PSAT Math Review
Combination Definition
Adding the two equations together to eliminate a variable -5x + 7y = 39 5x + 9y = 1 16y = 40
Quadratic Equation
An equation that contains a squared variable as the highest-order term
Polynomial
An expression comprised of variables, exponents, and coefficients, the only operations involved being addition, subtraction, multiplication, division, and integer exponents
(Exponents) When raising the entirety of an equation to a power, like (a x b)²...
Apply the power to all factors in a product
Multiplicity and Graphs
When a polynomial has a zero with an odd multiplicity, its graph will cross the x-axis When a polynomial has a zero with an even multiplicity, its graph will just touch the x-axis
(Radicals) When a fraction is under a radical...
You can rewrite it using two radicals: one containing the numerator and the other containing the denominator
Percent Formula
Percent = part/whole * 100%
(Functions) If you see f(g(x))...
Use result of g(x) as the x in f(x)
FOIL
(a + b)(c + d) = ac+ ad + bc + bd
Completing the Square
(a needs to equal 1 to use this process) -Move the constant to the other side of the equals sign -Divide b by 2 and square the quotient -Add the result to both sides -Take the square root from both sides -Solve both the positive and negative results from the equations
Factoring
-If there is a GCF for all of the factors, take it and divide the equation by it. For example, in 5x² + 10x + 15, the GCF is 5. Divide to make 5(x² + 2x + 3) -Find a * c -Find the factors of a and c that add to b -Re-write as binomials
Rational Expression
A fraction of polynomials For an expression to be rational, both the numerator and the denominator must both be polynomials Are often undefined for certain factors if a solution makes the denominator equal 0
(Exponents) When multiplying two terms with the same base...
Add the exponents
(Exponents) A base raised to a negative exponent...
Can be rewritten as a fraction, with 1 as the numerator and the base with a positive exponent as the denominator
(Radicals) Two multiplied factors under a single radical...
Can be rewritten as separate radicals multiplied together
Ratios Definition
Comparison of one quantity to another Can compare a part to a part or a part to a whole
When the systems of equations gives an answer like 6=6 or 4=4, then the equations are...
Dependent, meaning the system has an infinite amount of solutions
Rates Formula
Distance/work/result = rate * time` Be sure to remember what units you're working in when doing this!
Multiplying Polynomials
Distribute each term in the first set of parentheses to each term in the second set
What's one thing that you need to be careful of when doing probabilities?
Don't just add two percentages together, provided you're not doing P(AuB) problems.
Ways to Solve Quadratic Equations
Factoring Completing the Square Quadratic Equation
Percent Change Formula
Percent increase or decrease = (amount of increase or decrease/original amount) * 100%
Dividing Polynomials
Use polynomial long division, which is just regular division but with polynomials
Reflections
Flips the graph around an axis or line
Decreasing Functions
Functions that have y-values that decrease as the corresponding x-values increase
Increasing Functions
Functions that have y-values that increase as the corresponding x-values increase
Constant Functions
Functions that have y-values that stay the same as the x-values increase
Translations
Graphs moves up, down, left, or right
Radicals and Fractions
It's improper to leave a radical in the denominator of a fraction, so multiply the numerator and the denominator by the offending radical
Anatomy of Exponents
Made up of a base (the larger number) and an exponent/power
Piecewise Functions
Multiple pieces of equations duct-taped together If the x-input falls into one of the rules listed beside the function, it will equal a certain numerical value or equation
(Exponents) When raising a power to another power...
Multiply the exponents
The conversions from feet to yards and square feet to square feet to square yards are...
Not the same! Square feet * 1/3 * 1/3 = Square Yards
(Exponents) Any term raised to the 0 power equals...
One
Proper and Improper Rational Expressions
Proper rational expression has a lower-degree numerator than denominator Improper rational expression has a higher-degree numerator than denominator
Functions
Rules that transform inputs into outputs Differ from equations in the sense that each input only has one output Inputs, or domain, is represented by x Output, or range, is the result of an equation and is represented by f(x)
Systems of Equations Definition
Set of multiple equations with multiple variables that are interdependent Useful for modeling and simulaion
Extraneous Solutions
Solutions that don't satisfy the original equation and causes 0 in the denominator of any parts of the equation
Substitution Definition
Solve the simpler of the two equations for one variable and then substitute the result into the other variable r = 2x + 6 3r + 2x = 18 6x + 18 + 2x = 18
Kaplan Method for Math
Step 1: Read the question, identifying and organizing important information as you go Step 2: Chosen the best strategy to answer the question Step 3: Check that you answered the right question
Expansions/Compressions
Stretching or squashing a graph horizontally or vertically
Ways to solve Systems of Equations
Substitution Combination
(Exponents) When dividing two terms with the same base...
Subtract the exponents
Multiplicity
The number of times a factor in an equation, related to zeros (x-6)(x+6) makes the solution for (x-6), 6, a simple zero (x-6)(x-6) makes the solution for (x-6), 6, a double zero
Y-Intercept Definition
The value where the model begins
Zeros/Roots
The x-intercepts of a polynomial Can be found by setting each factor of the polynomials equal to 0
Combining Ratios
To combine a:b and b:c, make b a common value to create a:c 2:3 and 5:6, 10:15 and 15:18, 10:18
When your system has one solution, it will graph as...
Two lines that intersect at a single point
When your system has infinite solutions, it will graph as...
Two lines that overlap, making one
When your system has no solution, it will graph as...
Two parallel lines
Proportion
Two ratios set equal to each other Check the units of each ratios and double-check your work by cross-multiplying If a/b = c/d, then ad=bc, therefore b/a = d/a but not a/d=b/c
Rates and Polynomial. Radical, and Rational Equations
Typical rational equation that models a real-world scenario Uses rate equation (distance or work = rate x time)
Discriminant
b² - 4ac Its values determines the number of solutions Positive result = Two real solutions Result of 0 = One real solution Negative result = No real solutions
Polynomials are named based on their...
degree, or the highest power in the variable (for one-variable terms)/the highest sum of exponents on one term (for multi-variable terms)
(Functions) If you see (f * g)(x), convert it to...
f(x) * g(x)
(Functions) If you see (f + g)(x), convert it to...
f(x) + g(x)
(Functions) If you see (f - g)(x), convert it to...
f(x) - g(x)
Growth and Decay Equation
f(x) = f(0) * (1 + r)² r is the growth/decay rate
f(x) - g
f(x) moves down g units
f(x + g)
f(x) moves left g units
f(x - g)
f(x) moves right g units
f(x) + g
f(x) moves up g units
-f(x)
f(x) reflected over x-axis (up-facing functions become down-facing)
f(-x)
f(x) reflected over y-axis (left-facing functions become right-facing)
f(gx) (g > 1)
f(x) undergoes horizontal compression
f(gx) (0 < g < 1)
f(x) undergoes horizontal expansion
gf(x) (0 < g < 1)
f(x) undergoes vertical compression
gf(x) (g > 1)
f(x) undergoes vertical expansion
Function Notation
f(x)/any other letter(x) is the output, similar to y in slope equations K is the rate of change, like m in a slope equation f(0) is the y-intercept, like b in a slope equation f(x) = kx + f(0) y = mx + b
(Functions) If you see (f/g)(x), convert it to...
f(x)/g(x)
(Radicals) A radical can be written using a...
fractional exponent
Slopes of Lines Parallel to X-Axis
m = 0
Slopes of Lines Parallel to Y-Axis
m = Undefined, as the lines exist but can't be numerically calculated
(Radicals) The square root of a number can only be...
positive!
Slope Equation
slope = rise/run
What are linear equations are well-suited for...
solving for a single variable in terms of another that is clearly defined
Vertex Form
y = a(x - h)² + k Vertex is (h, k) k is the maximum/minimum of the graphed equation h = x is the axis of symmetry