in your computer, you're applying this concept.
#1 Write the quadratic function in vertex
form given a vertex of (-1,-2) and a
second point on the graph at (3,-10)
A. y = -3(x - 1)² + 2
B. y=-34 (x + 1)² + 2
C. y = -2(x - 1)²-2
D. y=-2 (x + 1)²-2
The quadratic function in vertex form is y = -1/2(x + 1)^2 - 2
How to determine the quadratic function in vertex form?The vertex is given as
(h, k) = (-1,-2)
The point is also given as
(x, y) = (3, -10)
Quadratic equations are second-order polynomial equations and they have the form y = ax^2 + bx + c or y = a(x - h)^2 + k
Substitute (h, k) = (-1,-2) in y = a(x - h)^2 + k
y = a(x + 1)^2 - 2
Substitute (x, y) = (3, -10)
-10 = a(3 + 1)^2 - 2
So, we have
-10 = a(4)^2 - 2
Add 2 to all sides
-8 = a(16)
Divide by 16
a = -1/2
Substitute a = -1/2 in y = a(x + 1)^2 - 2
y = -1/2(x + 1)^2 - 2
Hence, the equation is y = -1/2(x + 1)^2 - 2
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To determine the value of tangent of 7 times pi over 8, which identity could be used?
Given:
[tex]\tan \frac{7\pi}{8}[/tex]To Determine: The identity that is equivalent to the given tangent
Note that, the identity rule below would be applied
[tex]\tan \frac{\alpha}{2}=\sqrt[]{\frac{1-\cos \alpha}{1+\cos \alpha}}[/tex]Also,
[tex]\tan \frac{\alpha}{2}=\frac{\sin \alpha}{1+\cos \alpha}[/tex]And also,
[tex]\tan \frac{\alpha}{2}=\frac{1-\cos \alpha}{\sin \alpha}[/tex]From the given tangent, we can re-write it as below:
[tex]\begin{gathered} \tan \frac{7\pi}{8}\cong\tan \frac{\frac{7\pi}{4}}{2} \\ \text{Note} \\ \frac{7\pi}{8}=\frac{\frac{7\pi}{4}}{2} \end{gathered}[/tex]Therefore:
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\sqrt[]{\frac{1-\cos\frac{7\pi}{4}}{1+\cos\frac{7\pi}{4}}}[/tex]Also:
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\frac{\sin \frac{7\pi}{4}}{1+\cos \frac{7\pi}{4}}[/tex]And also,
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\frac{1-\cos \frac{7\pi}{4}}{\sin \frac{7\pi}{4}}[/tex]It can be observed from the option provided, the correct options is
I and III only
Look at the table. Is F(x) an exponential function? If so, Identify the base. if not, Why not?
ANSWER
YES, the base is 4 ......option B
Which one is the option to describe a piece wise function ?
.
A piece-wise function is a function that changes its value, based on the input
That is a function its range depends on the domain.
The correct answer is the third option
write the equation of the line that passes through the given points.
(4, 0) and (0, 2)
Answer:
Equation of line is given as y = mx + c, where m is the gradient and c is the y-intercept.
First find the gradient. Formula for gradient is given as (y2-y1)÷(x2-x1) or (y1-y2)÷(x1-x2).
Gradient = (2-0)÷(0-4) = -1/2
Equation of line is y = -1/2x + c
Substitute either one of the points into the equation to find c.
0 = -1/2(4) + c
c = 2
Hence, the equation of the line is y = -1/2x + 2.
QuestionWhich of the following expressions is equivalent to the verbal expression 'the quotient of 23x and 15t'?Select the correct answer below:23x · 15115123x0 23x + 151
Explanation
The quotient is the number obtained by dividing one number by another
for example:
the quotient of a and b is
[tex]a\text{ divided by b=}\frac{a}{b}[/tex]so
Step 1
Let
number 1=23x
number 2=15 t
so, the quotient would be
[tex]\frac{23x}{15t}[/tex]therefore, the answer is C:
[tex]\frac{23x}{15t}[/tex]I hope this helps you
1. Hay Story Problems Challenge Question Elliot delivered 630 newspapers in May. He delivered 35 more newspapers in June than May. Which equation can be used to find n, the number of newspapers Elliot delivered during these two months? A 630 + 35 x 2 = n B 630 + 35 = n C 630 + 630 - 35 = n D 630 + 630 + 35 = n Explain to your partner why your answer is correct.
Using elimination method, we get the result 630 + 630 + 35 = n
What is elimination method?
The elimination method involves removing one of the variables from a system of linear equations by adding or subtracting from the system and multiplying or dividing the variable coefficients.
You need an equation that can determine n, 630's total, and 35 more than 630.
More
The value "35 more in June than in May" refers to the value "35 more than 630." That value is represented by the sum (630 +35).
Total two months
In total, there will be two months' worth of delivered papers.
May deliveries + June deliveries = n
630 + (630 +35) = n
Eliminating parentheses, this expression is ...
630 + 630 + 35 = n
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A cylindrical candle is to be made from 18 in 3 of wax. If the candle’s height is twice its
diameter, what radius and height should it have, to the nearest tenth?
The radius is 1.13 inches and the height of the cylinder is 4.52 inches.
How to calculate the value?From the information, it's important to use the volume of a cylinder to illustrate the information.
Use the volume of a cylinder.
V = πr²h
V = volume.
r = radius
h = height
where:
V = 18
h = 2(2r) = 4r
Plug in the values into the equation and solve for r.
18 = πr²(4r)
18 = 4πr³
Divide both sides of the equation by 4π.
9 / (2π) = r³
r³ = 1.43
r = 1.13 inches.
Finally, multiply the radius by 4 to get the height. This will be:
= 4 × 1.13
= 4.52 inches.
In conclusion, the height is 4.52 inches.
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Find each indicated value or measure assume that all segments that appear to be tangent are tangent.
Answer:
6
Step-by-step explanation:
Using the intersecting chords theorem,
[tex]x^2=36 \\ \\ x=6 (x>0)[/tex]
A 25 ft ladder is leaning against a building.The base of the ladder is 6 ft away from the building.How high up is the ladder?
Applying the Pythagorean Theorem, the ladder's height from the ground is: 24.3 ft.
How to Apply the Pythagorean Theorem?If we know any two sides of a right triangle, the Pythagorean Theorem can be used to find the length of the third side, c, if c is the longest side, and a and b are the shorter sides of the right triangle, we ill have the equation:
c² = a² + b².
The ladder forms a right triangle with the wall of the building. Therefore:
c = length of the ladder = 25 fta = distance of base of the ladder from the building = 6 ftb = how high the ladder is up on the wall of the buildingSubstitute
25² = 6² + b²
b = √(25² - 6²)
b = 24.3 ft.
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QUIK ANSWER PLEASE!!! Solve the equationy^3 - 27 = 9y^2 - 27y
The first step is to simplify both sides of the equation. The equation can be written as
y^3 - 3^3 = 9y(y - 3)
For the left hand side, we would apply the difference of two cubes formula. it is expressed as
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
By comparing with the left hand side of the equation,
x = y and y = 3. It becomes
(y - 3)(y^2 + 3y + 3^2)
= (y - 3)(y^2 + 3y + 9)
The equation becomes
(y - 3)(y^2 + 3y + 9) = 9y(y - 3)
If we divide both sides of the equation by (y - 3), it becomes
(y - 3)(y^2 + 3y + 9)/(y - 3 = 9y(y - 3)/(y - 3)
y^2 + 3y + 9 = 9y
y^2 + 3y - 9y + 9 = 0
y^2 - 6y + 9 = 0
We would solve the quadratic equation by applying the method of factorisation. We would find two terms such that their sum or difference is - 6y and their product is 9y^2. The terms are - 3y and - 3y. The equation becomes
y^2 - 3y - 3y + 9
y(y - 3) - 3( y - 3) = 0
(y - 3)(y - 3) = 0
y - 3 = 0 twice
y = 3 twice
NEED HELP WITH THIS MATH QUESTION QUICK!
Answer:
[tex]\textsf{The quotient is $\boxed{10}\;x+\boxed{16}$}[/tex]
[tex]\textsf{The remainder is $\boxed{28}\:x^2+\boxed{10}\:x+\boxed{22}$}[/tex]
Step-by-step explanation:
Definitions
Dividend: The polynomial which has to be divided.
Divisor: The expression by which the dividend is divided.
Quotient: The result of the division.
Remainder: The part left over.
Long Division Method of dividing polynomials
Divide the first term of the dividend by the first term of the divisor and put that in the answer.Multiply the divisor by that answer, put that below the dividend and subtract to create a new polynomial.Repeat until no more division is possible.Write the solution as the quotient plus the remainder divided by the divisor.Given:
[tex]\textsf{Dividend}: \quad 10x^4-14x^3-10x^2+6x-10[/tex]
[tex]\textsf{Divisor}: \quad x^3-3x^2+x-2[/tex]
Therefore:
[tex]\large \begin{array}{r}10x+16\phantom{)}\\x^3-3x^2+x-2{\overline{\smash{\big)}\,10x^4-14x^3-10x^2+6x-10\phantom{)}}}\\{-~\phantom{(}\underline{(10x^4-30x^3+10x^2-20x)\phantom{-b)}}\\16x^3-20x^2+26x-10\phantom{)}\\-~\phantom{()}\underline{(16x^3-48x^2+16x-32)\phantom{}}\\28x^2+10x+22\phantom{)}\\\end{array}[/tex]
Solution:
[tex]10x+16+\dfrac{28x^2+10x+22}{x^3-3x^2+x-2}[/tex]
[tex]\textsf{The quotient is $\boxed{10}\;x+\boxed{16}$}[/tex]
[tex]\textsf{The remainder is $\boxed{28}\:x^2+\boxed{10}\:x+\boxed{22}$}[/tex]
Please help me I’ll mark u brainly
Answer:
a
Step-by-step explanation:
to rotate 180 degrees, simply invert both signs
Which statement correctly compares the function shown on this graph with the function y = 3x - 6?
The plotted function and the function y = 3x - 6 represent a straight line and have have same slope and hence, both the lines are parallel to each other.
What is the general equation of a straight line?The general equation of a straight line is of the form -
y = mx + c
where -
[m] is slope of line
[c] is y - intercept
Given is a equation of a straight line and a graph of a straight line.
We have the following equation -
y = 3x - 6
For this equation, the slope of the straight line will be [m] = 3 and the y - intercept would be [c] = - 6. Refer to the graph attached with green color for all the possible set of solution.
Now, the equation of the line plotted on graph -
m = (4 - 0)/(0 + 1.3)
m = 3.07
m = 3 (approx.)
Its y - intercept [c] = 4
Therefore, the equation of the plotted line -
y = 3x + 4
Now, both the lines are plotted on the graph. It can be seen from the graph and from the equation that they have same slope and hence, both the lines are parallel to each other.
Therefore, the plotted function and the function y = 3x - 6 represent a straight line and have have same slope and hence, both the lines are parallel to each other.
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[The complete question needs the plotted graph. It is attached at the end of the answer]
A carton of 12 eggs costs $3.00. A carton of 18 eggs costs $4.32. Suppose a supermarket wants to sell the eggs individually. Is there a price the supermarket can charge per egg that is between the prices per egg for the two different-sized cartons? Explain
The price the supermarket can charge per egg that is between the prices of per egg for the two different-sized cartons is $0.245 per egg.
According to the question,
We have the following information:
Cost of 12 eggs in a carton = $3.00
Cost of 18 eggs in another carton = $4.32
Now, to find the charge per egg that is between the prices of per egg for these different-sized cartons, we will first find the charge per egg of both of them.
We know that to find the cost of 1 product we divide the given cost by the number of products.
Cost of 12 eggs in a carton = $3.00
Cost of 1 egg = $ (3.00/12)
Cost of 1 egg = $0.25
Cost of 18 eggs in another carton = $4.32
Cost of 1 egg = $(4.32/18)
Cost of 1 egg = $0.24
Now, we have to find the price of egg between 0.24 and 0.25.
Note the difference between these two charges is 0.01.
We will divide it by 2 to find the exact middle charge:
0.01/2 = 0.005
We can add this in $0.24 or subtract it from$ 0.25.
$0.24 + $0.005 = $0.245
Hence, price the supermarket can charge per egg is $0.245.
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Write the equation of a line in the form y=Mx+b that passes through the point (3,6) and has a slope of -2/3. Sketch this line
Find an equation for the perpendicular bisector of the line segment whose endpoints are (3, -8) and (7,2).
ANSWER
[tex]y=-\frac{2}{5}x-1[/tex]EXPLANATION
We want to find the equation of the perpendicular bisector of the line segment with the given endpoints.
To do this, we first have to find the slope of the given line, since the slope of a line perpendicular to a given line is the negative inverse of the slope of the line.
Then, we have to find the midpoint of the given line since the line is a bisector, it passes through the midpoint of the given line segment.
To find the slope of the line, apply the formula for the slope of a line:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]where (x1, y1) and (x2, y2) are the two endpoints of the line segment
Hence, the slope of the line is:
[tex]\begin{gathered} m=\frac{2-(-8)}{7-3}=\frac{2+8}{7-3} \\ m=\frac{10}{4} \\ m=\frac{5}{2} \end{gathered}[/tex]The negative inverse of this is:
[tex]\begin{gathered} -(\frac{1}{\frac{5}{2}}) \\ \Rightarrow-\frac{2}{5} \end{gathered}[/tex]To find the midpoint of the endpoints, apply the formula for midpoint:
[tex](\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]Hence, the midpoint of the given endpoints are:
[tex]\begin{gathered} (\frac{3+7}{2},\frac{-8+2}{2}) \\ \Rightarrow(\frac{10}{2},\frac{-6}{2}) \\ \Rightarrow(5,-3) \end{gathered}[/tex]Now, we have the slope and an endpoint of the perpendicular bisector.
To find the equation of the line, we have to apply the point-slope method:
[tex]y-y_1=m(x-x_1)[/tex]Therefore, the equation of the perpendicular bisector of the line segment is:
[tex]\begin{gathered} y-(-3)=-\frac{2}{5}(x-5) \\ y+3=-\frac{2}{5}x+2 \\ y=-\frac{2}{5}x+2-3 \\ y=-\frac{2}{5}x-1 \end{gathered}[/tex]Sammy has 125 saved from his lifegaurding job
Using a linear function, it is found that:
a) The table is completed in the first image at the end of the answer.
b) The function is: s(w) = 125 - 10.50w.
c) The graph of the function is given by the second image at the end of the answer.
Linear functionThe slope-intercept representation of a linear function is shown by the rule presented below:
y = mx + b
The coefficients of the function are presented as follows:
m is the slope of the function, representing the rate of change of the output y in relation of the input x of the function, i.e, by how much y changes when x changes by 1.b is the y-intercept of the function, representing the numeric value of the function when the input x is of 0.In the context of this problem, the slope and the intercept are given as follows:
Slope of -10.50, which is the cost of the bus fare.Intercept of 125, which is how much he has saved from the bus fare.Hence the function is given as follows:
s(w) = 125 - 10.50w.
The numeric values to complete the table are given as follows:
s(6) = 125 - 10.5(6) = 62.s(10) = 125 - 10.5(10) = 20.A similar problem, also focused on linear functions, is presented at https://brainly.com/question/24808124
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Last year, Tammy opened an investment account with $8200 . At the end of the year, the amount in the account had decreased by 7.5% . How much is this decrease in dollars? How much money was in her account at the end of last year?
1. The decrease in dollars, based on a 7.5 percent decrease, is $615.
2. The balance in the account at the end of last year was $7,585.
What is a percentage?A percentage is a ratio or proportion of a variable in another.
For instance, the percentage decrease in the investment account was 7.5%, which in dollar terms amounts to $615.
The percentage reference gives an idea of the investment status and the fractional effect that the decrease had on it.
Initial investment = $8,200
Decrease in investment = 7.5%
Decrease in dollars = $615 ($8,200 x 7.5%)
Balance = $7,585 ($8,200 - $615)
Thus, Tammy's investment account decreased to $7,585 by $615, representing a 7.5% decrease.
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Find the values of x and y in the following right triangle. Enter square roots not decimals.
Recall the following trigonometric identities. If the legs of the right triangle have lengths a and b, the hypotenuse has length c, and the side a is adjacent to an angle θ, then:
[tex]\begin{gathered} \sin \theta=\frac{b}{c} \\ \cos \theta=\frac{a}{c} \end{gathered}[/tex]Then, for the given right triangle:
[tex]\begin{gathered} \sin (30º)=\frac{x}{8} \\ \cos (30º)=\frac{y}{8} \end{gathered}[/tex]Then, x and y are given by the expressions:
[tex]\begin{gathered} x=8\cdot\sin (30º)=8\cdot\frac{1}{2}=4 \\ y=8\cdot\cos (30º)=8\cdot\frac{\sqrt[]{3}}{2}=4\cdot\sqrt[]{3} \end{gathered}[/tex]Therefore, the answers are:
[tex]\begin{gathered} x=4\cdot\sqrt[]{3} \\ y=4 \end{gathered}[/tex]What are the coordinates of the foci of the conic section shown below?(y + 2)² /16 - (x − 3)² /9 = 1A. (3, -2±5)B.(-2+5,3)C. (-2,3±5)D.(-2+√7,3)
SOLUTION
Given the question on the question tab;
Explanation:
[tex]\frac{(y+2)^2}{16}-\frac{(x-3)^2}{9}=1[/tex][tex]h=3,k=-2,a=3,b=4[/tex][tex]The\text{ }standardform\text{ }is\frac{\text{ }\left(y+2\right)^2}{4^2}-\frac{\left(x - 3\right)^{2}}{3^{2}}=1.[/tex][tex]The\text{ }linear\text{ }eccentricity\text{ }is\text{ }c=\sqrt{b^{2} + a^{2}}=5.[/tex][tex]\begin{gathered} The\text{ first focus is:} \\ \left(h,k−c\right)=\left(3,−7\right). \\ The\text{ second focus is:} \\ \left(h,k+c\right)=\left(3,3\right) \end{gathered}[/tex]Final answer:
What is the ones digit in the number 2^2058. Hint: Start with the smaller exponents to find pattern
1) We need to find out a pattern for the powers that have a base 2:
[tex]\begin{gathered} 2^{1}=2 \\ 2^2=4 \\ 2^3=8 \\ 2^4=16 \\ 2^5=32 \\ 2^6=64 \\ 2^7=128 \\ 2^8=256 \\ (\ldots) \end{gathered}[/tex]Note that from 4 to 4 powers the last digit starts to repeat itself.
2) So, let's proceed with this dividing the exponent 2058 by 4:
Now, note that the remainder is 2, therefore we can state that:
[tex]2^{2058}=same\: ones\: digit\: as\colon2^2=4[/tex]what’s the correct answer answer asap for brainlist
Answer:
B
I hope that this helps
A performer expects to sell 5000 tickets for an upcoming concert.
They plan to make a total of $311000 in sales from these tickets.
Assume that all tickets have the same price.
How much money will they make if they sell 7000 tickets?
Answer:
$435,400
Step-by-step explanation:
Price of one ticket: $311,000/5,000
Price of one ticket: $62.20
Price of 7000 tickets: $62.20 x 7000
Price of 7000 tickets: $435,400
The expression 81 √ ⋅ 100√ represents the number of feet between home plate and first base. What is the distance, in feet, between home plate and first base? Answers are 19 30 80 and 810.
The distance between home plate and first base is -1 feet
what is distance ?
Distance is a numerical or occasionally qualitative measurment of how far apart objects or point are . in physics or everyday usage , distance may refer to a physical length.
How to determine the number
From the information given, we have that;
The expression for the number of feet between home plate and first base is given as;
√81-√100
Where;
√81 is the number of feet at home plate
√100 is the number of feet at the first base
To determine the distance, we take find the square root of the numbers and substitute, we have;
9 - 10 Find the difference
-1 feet
Thus, the distance between home plate and first base is -1 feet
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Find the equation of a parabola with a focus of (0, 9) and directrix y = –9.
Answer:
Step-by-step explanation:
Given that,
To find the standard form of the equation of the parabola with a focus at (0, 9) and a directrix y = -9.
What is a parabola?
A parabola is a cross-section cut out of the cone and represented by an equation
Focus of the prabola = (h , k + F ) = (0, 9)
Since the directrix, y = -9
F = -9
k + F = 9
k = 0
Vertex of the parabola = (h, k )
= (0, 0)
Standard equation of the parabola
( y - k ) = 4a (x - h)²
( y - 0 ) = 4a (x - 0)²
y = 4 * 9 x²
y = 36 x²
Thus, the required expression for the parabola with focus at (0, -9) and a directrix y = 9 is y = 36x².
write an equation, in factored form, of degree 6 polynomial that has four x-intercepts and a y-intercept of 64
ANSWER :
f(x) = (3x-2)(x-3)(x-4)(x-2)^3
EXPLANATION :
A polynomial with a degree of 6 has 6 factors.
f(x) = (x - a)(x - b)(x - c)(x - d)(x - e)(x - f) with 6 roots or x-intercepts.
But the problems states that it has 4 x-intercepts, so we will reduced the number of roots but maintaining the number of factors.
f(x) = (x - a)(x - b)(x - c)(x - d)(x - a)(x - a).
From here, we still have 6 factors but only 4 x-intercepts, the last two factors (x - a) is the same as the first factor.
So we can rewrite this as :
[tex]f(x)=(x-a)^3(x-b)(x-c)(x-d)[/tex]Next is to have a y-intercept of 64, y-intercept is the value of f(x) when x = 0
Substitute 0 to the function.
[tex]\begin{gathered} f(0)=(0-a)^3(0-b)(0-c)(0-d) \\ f(0)=a^3(b)(c)(d) \end{gathered}[/tex]Now we have f(0) = a^3bcd and f(0) = 64 as the definition from above.
We need to find the factors of 64,
64 = 8 x 4 x 3 x 2/3
And we can rewrite the equation as :
[tex]\begin{gathered} f(0)=a^3bcd \\ 64=a^3bcd \\ 8\times4\times3\times\frac{2}{3}=a^3bcd \end{gathered}[/tex]From here, we can observe that,
a^3 = 8 ⇒ a = 2
b = 4
c = 3
d = 2/3
So the function will be :
[tex]\begin{gathered} f(x)=(x-2)^3(x-4)(x-3)(x-\frac{2}{3}) \\ f(x)=(x-2)^3(x-4)(x-3)(3x-2) \end{gathered}[/tex]Explanation in 2/3
Since we only need 4 distinct factors of 64.
8 x 4 x 3 x 2/3
8 x 4 = 32
The product of the 3rd and 4th factor should be 2, in order to get 64.
Since from the first