Rates, Ratios, and Percents Flashcards
algebraic expression.
A combination of variables, constants, and arithmetical operations
term
a constant, a variable, or a product of simpler terms that are each a constant or a variable
constant term
A term with no variables
coefficient
A constant in a term with one or more variables
polynomial
an algebraic expression that’s a sum of terms and has exactly one variable. Each term in a polynomial is a variable raised to some power and multiplied by some coefficient.
first degree (or linear) polynomial
the highest power a variable is raised to is 1
second degree (or quadratic) polynomial
the highest power a variable is raised to is 2
linear equation
has a linear polynomial on one side of the equals sign and either a linear polynomial or a constant on the other side—or can be converted to that form
Different possibilities when evaluating two linear equations
- If they are equivalent, infinite solutions
- If they are not equivalent, one solution
- If they have a contradiction, there is no solution
Two strategies for solving linear equations with two unknowns
- Substitute the value of one equation into the other
- Make the coefficients of one unknown the same in both equations (ignoring the sign), then either add the equations or subtract one from the other
factoring
first add or subtract to bring all the expressions to one side of the equation, with 0 on the other side. Then try to express the nonzero side as a product of factors that are algebraic expressions
quadratic equation
ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0
How many roots can a quadratic equation have?
At most two, but can have just one or no root
Examples:
The equation x2 − 6x + 9 = 0 can be written as (x − 3)^2 = 0 or (x − 3)(x − 3) = 0. So, its only root is 3.
The equation x^2 + 4 = 0 has no real root. Since any real number squared is greater than or equal to zero, x^2 + 4 must be greater than zero if x is a real number.
How to factor a^2 - b^2
(a − b)(a + b)
Quadratic formula
How to solve a linear inequality with one unknown
The same as a regular linear equation, except if you divide or multiply by a negative number you have to flip the sign
Function Domain
The set of all allowed inputs for a function. This can be restricted by the function’s definition
Function range
The set of a function’s outputs
Linear equation to define a line
y = mx + b
Slope
the coefficient m in the formula y = mx + b
y-intercept
The constant b in the formula y = mx + b. Also the y-coordinate of the point where the line intersects the y-axis
Slope is 0
Line is horizontal, and has the equation y = b
Find x-intercept
Set the variable “y” to zero and solve for x
Given two points (x_1, y_1) and (x_2, y_2) with x_1 <> x_2, how to find slope?
ratio of the difference in their y-coordinates to the difference in their x-coordinates
m = (y_2 - y_1) / (x_2 - x_1)
How to find the equation for a line given a single known point (x_1, y_1) and slope m
m = (y - y_1) / (x - x_1), or (y - y_1) = m(x - x_1)
Three cases for two linear equations
- Unique solution - graphs are two lines intersecting at the point that is the solution
- Equations are equivalent - they both stand for the same line and have infinitely many solutions
- No solution - parallel lines that don’t intersect
Graphing a function f(x)
You can equate f(x) with y, as in y = f(x)
x-intercepts are solutions of the equation f(x) = 0
y-intercept is value f(0)
