Pre-requisite
Simplify the expression using the properties of operations: 3(x + 4).
The simplified version of 3(x + 4) is 3x + 12. We can get to this by multiplying x + 4 by 3.
Mia is painting her guest room before the holidays for her relatives.
Mia paints interior walls at a rate of 12 m² / Hour.
At what rate does Mia paint in cm² / Minute?
(Show the instructor the paper diagram).
2000 cm squared/ 1 minute
Explain Function Notation.
Take the equation for finding slope for example,
y = 2x + 8
Function notation is when you replace the y with F(x), so it would look like
F(x) = 2x + 8
This is just saying that the function of x is equal to y. Another way to say it is that y is a function of x. x is the input value and you take that and put it through a process to get y. We can see this when we input an x value into the equation.
F(3) = 2(3) + 8
F(3) = 14
This is a clear example of there being one output for an input, this means when you input 3 into this function the output will always be 14
What is the formula for calculating the slope of a line and the average rate of change when given two points (x₁, y₁) and (x₂, y₂).
(y₂ - y₁) / (x₂ - x₁)
This can be used to calculate a line's slope or the average change rate between two points.
Explain what a Polynomial is and the special types of Polynomials.
A Polynomial is an expression that contains several terms, terms are separated by plus or minus signs. (Multiplication and division can be shown in different ways such as 5x or 4/2).
A Monomial is an expression that contains one term.
A Binomial is an expression that contains two terms.
A Trinomial is an expression that contains three terms.
After that expressions are usually referred to as Polynomials.
What does the expression 4 ⋅ 3² simplify to?
(We can assume that 4 is raised to the power of 1).
4¹ ⋅ 3²
Since the bases and the powers are different, they'll be solved separately.
(4¹)(3²)
4 ⋅ 9
36
20z + 5 = 5(z-2)
Solve.
First, we can use the Distributive property on the right side of the equation for the expression in parentheses. To do this we multiply the z and the -2 by 5. This now leaves us with:
20z + 5 = 5z - 10. Now we can solve for z.
First, to get the variables on one side we can use the subtraction equality property and subtract the 5z from each side.
15z + 5 = -10
Now we can use the same Subtraction Equality Property for the constants, we will now subtract the positive five from both sides. Leaving us with:
15z = -15
Now we can divide both sides by 15 to get z = -1
f(x) = 4x + 5
What is the value of f(-5)
f(x) = 4x + 5
f(-5) = 4(-5) +5
f(-5) = -20 + 5
f(-5) = -15
Given the table of values for a function f(x), identify the vertex coordinates and the y-intercept of the parabola.
(Show table on paper).
(Point out the vertex and y-intercept).
The vertex coordinates are (2,6)
The y-intercept of the parabola is (0, 2)
What type of Polynomial is the expression below.
23x² + (-5)
This is a Binomial.
It contains two terms that are separated by a plus sign.
Simplify and combine like terms in the expression: 2(3x + 4) - 5(x - 2).
2(3x + 4) - 5(x - 2)
Start by factoring out the expressions.
6x + 8 - 5x + 10
Now combine alike terms
(6x - 5x) + (8 + 10)
x + 18
x + 18 is the simplified expression of
2(3x + 4) - 5(x - 2).
Which equation has infinite solutions?
1. 32x + 52 = 78x - 42
2. 25x - 40 = 5 (5x - 8)
3. 71x + 34 = 71x - 34
4. 122x - 4 = 89x - 4
We want to find the equation that has infinite solutions, this means that any value of x would make the equation true.
The answer is No.2
This is because the leading coefficients are the same on both sides of the equation and the constant is the same on both sides as well.
As for the other equations we can also label them as having no solution, one solution, or infinite solution. To quickly label each one we can look at the coefficients and constants in the equation.
If the coeffecients are equal on both sides of the equation but the constants are different, this means there is no solution to the equation. This is means that for no value of x (ex. 1) will make the equation true. The equation with equal coeffecients but differing constants on this list is No.3
71 (1) + 34= 71 (1) - 34
71 + 34 = 71 - 34
105 = 37
This is incorrect as 105 does not equal 37, therefore No.3 has no solutions.
Next is to find the equations with one solution. This means only one x value will make the equation true. To find them we have to look for equations with differing coefficients on each side. That way no matter the constants there will always be one solution. The equations with differing coefficients are No.1 and No.4.
No.1
32x + 52 = 78x - 42
-32x -32x
52 = 46x - 42
+42 +42
94 = 46x
94/46 46x/46
94/46 = x
Here we get an answer of:
45/23 = x
Proving that No.1 has one solution, with this we can safely assume that equations that have differing coefficients on the same variable in this context have one solution. So we then can assume that No.4 also has only one solution.
122x - 4 = 89x - 4
-89x -89x
33x - 4 = - 4
+4 +4
33x = 0
33x/33 = 0/33
x = 0
What is the Average Rate of Change over the interval -1 ≤ x ≤ 3
Is it positive or negative?
Refer to graph on paper.
Refer to the graph on paper.
The ARC over the interval -1 ≤ x ≤ 3 is 1/2. The work is provided below:
How we find the slope with two coordinates with linear equations is by subtracting the y values & x values and then putting the y / x to find the slope.
We have the coordinates
(-1 , -2) & (3 , 0)
0 - (-2) = 2
_____ __
3 - (-1) = 4
2 / 4 can be simplified to 1 / 2
Write a function that describes the relationship between the number of hours studied (h) and the score achieved on a test (s) if for every hour studied, the test score increases by 5 points, starting from a base score of 50.
The answer can be written as,
s(h) = 5h + 50
You may recognize this is in the y = mx + b format, we can break each part down into this: the number of hours (h) is the input, 5 is the constant of proportionality, and 50 is the y-intercept.
Which of the following would not simplify to 24x⁵
1.) 3x² ⋅ 8x³ 3.) 2x³ ⋅ 12x²
2.) 4x ⋅ 6x⁴ 4.) 6x⁵ ⋅ 4x
Short answer: 4.)
Long answer: To solve this problem, multiply the coefficients together, (6)(4).
This gets you 24.
Overall, that gives you: 24x⁵ ⁺ ¹
This simplifies to 24x⁶
This is tricky as we are inclined to think that the x in 4x is raised to the power of 0 as it is not shown in the problem.
This is wrong if something is raised to the power of 0 it is equal to one. (Ex. 3⁰ is 3 times itself 0 times, (1).
Even though the exponent isn't shown we can assume that x is raised to the power of 1 because it equals itself. (x ⋅ 1)
If something is raised to the power of one, it is equal to itself.
If we go through we can see that all the other options are equal to 24x⁵.
1.) 3x² ⋅ 8x³
(3 ⋅ 8)(x² ⋅ x³)
24x⁵
2.) 4x ⋅ 6x⁴
(4 ⋅ 6)(x¹ ⋅ x⁴)
24x⁵
Here we can assume that the x in 4x is raised to the power of 1. (x ⋅ 1)
3.) 2x³ ⋅ 12x²
(2 ⋅ 12)(x³ ⋅ x²)
24x⁵
After going through and simplifying all of the options, we can see that No.4 is the only one that does not simplify to 24x⁵.
Simplify and solve the expression using the properties of integer exponents: ((2³ \ 2⁴) \ 2²).
First, start by simplifying the inside set of parentheses. Thats (2³ \ 2⁴).
2³ \ 2⁴ can be simplified to 2³ ⁻ ⁴.
2⁻¹ or 1 / 2¹
Now we can divide 2⁻¹ by 2².
2⁻¹ / 2² can be simplified to 2⁻¹ ⁻ ² or 2⁻³
Or to make it even simpler 1 / 2³
Which is equal to 0.125
Remember when dividing with exponents, subtract the exponents from one another. When multiplying add the exponents together.
(Ex. 1ⁿ / 1ᵐ can be simplified to 1ⁿ ⁻ ᵐ )
(Ex. (1ⁿ) (1ᵐ) can be simplified to 1ⁿ ⁺ ᵐ)
When dealing with negative exponents such as 2⁻¹ it's easier to think about them as 1 / 2¹.
When working with the same bases in math, leave the bases be but when the bases are different combine them.
(Ex. aⁿ / aᵐ = a ⁿ ⁻ ᵐ)
(Ex. bⁿ / aᵐ = (b / a)ⁿ ⁻ ᵐ)
Multiplication is like this
Same base: aⁿ × aᵐ = aⁿ ⁺ ᵐ
Diff. bases, same exponent: aⁿ × bᵐ = (a × b) ⁿ ⁺ ᵐ
Different everything: aⁿ × bᵐ = (aⁿ) × (bᵐ )
Another way to do this is to just simplify each component and then solve
((2³ \ 2⁴) \ 2²).
2³ = 8
2⁴ = 16
2² = 4
((8 / 16) / 4)
(0.5 /4)
0.125
Sharah wants to buy some pies from her local yard sale, her families are coming up for the holidays and she needs enough pies for all of them. But she wants to spend the least amount of money possible.
Sharah needs at least 24 pies; Sharah has already baked 9 pies at home. There are boxes of pies for $20 per box. Each box contains 5 pies.
Let x be the number of boxes Sharah buys.
Help Sharah out! You can do this by making an inequality and solving it to find out the least amount of boxes she needs to buy and how much money it will cost Sharah.
Let x be the number of boxes Sharah buys.
To make the inequality we can start off with the information that Sharah needs more than 24 pies. So, x's value should be greater or equal to 24.
x ≥ 24
We can now add that she already baked 9 pies
9 + x ≥ 24
We can add 5 as a coefficient to x to show that for one box you're getting 5 pies.
9 + 5x > 24. Now we can solve this as if we were solving an equation.
9 + 5x > 24
-9 -9
5x > 15
5x/5 15/5
x ≥ 3
This is simply showing that Sharah must buy at least 3 boxes of pie.
Ex. (5 pies in a box)(3 boxes) + (9 original pies)
15 + 9 = 24
Now to find out how much money she needs to spend we take the least number of boxes she needs to buy & multiply it buys 20.
(3 Boxes) (20 Dollars)
60 dollars is the least amount of money Sharah can spend if she wants to get at least 24 pies.
What is the domain of f(x)?
Refer to paper copy.
The domain is -5 ≤ x ≤ 7
This shows all of our possible inputs.
Calculate the average rate of change of the function g(t) = t² + 2t, over the interval of (t = 1) to (t = 3).
First, we need to evaluate g(1) and g(3).
g(1) = 1² + 2 (1)
1 + 2
g(1) = 3
g(3) = 3² + 2 (3)
9 + 6
g(3) = 15
So now we have our two coordinates points,
(1, 3) & (3,15)
(The input is the x value and the function of the x value is the y value).
15 - 3 12
-------- = ----
3 - 1 2
The average rate of change over the interval of g(1) and g(3) is 6.
Rewrite the exponential expression
g(x) = 2ˣ ⁺ ³ ⋅ 2²ˣ ⁻ ¹
in its simplest form, using properties of exponents.
And then evaluate g(17)
To do this we can follow the properties of exponents, where aⁿ ⋅ aᵐ = aⁿ ⁺ ᵐ
Since both of our bases and are the same but our powers are different we can use this rule.
g(x) = 2ˣ ⁺ ³ ⋅ 2²ˣ ⁻ ¹
Now we can rewrite it to be able to see it easier.
g(x) = 2⁽ˣ ⁺ ³⁾ ⁺ ⁽²ˣ ⁻ ¹⁾
g(x) = 2³ˣ ⁺ ²
Now that we have a simplified version, we can go ahead and plug in our input.
g(17) = 2³⁽¹⁷⁾ ⁺ ²
3(17) + 2 = 53
g(17) = 2⁵³
2⁵³ ~ 9,007,199,254,740,992
A rectangle has a perimeter of 102. If a rectangle's length is represented by x and its width is represented by y. The width is 2 times the length plus five.
Simplify the expression.
Let x be one side of the length of the rectangle.
Let y be one side of the width of the rectangle.
y = 2x + 5
We have
2(x) + 2(y) = 102
So now we can substitute it in.
(Remember that one side can be multiplied by two as you need to add all four sides together to get the perimeter.)
2(x) + 2(2x +5) = 102
So, we can simplify it to
2x + 4x + 10 = 102
6x + 10 = 102
Solve for x.
5x + 35 ≥ 50 & 16x +15 > 31
Note: Do not put x = blank,
it should be x < or > blank.
Ex. x < 990
Let's start off with the first given inequality,
5x + 35 ≥ 50 We can solve as an equation.
-35 -35
5x ≥ 15
5x/5 15/5
x ≥ 3
(Draw the number line in real life with an open circle and the line going right.)
Now for the next inequality,
16x + 15 > 31
-15 -15
16x > 16
16x/16 16/16
x > 1
(Now draw on paper, a closed circle with the line going right)
Now to find the solution to both equations we can look at our number line and find if there is a value of x that makes both inequalities true.
The answer is x ≥ 3. We can try this out.
5(3) + 35 ≥ 50
15 + 35 ≥ 50
50 ≥ 50
It satisfies the first inequality now let's try to the next.
16(3) + 15 > 31
48 + 15 > 31
63 > 31
It makes both inequalities true therefore the answer is x ≥ 3.
Describe what a function is.
A function is when you take an input, such as the variable x and you get one output, such as the variable y.
(Show mapping idea paper.)
Two inputs can have the same output, a common example of this is a parabola which can have two 'roots'. These are the graph's two x-intercepts (When y is equal to zero).
(A parabola can have only one or no roots)
Given two points on a linear function:
(19, 56) and (17, 32), construct the linear function.
We can do this by finding the slope (12) and y-intercept (-172). We can now arrange these components in the y = mx +b form.
g(x) = 12x -172
Work for finding the slope and y-intercept below.
32 - 56 = -24
17 - 19 = -2
-24 / -2 = 12
For finding the intercept we can use the slope and one coordinate point and insert it into the
y = mx + b format.
32 = 12(17) + b
32 = 204 + b
-204 on both sides
-172 = b
(x² + 5x -54) + (23x² -80x + 102)
Arrange them vertically to think about it easier and then you can add/subtract alike terms.
(x² + 5x - 54)
+ (23x² - 80x + 102)
-------------------------
24x² - 75x + 48
Simple and easy!