Radicals & Irrational numbers
Is the square root of five a rational or irrational number?
The square root of five is an irrational number, it is approximately 2.23606797749979. It is not a perfect square.
It cannot be expressed as a simple fraction, and it goes on forever without repeating itself.
What is the key feature distinguishing an exponential function from a linear function?
An exponential function either rapidly increases or decreases due to it being raised to a constant power. You can think of it as a rocket going up or a sled going down a steep hill. (Ex. f(x) = 2ˣ)
Whereas a linear function has a constant rate of change therefore remains in a straight line.
(Ex. f(x) = 2x)
Explain the difference between an Arithmetic sequence and a Geometric sequence.
An Arithmetic sequence is when you add or subtract the same thing from the previous number. An example of this is 3, 7, 11, 15, 19...
Each time you add the integer 3 to each number continuously.
A Geometric sequence is when you multiply or divide the previous number by the same value.
For example, 3, 6, 18, 24, 48...
Each previous number is multiplied by 3 to get the next number.
Rewrite the expression 3(x + 4) by distributing. What is the result?
3x + 12
Solve the equation x² = 144 by taking square roots.
√x² = √144
x = ± 12
Is the sum of the √2 and 5, rational or irrational?
The sum of an irrational number and of rational number is irrational.
Here the square root of 2 is irrational. The number 5 is rational. Therefore, the sum of the √ 2 and 5 is irrational.
Which real-world situation can be modeled by a exponential growth function?
The cost of rent each month after the initial deposit which can be expressed as c(m) = 230 = 350x
Or.
The amount of money you have in a bank account after the initial deposit, with annually interest.
c(y) = 230 ⋅ 2ʸ
c(y) = 230 ⋅ 2ʸ represents an exponential growth function.
A car rental company charges a flat fee of $50 plus $20 for each day the car is rented. Write a function to represent the total cost c(d) as a function of the number of days d.
c(d)=50 + 20d
If you think about the problem, it shows you that sequences are just functions, you input the number in the sequence you want to find and find the numeral value. We can see that in this function above, 50 is the first number in the sequence and the common difference is 20. Though it is different from the usual aₙ = a₁ + d (n - 1). We can manipulate the function above to look like the average arithmetic sequence formula.
c(d)= 50 + 20d
We can think by how can we get our function to the average arithmetic sequence formula format, this would be by adding 20, why? Because if we expand average arithmetic sequence formula it is equivalent to aₙ = a₁ + dn - d. So it makes sense to add the missing piece of it.
c(d) = 50 + 20d + 20
c(d) = 70 + 20(n - 1)
This shows how a sequence is really just a function.
Given the expression 4x² + 12x, factor out the greatest common factor and provide the equivalent expression.
We can start by dividing both by the common factor of 4x.
4x (x + 3), this is the equivalent expression of
4x² + 12x.
Factor the quadratic:
(x - 2) (x - 3).
Short answer: x² - 5x + 6
Long answer:
x (x - 2) - 3(x - 2)
Now distribute.
x² - 2x - 3x + 6
x² - 5x + 6
√50 / √ 5
Is the product rational or irrational?
We can start by rearranging the problem to be
√ 50 / 5
√10
The square root of ten isn't a perfect square, it is infinite and never repeats.
Therefore, the product of √ 50 / √ 5 is irrational.
Work below shown.
√10 ≈ 3.16
√50 ≈ 7.07
√5 ≈ 2.24
7.07 / 2.24 ≈ 3.15625
This then can be rounded up to 3.16.
f(x) = 4x - 6
g(x) = 9x - 4
Find the solution set.
We can see that y = f(x) and y = g(x) therefore we can set this up like a simple system of equations problem.
4x - 6 = 9x - 4
+ 6 + 6
4x = 9x + 2
-9x -9x
-5x = 2
/-5 /-5
x = -5/2
So the solution set would be {-5/2}
A biologist observes that a bacterial culture doubles in size every hour, starting with 100 bacteria. Define a function that expresses the number of bacteria N(t) after t hours and identify whether this relation represents a sequence.
Short answer: N(t) = 50 ⋅ 2ᵗ
With the information provided, we can make an Explicit formula as our function. We can determine that this is an geometric sequence based on that we are multiplying the previous number by the common difference to get the next number.
We can view N(t) as aₙ.
Starting we can see our first number in the sequence is 100.
N(t) = 100 ⋅ dᵗ ⁻ ¹
We know that the bacteria double each hour, that's our common difference.
N(t) = 100 ⋅ 2ᵗ ⁻ ¹
N(t) = 100 ⋅ 2ᵗ ⋅ 2⁻ ¹
Which is equivalent to,
N(t) = 50 ⋅ 2ᵗ
Just in case it helps your brain with it better like it does mine.
Now to test out our formula, if we kept doubling 100 it would go like this.
100, 200, 400, 800, 1600, 3200, 6400, 12,800, 25,600... And so on
So let's try it with the formula.
N(1) = 50 ⋅ 2¹
N(1) = 100
N(9) = 50 ⋅ 2⁹
N(9) = 25,600
We can see this function is like a geometric explicit sequence as there is an input (t) like you would input the number in the sequence you want to find.
Rewrite the expression 3(x - 2) + 5(2 - x).
3(x - 2) + 5(2 - x)
Expand.
3x - 6 + 10 - 5x
Now combine like terms
4 - 2x
Given the function; f(x) = 2x² + 8x + 6, rewrite it in the form easiest to see the vertex coordinates.
The easiest version of the function to view the vertex coordinates is vertex form (f(x) = a (x - h) ² + k).
f(x) = 2x² + 8x + 6
Divide by two.
f(x) = 2(x² + 4x) + 6
We can realize that,
x² + 4x + 4
can be factored in as (x + 2)²
But that isn't what we want so, we can subtract 4 from the equation (Don't forget to subtract 8 from 6 as the multiplier is 2!)
f(x) = 2(x² + 4x + 4 - 4) + 6 - 8
f(x) = 2((x+2) ² - 4) + 6 - 8
f(x) = 2(x+2) ² -2
So, the coordinate of the vertex is (-2, -2)
Without calculating directly, determine whether
√ 225 + √ 196
is a rational or irrational number and explain your reasoning.
We know that √ 225 is a perfect square and is therefore rational. (It's equal to 15²).
√196 is another perfect square, therefore rational. (it is equal to 14².)
So now we know that this is a rational plus a rational, therefore it is equal to a rational number.
Find the output of f(4).
(Show paper copy on No.2)
f(4) = -5
Write a recursive arithmetic sequence defined by the first term of 5 and a common difference of 3. Write the recursive formula for the n-th term of the sequence.
a₁ = 5
aₙ = aₙ ₋ ₁ + 3
Ex. If I want to find a₃, I have to start by finding a₂.
a₂ = a₂ ₋ ₁ + 3
a₂ = a₁ + 3
a₂ = 5 + 3
a₂ = 8
Now we can start finding
a₃ = aₙ ₋ ₁ + 3
a₃ = a₃ ₋ ₁ + 3
a₃ = a₂ + 3
a₃ = 8 + 3
a₃ = 11
And so on, we continue with this to generate an arithmetic sequence.
Factor the expression x² + 5x + 6 to show its structure.
We can factor this by understanding that it will follow the (x + a)(x + b) pattern, to properly factor this we must understand that a + b must equal 5 and (a)(b) must equal 6. So we can list what adds up to 5.
1 5
2 3
The only two numbers that fit both constraints is 2 & 3.
So we can now assume that we can factor our original expression out to (x + 2)(x + 3).
We can test this out now.
x(x + 2) + 3 (x + 2)
x² + 2x + 3x + 6
x² + 5x + 6.
Done!
Solve for x, using the quadratic equation.
2x² + 3x - 5 = 0
Short answer:
x = (-3 ± 7) / 4
Long answer:
First identify the a, b, and c.
a = 2
b = 3
c = -5
Now we can plug these numbers into the quadratic formula.
(Show it on paper).
Simplify this expression. x = √50 + 3 / 4 - √32
Is x rational or irrational
Justify your answer.
First, let's simplify both of the square roots.
Starting with √50.
√50
5 10
2 5
So now we have simplified it to √(5² + 2)
5√2
Now moving on to √32
√32
4 8
2 4
√(4² + 2)
4√2
So, we have simplified our two squares.
5√2 + 3/4 - 4√2 = x
Now combine alike terms
(5√2 - 4√2) + 3/4 = x
√2 + 3/4 = x
We can now say that x is irrational just based on looking at the expression.
We can justify this as an irrational number (√2) plus a rational number (3/4) is equal to an irrational number.
(Show piecewise function on paper).
Without graphing, is this a function? If not, how could you make it a function?
Looking at the piecewise function, we can see that the input 0 has two outputs. Making this not a function, we can make this function by changing the input (0) only to have one output.
Give the closed-form (Or Explicit) formula for Arithmetic and Geometric sequences and explain what each part means.
Meaning don't give the recursive formula. (Where you give the first number in the sequence and give the formula where you can then find a₂ and so on.)
Ex.
a₁ = 5
aₙ = aₙ ₋ ₁ + 3
Keep in mind that sequences are like functions, the formatting is typically aₙ but, it can also be presented as a (n) where n is the number position in the given sequence.
(Ex. a₃ refers to the third number in a given sequence.)
The short answer is:
aₙ = a₁ + d (n - 1) (Arithmetic)
aₙ = (a₁) (d)ⁿ ⁻ ¹ (Geometric)
A longer explanation is given below.
a₁ is the value of the number first in the sequence.
d is the same value being subtracted or added.
How this formula works is you input the common difference (d), the number position in the sequence that you want to find (n & aₙ), and the first number of the sequence (a₁).
Now this is where the formulas differ,
aₙ = a₁ + d (n - 1) The Arithmetic sequence factored out is equivalent to aₙ = a₁ + (dn) - d. This means you multiply the common difference by the position number in the sequence. Once that's finished, we can now add the numerical value of the first number in the sequence. After we can now subtract the common difference.
We can think of this like this: (d)(n) is a common difference times the number of the position in the sequence that we want to find. So now we know how much common difference there is between a₀ and aₙ. We can now add a₁ 's value, but now we have the common difference between a₀ and a₁, we can get rid of that by subtracting d from the equation. Yay! Now we understand Arithmetic Closed-Form Formulas!
Now onto Geometric Closed-Form Formulas (What a mouthful)
We are presented with the closed form for Geometric sequences below.
aₙ = (a₁) (d)ⁿ ⁻ ¹
By now you know what the variables mean, n - 1 has stayed the same but instead, we are raising d to the power of n - 1. We can now view the formula like this: aₙ = (a₁) (d)ⁿ - (d)¹. We are saying that we are raising the common difference to the power of the number position in the sequence we are looking to find. If you are having trouble imagining this here is an example below.
(d)4 = (d ⋅ d ⋅ d ⋅ d)
This counts for all the common differences between a₀ and a₁, we add a₁ and subtract d¹ (Which is d) and then you're done!
Identify the structure of the expression a² - b².
Short answer: (a - b) (a + b)
We can see this by expanding.
a (a - b) + b (a - b)
a² - ab + ba - b²
We can combine similar terms to see
a² - b².
Solve for x by completing the square.
x² + 6x - 10 = 0
Short answer: x = -3 ± √ 19
Long answer:
x² + 6x - 10 = 0
x² + 6x = 10
x² + (3)2x = 10
x² + 6x + (3)² = 10
x² + 6x + 9 = 19
(*Remember to also add 9 to the other side to maintain equality*)
x² + 6x + 9 = 19
(x + 3)² = 19
√(x + 3)² = √19
x + 3 = √19
x = -3 ± √19