Notes
Important Things to Remember on the SOL
Evaluating Expressions
*To evaluate an expression, replace all of the variables in the given problem with the
replacement values and use ______________(order of operations)
*Absolute value bars | | take the answer inside and make it _______________.
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*Treat | |, √ , and √ like parentheses when deciding what order to solve.
Laws of Exponents
*To multiply terms with the same base, _____exponents, __________ the numbers out
front
- Example: 8𝑥 2 ∙ 2𝑥 6 = 8 ∙ 2 ∙ 𝑥 2+6 = 16𝑥 8
*To raise a term to a power, ________________ the exponent to each thing on the
inside, making sure to ________________ exponents.
- Example: (4𝑥 3 )2 = 42 𝑥 3∙2 = 16𝑥 6
*To divide terms, _______________ exponents and _____________ the numbers
out front.
- Example:
25𝑥 8 𝑦 9
−5𝑥 4 𝑦
25
= −5 𝑥 8−4 𝑦 9−1 = −5𝑥 4 𝑦 8
*Whenever you see a negative exponent, SWITCH where the term goes in the fraction.
Once you switch, make the exponent _______________.
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- Example: 4𝑥 5 ∙ 2𝑥 −12 = 8𝑥 −7 = 𝑥 7
Polynomial Operations
*To add/subtract polynomials, find terms that have the same letters attached and
combine the coefficients (numbers out front) using adding/subtracting
- Example: (4𝑥 3 − 5𝑥 + 3) + (6𝑥 + 2𝑥 3 ) = (4𝑥 3 + 2𝑥 3 ) + (−5𝑥 + 6𝑥) + 3 = 6𝑥 3 + 𝑥 + 3
*To multiply two binomials, use _____________ (First Outside, Inside Last)
- Example: (2𝑥 − 3)(5𝑥 + 6) = 10𝑥 2 + 12𝑥 − 15𝑥 − 18 = 10𝑥 2 − 3𝑥 − 18
*To divide a trinomial by a binomial, _________________ the top, then cancel the
matching factors.
- Example: (𝑥 2 + 2𝑥 − 15) ÷ (𝑥 + 5) = (𝑥 + 5)(𝑥 − 3) ÷ (𝑥 + 5) = (𝑥 − 3)
*To multiply a polynomial by a monomial, _______________ to each term on the inside.
- Example: 3𝑥 2 (−4𝑥 + 𝑥 2 − 1) = 3𝑥 2 (−4𝑥) + 3𝑥 2 (𝑥 2 ) + 3𝑥 2 (−1) = −12𝑥 3 + 3𝑥 4 − 3𝑥 2
Factoring Polynomials
*If a problem asks you to factor, always check for a _________________________
(GCF) first!
*If there is a number in front of the 𝑥 2 term after taking out a GCF, make sure you
____________ factor (Multiply the first number by the last number and proceed as
normal)
*When factoring, you’re always looking for two numbers that ________________ to
get the last number, but ___________ to the middle number.
*Insert a ________ if a term is missing in the middle.
* The word ___________ means a polynomial cannot be factored.
Simplifying Radicals
*Make a factor tree for the number underneath the √
*√ = circle _____________
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* √ = circle _____________
*______ thing comes to the outside from each circle
Solving Literal Equations
*Use ________________ to get the desired letter by itself.
Solving Quadratic Equations
*Quadratic equations are of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
*The solutions, roots, or __________ are always put in { }.
*You can ALWAYS use the quadratic formula to solve, which is:
*The solutions, __________, or zeros of the graph of a quadratic equation are where
the graph crosses the _______-axis.
Solving Multistep Equations
*Multistep equations have all one variable with no 𝑥 2 term.
*Follow DCMAD to solve:
D________________________________________
C_________________________________________
M_________________________________________
A__________________________________________
D__________________________________________
*If the variables disappear and you get DIFFERENT numbers on each side of =,
write _______________________________.
*If the variables disappear and you get the SAME number on each side of =,
write _______________________________.
Solving Systems of Equations
*A system of equations is two equations of the form 𝐴𝑥 + 𝐵𝑦 = 𝐶. We are looking for the
ordered pair (𝑥, 𝑦) that we can plug in to BOTH equations and BOTH are true.
*We learned three methods of solution: substitution, elimination, and graphing.
1) Substitution – Get a variable by itself in one of the equations
Plug the expression into the OPPOSITE equation
Solve for the variable
Plug your answer back into the equation to solve for the other letter
2) Elimination – Stack the equations so the letters and numbers match
See if you can eliminate one of the letters by adding or ___________
If you can’t, multiply the equations by a number to get them to match
Eliminate one of the variables using ____________ or subtracting
3) Graphing – Graph both equations. Where the lines cross is the solution.
If the lines DON’T cross, it’s ______________________________.
If you get the SAME line, it’s ______________________________.
Solving Multistep Inequalities
*Inequalities use the following four symbols: _____, _____, _____, or _____.
*Solve a multistep inequality like you’re solving an equation
- Use ________________ (our rule for solving equations)
*Flip the way the inequality sign faces if you ____________ or ___________ both
sides by a negative number.
*A solution to an inequality is any value that you can plug in to the inequality and get
a true statement
Solving Systems of Inequalities
*To solve a system of inequalities, graph the inequality as if it was in 𝑦 = 𝑚𝑥 + 𝑏 form.
- Use a dotted line ---- for the symbols _____ or _____
- Use a solid line
for the symbols _____ or _____
- To decide which side of the graph to shade in, plug in a point to see if you get a
true statement.
*As long as it’s not on the line, ________ is the easiest to use.
- Complete the same steps for BOTH inequalities. Solutions are points where the
shading of both graphs overlap.
Finding Slope
*To find the slope given points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ), use the formula:
*The slope of a line using a graph can be remembered as ________ (change in 𝑦 over
change in 𝑥)
* Don’t forget HOYVUX! HOYVUX stands for
*If the equation is not in 𝑦 = 𝑚𝑥 + 𝑏 form and it asks you for slope, get 𝑦 by itself
FIRST before you determine the slope!
*A negative slope of a graph looks like it’s going _______ from left to right, while
A positive slope looks like it’s going ________ from left to right.
Writing the Equation of a Line in 𝑦 = 𝑚𝑥 + 𝑏
*The equation for slope-intercept form is _________________ where ______ is
slope and 𝑏 is the __________________________.
* To write the equation, determine your ____ (slope) first, then use the given point
(𝑥, 𝑦) to solve for ____ (y-intercept)
* Your final answer should only have numbers replacing the ____ and the ____.
Parallel and Perpendicular Lines
*Parallel lines have the ______________ slope.
*Perpendicular lines have ________________ __________________ slope (Flip
and Switch)
*Regardless of whether or not you have to change the slope, you always need to solve
to find a new _____ (y-intercept).
When is a Relation a Function?
*A relation is a function if all of the _____ values are unique.
*The graph of a function passes the ________________ line test.
*Function notation is written as 𝑓(𝑥), although they may use different letters than 𝑓.
Domain and Range
*The domain of a function is the ____ values (the inputs)
*The range of a function is the _____ values (the outputs)
*The domain of a graph contains all of the x (horizontal) values
- If the graph is a line rather than individual points, the domain is
smallest x-value ≤ x ≤ biggest x-value
*The range of a graph contains all of the y (vertical) values
- If the graph is a line rather than individual points, the range is
smallest y-value ≤ y ≤ biggest y-value
*To find the range of 𝑓(𝑥) given a domain, plug in each value and use ____________
to solve. The range is each one of your answers put into { }
Zeros of a Function
*The zeros of a function are values you can plug in and get _________ as an answer.
-DO NOT JUST PLUG IN ZERO! Zero is the answer you should get, not the input!
*The zero of a graph is where the graph crosses the _____-axis.
*Zeros are also called _____________ or _________________.
Intercepts
*On a graph, the 𝑥-intercept is where a graph crosses the _____-axis
- it is represented by the point (𝑥, 0)
* On a graph, the 𝑦-intercept is where a graph crosses the _____-axis
- it is represented by the point (0, 𝑦)
*If the problem gives you the equation in standard form _______________ and asks
for the intercepts, use _______________ (but don’t smack the paper!)
Direct Variation
*”When 𝑦 varies directly as 𝑥” means to use the formula ____________, where 𝑎 is
called the constant of _________________.
*Every graph of direct variation has two things in common:
1) It’s a _____________ line (not curvy)
2) It passes through __________ (also called the ___________)
Inverse Variation
*”When 𝑦 varies inversely as 𝑥” means to use the formula ____________, where 𝑘 is
called the constant of ___________________.
Line of Best Fit
*If a question asks you for the line of best fit, it wants you to find the equation
that BEST fits the points on the graph or in the table. IT WON’T BE EXACT!
*If the question asks for a line of best fit, the answer will be in 𝑦 = 𝑚𝑥 + 𝑏 form.
- We can actually find this on the calculator. I will give you a sheet for this.
*If the question asks for a curve of best fit, the answer will be in 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
form.
- We can actually find this on the calculator too. I will give you a sheet for this.
*You can use these equations to help you predict other values by plugging in the
given values into the equation and using PEMDAS to solve.
Box-and-Whisker Plots
*Range = ______________________ - __________________
*Interquartile range = _____________________ - _______________________
Standard Deviation, Mean Absolute Deviation, and Z-Scores
*Variance, standard deviation, and mean absolute deviation describe the data values’
relationship to the _________ (average)
*The bigger the standard deviation, the _____________ away from the mean the
data is.
*If the data has outliers, mean absolute deviation is a better measure than standard
deviation or variance.
* A z-score tells you how many __________________________ a data point is away
from the mean. (Positive means __________ the mean, Negative means _________ )