A pyramid can be formed using equal-size balls. For example, 3 balls can be arranged in a triangle, and then a fourth ball placed in the middle on top of them. The function p(n) = -n(n + 1)(n + 2) gives the number of balls in a pyramid, wheren is the number of balls on each side of the bottom layer. (For the pyramid described above, n = 2. For the pyramid in the picture, n = 5.) If you had 1000 balls available and you wanted to make the largest possible pyramid using them, what would be the size of the bottom triangle, and how many balls would you use to make the pyramid? How many balls would be left over?

Q: Determine the value of k such that when f(x)=x* - kx +7x-6 is divided by x - 2, the remainder is 4.

A: We have to find the value of k. To solve this problem we will use 'Remainder theorem' which states…

Q: Find the domain of the function: h(x) = √(3x + 12)

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Q: A degree five polynomial with a negative leading coefficient is called a _function and its right arm…

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Q: The Range of the quadratic function f(x) = -x² + 18x + 52 is O [-9, 00) O [52, 0) O (0,9] O (-∞0,…

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Q: If f(-6) = 6 and f '(x) < 8 for all x, what is the largest possible value of f (3)?

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Q: The graph of the polynomial f(x) is given below. If f(x) has degree 3, find the factored equation…

A: Using a graph, it can be found out whether a function has solutions if the graph cuts the x-axis at…

Q: Consider the function f (x) = x - 12x + 40x². If there is a remainder of 727,875 when the function…

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Q: 6. Determine the vertex and the axis of symmetry for f(x) = 3x2 – 5x + 12

A: We have given a quadratic function.  We have to find the vertex and line of symmetry

Q: 1. The CEO of a furniture factory finds that the cost to manufacture beds is given by the function…

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Q: Find the values of k for which f(x) = kx3 -9kx2+9x+3 is increasing on R

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Q: 6. For the function a 10th degree polynomial. a) Up to how many x intercepts can there be? b) Up to…

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Q: For the polynomial function in FACTORED form, f (z) = –5a* (x + 3)* (z² + 4)*(z – 1)*. find the END…

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Q: 2. Find all the values of x for which the function 3x – 2 - g(x) = is undefined. %3D 3x3 + 13x2 –…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: For the polynomial p(x) = -(x – 10)²(x + 4) a. Graph p(x) by analyzing end behavior and…

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Q: If g(x) = x2 + 8x with x 2 -4, find g(33). 5=

A: Given g(x)=x2+8x, x≥-4 To find g-133

Q: If g(x) = - 1 where f(3) = 4 and f'(3) = - 3, find g'(3). f(x) Express your answer as an integer, or…

A: It is given that,the function             g(x)=1f(x)         where f(3)=4, f'(3)=-3 we have to find…

Q: For what values of k will the function g(x)=x2-kx+9 have only one zero

A: compare this polynomial with the standard form of a polynomial function    so here we have

Q: Find the largest damain af the function : 2X+2X-1 x²+3X-10

A: Given  y = 2x2+2x-1x2+3x-10 To find the domain of the above function, we have to find the value of x…

Q: 13. If the polynomial function is of degree 4, how many turning points will there be? B. at most 5…

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Q: Find the asymptotes of the functions: 4x2 1- f(x) = x² +8 x2-6x+7 2- f(x) x+5

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Q: A graph of y=f(x) is shown, where f(x)=2x^5-3x^4+x^3-8x^2+5x+3 and f(-x)= -2x^5-3x^4-x^3-8x^2-5x+3.…

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Q: pretty dish can be determined by the function below.  b(t)=2(3)^t Which statement represents the…

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Q: Sketch the function. 4 k(x) = -x² +6x²-8 Part 1 of 4

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Q: The profit from producing n quilts per year is given by the function P(n) = -15n² + 240n – 585,…

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Q: Consider the following quadratic function. f(x) = 3x^2 - 30x  + 73   plot five points on the graph…

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Q: 3. For the graph of the function f(x) = x² – 20x + 102, find the following: a. Use completing the…

A: To Determine: for the graph of the following a) use completing the square to find the axis of…

Q: 2. f(x) = 14x – 10x² + 2x – 5

A: Let's find.

Q: for g(x)=24x3+24x26x and k(x) = 8x3+1, find an equation of k/g, evaluate  (k/g) (-1) (k/g)(x)=…

A: Given that,  g(x)=24x3+24x2+6x  and k(x) = 8x3+1. By using the definition of f/g, fg(x)=f(x)g(x), if…

Q: the graph of the quadratic function f(x) = x2 + dx + 5d has its vertex on the x-axis, what are the…

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Q: Suppose that 3 ≤ f '(x) ≤ 5 for all values of x. What are the minimum and maximum possible values of…

A: Given problem:- Suppose that 3 ≤ f'(x)  ≤5 for all values of x. What are the minimum and maximum…

Q: If g(x) 5 x + 6 f(x), where f(- 5) = 3 and f'(- 5) = - 10, find g'(- 5). Express your answer as an…

A: Given gx=5x+6fxf-5=3f'-5=-10

Q: If g(x) = - 3 f(x), where f(6) = 10 and f'(6) = - 4, find g'(6). Express your answer as an integer,…

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Q: The graph of f(x) = x4 + x3 - 7x2 - x + 6 is shown below. Use the graph to factor f(x). O A. f(x) =…

A: From the graph, the x-interceptions are at x=-3,-1,1 and 2. so, the roots of the equation f(x)=0 are…

Q: If f(2) 5 and f'(x) > 2 for 2

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Q: If f(4) = 9 and f'(x) > 2 for 4

A: Solution: Given that f(4) = 9 and f'(x) 2 for all 4  x  6 If a function is continuous on some…

Q: Given h(x) V2x + 4, find four points on the graph of the function.

A: Given that: hx=2x+4                                                                 ...........(1)…

Q: Find the asymptotes of the functions: 4x2 1- f(x) = x² +8

A: As per our guidelines we need to solve only one question.Please repost the other remaining question…

Q: If f(-9) = 2 andf'(x) < 9 for all x, what is the largest possible value of f(1)?

A: Lagrange's Mean Value Theorem : If f(x) be continuous on [a, b] and differentiable on (a, b) then…

Q: 1. Construct a polynomial function f(x) that might have the given graph shown. ( More than one…

A: For the given graph we, need to construct polynomial function fx

Q: Find the function that corresponds to the graph. -3 %3D O f(x) = 2x - 2 x) = + 4 %3D O f(x) = 2x + 4…

A: The graph is given and we need to find the corresponding function.

Q: If the graph of the quadratic function f(x) = x2 + 5dx + d2 + 1 has its vertex on the x-axis, what…

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Q: Find o for the function g(X) - 4x +5.

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Q: olynomial function f ( x ) = ( x + 2 ) 8 ( x − 2 ) 7 ( x − 4 ) 2 has 2 as a zero. What is the…

A: Given polynomial function :

Q: Consider the function f(x) = x + 16x² + 60x + 40. %3D If there is a remainder of-5 when the function…

A: Remainder theorem : It says that if you divide a polynomial, f(x), by a linear expression, x-A,…

Q: SEE EXAMPLE 2 Find the zeros of each function by factoring. 7. h(x) =x² + 4x 10. h(x) = 3x² + 13x +4…

A: We solve question number 9 by finding root we just equate given function with zero

Q: Determine the domain of the function f(n) = n/6-2n

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Q: In the function:  f(x)= (7x+B)/(4x+b) , with 7b&gt;4B When is the function increasing and when is…

A: Function is increasing where slope f’(x) &gt;0 and decreasing where f’(x) &lt;0. Find the derivative…

Q: Given n(x) = 7|x| + 3x - 1 (a) find n(-x). (b) find -n(x). (C) Is n(-x) = -n(x)? (d) is n(-x) =…

A: Given: n(x) = 7 |x| + 3x - 1

Q: The function W(x) = 0.5x² gives the number of servings of wedding cake that can be obtained from a…

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Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.