Chebyshev's inequality is a statistical tool that provides a bound on the probability that a random variable deviates from its mean by a certain amount. It can be used to determine whether there is statistical evidence to support a hypothesis based on sample data.
Suppose we have a random variable D that represents the proportion of defective lightbulbs in a population. We want to test the hypothesis that the average proportion of defective lightbulbs in the population is greater than 0.10, i.e., H₀: D ≤ 0.10 vs H₁: D > 0.10. Let's assume that we have a sample of n lightbulbs and that the sample proportion of defective lightbulbs is W.
Chebyshev's inequality states that for any random variable X, the probability that X deviates from its mean by k standard deviations is at most 1/k². In other words,
P(|X - μ| ≥ kσ) ≤ 1/k²,
where μ and σ are the mean and standard deviation of X, respectively.
Now, let's apply Chebyshev's inequality to our sample proportion W. Since we don't know the true mean and standard deviation of D, we can use the sample mean and sample standard deviation as estimates. The sample mean is W/n and the sample standard deviation is √[W(1-W)/n]. We want to find the probability that D is greater than 0.10, which is equivalent to finding the probability that W/n is greater than 0.10.
Let k = (0.10 - W/n)/(√[W(1-W)/n]). Then,
P(D > 0.10) = P(W/n > 0.10) = P(W - 0.10n > 0) = P(W - μ ≥ (0.10 - μ)n) ≤ σ²/[(0.10 - μ)n]²,
where μ = E(W) = D and σ² = Var(W) = D(1-D)/n. Thus,
P(D > 0.10) ≤ D(1-D)/n[(0.10 - D)n]².
To decide whether there is statistical evidence to support the hypothesis H₁, we need to compare this upper bound on the probability of D being greater than 0.10 to the significance level α = 0.05. If the upper bound is less than α, then we reject the null hypothesis H₀ in favor of the alternative hypothesis H₁.
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Write a form of 1 that you can use to rationalize the denominator of the expression.
A form of 1 that you can use to rationalize the denominator of the expression 8/(∛4), the rationalizing factor is 9.
Describe Rationalization?In mathematics, rationalization refers to the process of eliminating radical or irrational expressions from the denominator of a fraction. This is done by multiplying both the numerator and the denominator of the fraction by a suitable expression that will result in a rational denominator.
The resulting fraction has a rational denominator, which makes it easier to work with and manipulate algebraically. Rationalization is a useful technique in algebra, trigonometry, and calculus, and is often used to simplify expressions and solve equations.
To rationalize the denominator of the expression 8/(∛4), we need to multiply the numerator and the denominator by a rationalizing factor that will eliminate the radical in the denominator.
Since the root 4 is equal to 2, we can rewrite the expression as:
8/3²
The square of 3 is 9, so we can use 9 as the rationalizing factor.
Multiplying the numerator and denominator by 9, we get:
(8/3²) x (9/9) = 72/9
Simplifying, we get:
72/9 = 8
Therefore, the rationalizing factor is 9.
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prove that:
(P+1/P-1 + P+1/P+1)-(P-1/P+2 + P+1/P-2)=-4(2p² + 1)/(p²-4) (p² - 1)
The given identity (P+1/P-1 + P+1/P+1)-(P-1/P+2 + P+1/P-2)=-4(2p² + 1)/(p²-4) (p² - 1) is proven below
How to prove the given identityGiven the following equation
(P+1/P-1 + P+1/P+1)-(P-1/P+2 + P+1/P-2)=-4(2p² + 1)/(p²-4) (p² - 1)
We start by simplifying the left-hand side of the equation:
(P+1/P-1 + P+1/P+1)-(P-1/P+2 + P+1/P-2)= [(P²+1)/(P(P-1))] + [(P²+1)/(P(P+1))] - [(P²-1)/((P+2)P)] - [(P²+1)/((P-2)P)]
Next, we have the following steps to simplify the expression
= [(P³ + P² + P + 1)/(P(P²-1))] - [(P³ - 3P)/(P(P²-4))]
= [(P³ + P² + P + 1)(P²-4) - (P³ - 3P)(P²-1)]/[(P(P²-1))(P(P²-4))]
= [(P⁵ - 3P³ + P³ - 3P² + P² - 4P + P² - 4P - 4 + P³ - 3P)/(P⁴ - 5P² + 4)]
= [P⁵ - 2P³ - 6P² - 8P - 4]/[(P²-4)(P²-1)]
= -4(2p² + 1)/(p²-4) (p² - 1)
Hence, the given identity is proven.
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How many quarters do you need to add to 31/4 to get 41/2
Answer:
51/4 quarters
Step-by-step explanation:
To add fractions with different denominators, we need to find a common denominator. In this case, the common denominator for 4 and 2 is 4x2=8.
So we need to convert both fractions to have a denominator of 8:
31/4 = (31/4) x (2/2) = 62/8
41/2 = (41/2) x (4/4) = 164/8
Now we can add the fractions:
62/8 + x/4 = 164/8
Subtracting 62/8 from both sides:
x/4 = 102/8
Simplifying:
x = 51/4
Therefore, we need to add 51/4 quarters to 31/4 to get 41/2
diegos family spend 130$ total on a night ouy. they purchases 5 tickets to the fair and a family dinner for 55$ how much did each ticket to the fair cost
If Diego's family spent a total of $130, on a night out, then the each ticket for the fair cost's $15.
In order to find the "total-cost" of the tickets, we subtract the cost of the family dinner from the total amount spent;
⇒ Total cost of tickets = Total amount spent - Cost of family dinner,
⇒ Total cost of tickets = $130 - $55,
⇒ Total cost of tickets = $75.
Next, we divide the total cost of the tickets by the number of tickets purchased to find the cost of each ticket:
So, Cost per ticket = (Total cost of tickets)/(Number of tickets),
⇒ Cost per ticket = $75/5,
⇒ Cost per ticket = $15,
Therefore, each ticket to the fair cost $15.
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Find f(g(1)) and g(f(1))
F(x)=x^2+2;g(x)=2x-5
Answer:
Step-by-step explanation:
Given the functions f(x) = x^2 + 2 and g(x) = 2x - 5, we can find f(g(1)) and g(f(1)) by evaluating the inner function first and then using its result as the input for the outer function.
First, let’s find f(g(1)). We start by evaluating the inner function g(1):
g(1) = 2 * 1 - 5 = -3
Now we can use this result as the input for the outer function f(-3):
f(g(1)) = f(-3) = (-3)^2 + 2 = 9 + 2 = 11
Next, let’s find g(f(1)). We start by evaluating the inner function f(1):
f(1) = 1^2 + 2 = 3
Now we can use this result as the input for the outer function g(3):
g(f(1)) = g(3) = 2 * 3 - 5 = 6 - 5 = 1
So, f(g(1)) = 11 and g(f(1)) = 1.
The figure on the right is a scaled copy of the figure on the left
Which side in the figure on the right corresponds to segment UV?
What is the scale factor
The side that corresponds to uv is LK
The scale factor is 3 : 1
How to solve for the scale factorTo solve for the scale factor between two geometric figures, follow these steps:
Identify corresponding sides or corresponding lengths between the two figures.
Choose one pair of corresponding sides and write a proportion using the lengths of those sides.
Solve for the scale factor by simplifying the proportion.
The shape in UV occupies the space of 6 boxes
The space in LK is made of 2 boxes
Hence we have 6 : 2
= 3 : 1
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solve the rational equation 2/x-2+3/x-4=1/x^2=6x+8
the solution to the original rational equation is: [tex]x = 4.678[/tex] (rounded to three decimal places)
What is the rational equation?A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not zero. In other words, it is a number that can be written in the form of a/b, where a and b are integers and b is not equal to zero
According to InformationThere are two equal signs in the equation, which is incorrect. Assuming you meant to write:
[tex]2/(x-2) + 3/(x-4) = 1/(x^2 + 6x + 8)[/tex]
We can start by finding a common denominator for the left-hand side of the equation:
[tex]2/(x-2) + 3/(x-4) = 1/[(x+2)(x+4)][/tex]
Multiplying both sides of the equation by (x-2)(x-4)(x+2)(x+4), we get:
[tex]2(x-4)(x+2)(x+4) + 3(x-2)(x+2)(x+4) = (x-2)(x-4)[/tex]
Expanding and simplifying, we get:
[tex]5x^3 - 19x^2 - 39x + 56 = 0[/tex]
This polynomial equation does not factor nicely, so we can use the rational root theorem or numerical methods to find approximate solutions. Using a calculator or computer, we find that there is one real solution to the equation:
[tex]x \approx 4.678[/tex]
Therefore, the solution to the original rational equation is:
[tex]x = 4.678[/tex] (rounded to three decimal places)
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I deposited #300.00 in a bank for
four years. If it earned simple
interest at the rate of 6% per annum,
how much interest did I get for the
four years?
Answer:
Simple interest= PRT/100
parameters
price =300
Rate=6%
Time=4years
300*6*4/100
7200/100
=72
Angle 3 is 120°.
What is the
measure of 25?
m45 = [?]°
Answer in degrees.
[?]°
1/2
4/3= 120°
5/6
8 7
Enter
Answer:
angle 5 is 120 since there is Z shape with angle 3
Which shows one way to determine the factors of x³ + 5x² - 6x - 30 by grouping?
Ox(x²-5) + 6(x² - 5)
Ox(x²+5)-6(x²+ 5)
O x²(x - 5) + 6(x - 5)
O x²(x+5)-6(x+ 5)
Answer:
x^2(x+5)-6(x+5)!!!!!!!!
A glass jug can hold (p+6) quarts less water than a plastic container. 2 glass jugs and 2 plastic containers contain 6p quarts of water in all.
How much water can the plastic container hold? Give your answer im terms of p.
The amount of water that the plastic container can hold is -p + 3.
How to determine the amount of water that can be heldTo determine the amount of water that can be held, we will first assume that the container can hold x quantity of water.
So, the glass jug can hold:
p + 6 - x
2 glass jugs and 2 plastic containers can hold 6p quarts of water.
= 2(p + 6 - x) - 2x = 6p
2p + 12 - 2x -2x = 6p
2p + 12 -4x = 6p
12 - 4x = 6p - 2p
-4x = 6p -2p - 12
-4x = 4p -12
x = 4p - 12/-4
x = -p - 3
or -p + 3
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McKenzie painted a yellow rectangle that measured 8 1/2 in. wide and 11 in. long for part of a mural she was working on. She also painted a
purple rectangle that was 1/4 the size of the yellow rectangle. What is the area of the purple rectangle that McKenzie painted?
Answer:
23.375 in^2
Step-by-step explanation:
To find the area of a rectangle or square, you need to multiple the length by the width.
The length: 8.5
Width: 11
8.5*11 = 93.5
Now we divide by 4
93.5/4
23.375
(x+3) (x-2) =2x
find the valűe of x
Step-by-step explanation:
[tex](x + 3)(x - 2) = 2x\\ {x}^{2} + x - 6= 2x \\ {x}^{2} - x - 6 = 0 \\ (x - 3)(x + 2) = 0 \\ x = 3 \: or \: x = - 2[/tex]
What is the end behavior of this radical function?
The end behavior of this radical function is "as x approaches positive infinity, f(x) approaches positive infinity".
As we know that the function f(x) = 4√(x − 6) is a radical function with an even index (4), which means that the function is defined for all non-negative values of x.
As x approaches positive infinity, the value of x − 6 also approaches positive infinity, and the square root function grows without bound.
Since the function is multiplied by a positive constant (4), the entire function f(x) also grows without bound as x approaches positive infinity.
Therefore, the end behavior of the function is that as x approaches positive infinity, f(x) approaches positive infinity.
Hence, option A correctly describes the end behavior of the function.
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PLS HELP ASAP 90 POINTS Kim and her friends watched the server making smoothies. The table shows the number of mangos that were used for each of the different sizes of smoothies that the friends ordered.
Mangos Used Smoothie Size
Bri 1 7 oz
Kim 3 21 oz
Angela 4 28 oz
Which statement is correct based on the data?
The ratio of smoothie size to mangos used for Kim is 1:7, and the ratio of smoothie size to mangos used for Bri is 3:28.
The ratio of smoothie size to mangos used for Angela is higher than the ratio of smoothie size to mangos used for Kim.
The ratio of smoothie size to mangos used for Bri is 7:1, and the ratio of smoothie size to mangos used for Angela is 4:28.
The ratio of smoothie size to mangos used for Kim is the same as the ratio of smoothie size to mangos used for Angela.
Answer:
The correct statement is "The ratio of smoothie size to mangos used for Kim is the same as the ratio of smoothie size to mangos used for Angela".
The correct answer option is option A
How to solve ratio?
Mangos Used Smoothie Size
Angela 1 9 oz
Kim 3 27 oz
Bri 4 36 oz
Check the given options to determine which statement is correct;
The ratio of smoothie size to mangos used for Kim is the same as the ratio of smoothie size to mangos used for Angela.
Kim = 27 : 3 = 9 : 1
Angela = 9 : 1
True
The ratio of smoothie size to mangos used for Kim is 1:9, and the ratio of smoothie size to mangos used for Bri is 4:27.
Kim = 27 : 3 = 9 : 1
Bri = 36 : 4 = 9 : 1
False
The ratio of smoothie size to mangos used for Angela is higher than the ratio of smoothie size to mangos used for Kim.
Angela = 9 : 1
Kim = 27 : 3 = 9 : 1
False
The ratio of smoothie size to mangos used for Bri is 9:1, and the ratio of smoothie size to mangos used for Angela is 4:36.
Bri = 36 : 4 = 9 : 1
Angela = 9 : 1
False
Therefore, Kim and Angela has equal ratio of smoothie size to mangoes used.
please help for this question
The dilation of the object is given by the coordinates:
(-6,-8)
(-2, 0)
(-2, -8)
A dilation is a function f from a metric space M into itself that fulfills the identity d=rd for all locations x, y in M, where d is the distance between x and y and r is some positive real integer.
For the first dilation
Original points are
(2,1)
(4,5)
(4,1)
Multiply by scale factor
(2,1) x 2 = (4,2)
(4,5) x 2 = (8, 10)
(4,1) x 2 = (8, 2)
This given us the coordinates of triangle B).
For the second dilation
(2,1)
(4,5)
(4,1)
Adjust for the center of dilation which is (5,5)
(2,1) less (5,5) = (-3, -4)
(4,5) less (5,5) = (-1, 0)
(4,1) less (5,5) =(-1, -4)
Multiply the New original point by scale factor
(-3, -4) x 2 = (-6,-8)
(-1, 0) x 2 = (-2, 0)
(-1, -4) x 2 = (-2, -8)
Thus, the new coordinates of the dilated triangle C are:
(-6,-8)
(-2, 0)
(-2, -8).
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Find mJKM
J= 3x
L=64
JKM=(5x+4)
The measure of angle JKM is 26°
What is exterior angle theorem?The exterior angle theorem states that the exterior angle of a triangle is equal to the sum of the opposite angles in the triangle.
Also the sum of angle in a triangle is 180°
Therefore ;
3x+64 = 5x+4
collecting like terms
5x -3x = 64-4
2x = 60
x = 60/2
x = 30
since x = 30
value of the exterior angle = 5x+4
= 5×30+4
= 150+4 = 154
therefore angle JKM = 180-( 154)
= 26°
Therefore the measure of angle JKM is 26°
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Explain how you know what a fraction was multiplied by when the product is greater than a factor.
When the product of a fraction and a factor is greater than the factor, it means that the fraction is greater than 1.
Why is this true of fractions ?Due to the principles of multiplication, when multiplying a value greater than 1 with a given amount, the product will be larger than the original number. To provide an example, if we multiply 5 by 2, the result will be 10, which is greater than 5.
By extension, if we multiply a fraction with a factor that's greater than 1, the resulting product will be greater in size as compared to the initial quantity. For instance, when we calculate 1/2 multiplied by 3, the outcome is 3/2, which surpasses the worth of 1/2. Hence, it can be deduced that any result which exceeds its own source was obtained through multiplication by value greater than 1.
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Can someone help me with this? I can't figure it out
In linear equation, 9 is the constant of variation k.
What in mathematics is a linear equation?
An algebraic equation with simply a constant and a first-order (linear) component, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Equations with variables of power 1 are referred to be linear equations. axe+b = 0 is a one-variable example in which a and b are real numbers and x is the variable.
given x varies inversely with y then
xy = k ← k is the constant of variation
to find k use the condition x = - 4 when y = - 9, hence
k = -4 × -9 = 36
x = 36/y
when x = 4 , then y = 36/4
x = 9
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help!!!
in the fridge below, m WXZ =72, abs m 1 is 6 degrees more than m 2. find m2
Step-by-step explanation:
m ∠WXY = 72°
m ∠ 1 = m ∠ 2 + 6°
m ∠WXY = m ∠ 1 + m ∠ 2
72° = (m ∠ 2 + 6°) + m ∠ 2
72° = 2 × (m ∠ 2) + 6°
2 (m ∠ 2) = 72° - 6°
2 (m ∠ 2) = 66°
(m ∠ 2) = 33°
#CMIIWfill in "blanks"
blank g = blank kg = 3/10 kg
The amount of grams that is equivalent to 3/10 of a kg is given as follows:
300 grams.
How to obtain the amount of grams?The amount of grams that is equivalent to 3/10 of a kg is obtained applying the proportions in the context of the problem.
The amount of kg is 3/10 of a kilogram is given as follows:
3/10 = 0.3kg.
(conversion of a fraction to decimal, divide the numerator by the denominator).
Each kg is composed by 1000 grams, hence the amount of grams in 0.3 kg is given as follows:
0.3 x 1000 = 300 grams.
(proportion applied to obtain the conversion).
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A bag contains 19 blue, 28 purple, 21
red, and 29 orange balls. You pick
one ball at random. Find the
probability that it is purple or orange.
P(purple or orange) =
Simplify your answer completely.
Thus, the probability for the outcome of either purple or orange: P(purple or orange) = 57/97.
Explain about the term random selection:a choice that is made at random (entirely by chance, with no predictability). Every person in the population under study need to have an equal probability of getting chosen. Every person who is under investigation has an equal probability of being chosen at random for the sample.
Given bag with balls:
19 blue, 28 purple, 21 red, and 29 orange ballsTotal = 97 ballsprobability = favourable outcome / total outcome
P(purple) = 28/97
P(orange) = 29/97
Thus,
P(purple or orange) = 28/97 + 29/97
P(purple or orange) = 57/97
Thus, the probability for the outcome of either purple or orange ball : P(purple or orange) = 57/97.
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Please Factorise 8x - 6
Answer:
2(4x-3)
Step-by-step explanation:
Find common numbers between 8 and 6, which is 2. 2x4= 8 2x3=6
x is only on 8 so you can't put x on the outside of the bracket. so you put it with the 4 so it comes out correct
Suppose that $2000 is invested at a rate of 4.6%, compounded quarterly. Assuming that no withdrawals are made, find the total amount after 6 years
Do not round any intermediate computations, and round your answer to the nearest cent.
Answer:
$2,631.55
Step-by-step explanation:
To find the total amount in the account after 6 years, we can use the compound interest formula.
Compound Interest Formula[tex]\boxed{\sf A=P\left(1+\dfrac{r}{n}\right)^{nt}}[/tex]
where:
A = Final amount.P = Principal amount.r = Interest rate (in decimal form).n = Number of times interest is applied per year.t = Time (in years).Given values:
P = $2,000r = 4.6% = 0.046n = 4 (quarterly)t = 6 yearsSubstitute the given values into the formula and solve for A:
[tex]\implies \sf A=2000\left(1+\dfrac{0.046}{4}\right)^{4 \cdot 6}[/tex]
[tex]\implies \sf A=2000\left(1+0.0115\right)^{24}[/tex]
[tex]\implies \sf A=2000\left(1.0115\right)^{24}[/tex]
[tex]\implies \sf A=2000\left(1.3157739...\right)[/tex]
[tex]\implies \sf A=2631.54794...[/tex]
Therefore, the total amount after 6 years is $2,631.55 rounded to the nearest cent.
I would love some help please 18-20
18. C
(m / 2) - 6 = (m / 4) + 2
---Multiply everything by the LCM of the denominators
---LCM = 4
2m - 24 = m + 8
m - 24 = 8
m = 32
19. A
k / 12 = 25 / 100
---We can simplify 25/100
---We want to simplify enough to where the denominator of 25/100 is a multiple or factor of 12
k / 12 = 1 / 4
---4 x 3 = 12, 1 x 3 = 3
k = 3
20. A
9 / 5 = 3x / 100
---Cross multiply and solve algebraically
(5 * 3x) = (9 * 100)
15x = 900
x = 60
Hope this helps!
Find the circumference and the area of a circle with radius 4 m.
Use the value 3.14 for , and do not round your answers. Be sure to include the correct units in your answers.
4m
Circumference:
Area:
On a scale drawing of a soccer field, 0.5 cm equals 8 m.
If the drawing has dimensions 6.5 cm X 3.25 cm, what is the actual length of the soccer field, in meters?
[tex]\sf Length\, of\,the\,field=\boxed{\sf 104(m)}}.[/tex]
Step-by-step explanation:1. Create a conversion factor.A conversion factor is just a fraction that contains an equivalence. We make the numerator be the unit we want to have as a result and the denominator is the current unit that we have.
The problem states that 0.5 cm equals 8 m, and we want to convert from cm to m, therefore, a conversion factor for this problem is:
[tex]\sf \dfrac{8(m)}{0.5(cm)}[/tex]
2. Use the conversion factor to convert each unit,To use the conversion factor, just multiply each measure by the fraction:
[tex]\sf 6.5(cm) \dfrac{8(m)}{0.5(cm)}[/tex]
Here the centimeters (cm) cancel out each other and the ending answer is expressed in meters:
[tex]\sf 6.5(cm) \dfrac{8(m)}{0.5(cm)}=\boxed{\sf 104(m)}.[/tex]
For the other measure:
[tex]\sf 3.25(cm) \dfrac{8(m)}{0.5(cm)}=\boxed{\sf 52(m)}.[/tex]
So a soccer field, as you may already know, is always longer than it is wide, therefore, the greatest measure (104m) should be the length of the field.
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Pls, HELP!!
Law of Cosines
Solve for c. Round your final answer to the nearest tenth
The value of side c to the nearest tenth is 4.2.
What is the value of side c?The law of cosines signifies the relation between the lengths of sides of a triangle with respect to the cosine of its angle.
It is expressed as:
c² = a² + b² - ( 2ab × cosC )
From the diagram:
a = 7
b = 8
Angle C = 32 degrees
Plug these values into the above formula and solve for c.
c² = a² + b² - ( 2ab × cosC )
c = √( a² + b² - ( 2ab × cosC ) )
c = √( 7² + 8² - ( 2 × 7 × 8 × cos32 ) )
c = √( 49 + 64 - ( 112 × cos32 ) )
c = √( 113 - 94.98 )
c = √18.02
c = 4.2
Therefore, the value of c is 4.2.
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Find the surface area of each composite figure. Use 3.14 for π. Round to the nearest tenth. 12m. 6cm. 6cm. 4cm.
The Surface area of the composite figure is calculated as approximately: 234 sq. cm
How to Find the Surface Area of a Composite Figure?The surface area of the composite figure is the area surrounding the faces of the solid as a whole. Therefore, we have:
Surface area (SA) = Surface area of the square prism + surface area of the square pyramid - 2(area of base)
Area of base = area of square = 6 * 6 = 36 sq. cm.
Surface area of the square prism = 2a² + 4ah
a = 6 cm
h = 4 cm
Plug in the values:
Surface area of the square prism = 2(6²) + 4*6*4
= 72 + 96
= 168 sq. cm.
Surface area of the square pyramid = 2bs + b²
b = side length = 6 cm
s = slant height = √(8² + 3²) = 8.5 cm
Plug in the values:
Surface area of the square pyramid = 2 * 6 * 8.5 + 6² = 138 sq. cm.
Surface area of the composite figure = 168 + 138 - 2(36) = 234 sq. cm
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PLS I NEED HELP WITH THIS I WILL MARK YOU AS THE BRAINLIEST!!
Answer:
linearlinearquadraticexponentialStep-by-step explanation:
You want to classify the functions shown in the tables as linear, quadratic, or exponential.
DifferencesWe notice all of the tables have x-values that are evenly spaced. this means we can look at the differences between y-values to determine the kind of function the table represents.
The differences have the following interpretation:
differences are constant — lineardifferences have a constant difference — quadraticdifferences (and terms) have a constant ratio — exponential5, 3, 1, ...The differences of y-terms are constant at 3 -5 = -2.
The function is linear.
2, 5, 8, ...The differences of y-terms are constant at 5 -2 = 3.
The function is linear.
5, 1, 5, ...We observe that the y-values have a minimum. We don't need to take differences to know this is not linear or exponential. Of the offered choices, the only one that makes sense is "quadratic."
The differences of y-terms are ...
1 -5 = -4, 5 -1 = 4, 17 -5 = 12, 37 -17 = 20
The differences of differences are ...
4 -(-4) = 8, 12 -4 = 8, 20 -12 = 8
The second differences are constant.
The function is quadratic.
1, 3, 9, ...The first differences are ...
3 -1 = 2, 9 -3 = 6, 27 -9 = 18, 81 -27 = 54
The second differences are
6 -2 = 4, 18 -6 = 12, 54 -18 = 36
We note that the first and second differences are not constant, but the ratio of terms at every level is 3/1 = 6/2 = 12/4 = 3.
The function is exponential.
