Unit 2 Test
Ordered pair
(x, y)
Horizontal Lines
A Horizontal Line has an equation of y = a number. It has a slope of zero. (HOY - horizontal, zero slope, y)
Vertical Lines
A Vertical Line has an equation of x = a number. It's slope is undefined (VUX - vertical, undefined, x)
Piecewise Function
A piecewise function is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain.
Vertical line test
A test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation
A relation
Any one pair or set of pairs of data points. X is the input, y is the output. The set of all possible outputs. Numerically or alphabetically. Use {}.
Domain and range of a regular graph of y = x²
Domain: (-∞, ∞) Range: [0, ∞)
Domain and range of a regular graph of y = |x|
Domain: (-∞, ∞) Range: [0, ∞)
How to plot y = x²
Find the vertex. Move one up, one to the side on both sides. Then move four up and two to the side from the vertex. Continue the parabola. Pay attention to the possible domain limitations
How to plot y = √x
Find the vertex. Move one up, one to the side. Then move two up, four to the side from the vertex. Continue this line. Pay attention to the possible domain limitations
How to plot y = |x|
Find the vertex. Then use the fact that it has a slope of 1 to continue the graph. Pay attention to the possible domain limitations
A function
Has only one output per input. Two different inputs can have the same output, but the same inputs cannot have different outputs. Can be expressed in different ways including arrow diagrams, sets of ordered pairs, graphically, or in "f of x" notation.
When to use ( ), [ ], ( ], or [ ) instead of {} when writing the domain or range
If it is from a number to infinity.
Linear Pattern
If the plotted points make a pattern, then the coordinates of each point may have the same relationship between the x and y values. In such a case, the x and y values are connected by a certain rule. A linear pattern is said to exist when the points examined form a straight line. Define your variables. Remembers that x is the dependent variable and y is the independent variable.
How many x-intercepts can a function have?
Infinitely many (explain why)
Parallel Lines
Lines that are parallel have the same slope and do not intersect.
Perpendicular Lines
Lines that are perpendicular have a negative reciprocal slope.
How many y-intercepts can a function have?
Only up to one (explain why)
Point-Slope Form of a Line
Point-Slope form is y-y₁ =m(x-x₁) where the point is given as (x₁ , y₁ ) and m is the slope.
How to find the domain of the graph y = √x without graphing
Set x + _ ≥ 0. If it is -x + _, remember to flip the sign. Whatever else this is added to or multiplied by outside the radical is not important to the domain. (?)
Slope Intercept Form of a Line
Slope Intercept form of a line is given as y=mx+b where m = the slope of the line and b = the y-intercept of the line.
A domain
The set of all possible inputs. Numerically or alphabetically. Use {}.
Slope
The slope of the line can be calculated by finding the change in y over the change in x. The slope represents the rate of change in the data (steepness, rise/run, change in y over change in x). Positive if the graph moves upward from left to right and negative if it moves downwards from left to right
The Standard Form of a Line
The standard form of a line is in the form Ax + By = C where A is a positive integer, and B, and C are integers. No fractions The standard form of a line is just another way of writing the equation of a line. We can graph the standard form of a line by finding the x-intercept and the y-intercept. To do this, just set either variable to zero to calculate the opposite one.
Why is a vertical line (or any line that is not a function) not a function?
There are multiple y points (outputs) per x point (input)
Plug values into an equation of a relation...
To check if they are functions. If one x value equals more than one y value. For instance, x = y² is not a function because if x is 9 y can be 3 or -3.
Break even point
Write both equations equal to each other and check when they are equal. Often used to check when something becomes is a better deal than something else.
Writing Equations from Word Problems
Y intercept typically is: a. a starting amount b. a one time fee c. something that doesn't change based on use Slope typically is: a. a rate of change b. a recurring cost c. something that changes based on use Ordered pairs look like: a. after three weeks, he had two fish (3,2). Just remember that x is the independent variable and y is the dependent variable.
Y is the...
dependent variable
All linear equations are functions...
except those whose graphs are vertical lines (?)
Transformations of Function
f (x) + k shifts the function k units upward. f (x) - k shifts the function k units downward. f (x + h) shifts the function h units to the left. f (x - h) shifts the function h units to the right. -f (x) reflects the function in the x-axis (that is, upside-down) The vertex is now at (h,k). If the whole function is made negative [say if it is -(x² + 2) not just -x² + 2] then the vertex is also made negative. In this situation, the first equations would have a parabola going downwards from (0,-2), but the second would only have it going down from (0,2).
X is the...
independent variable
Function notation
using f(x) = blank to represent y = blank. It means y is the function of x/a function that has an output of _ (like x + 4 for instance). How to solve: if in y = 4x + 3 y = f(1) then f(1) = 4(1) + 3 so f(1) = 7 (the value next to the f replaces all x's. Solve from there). The order pair answer is the x value, the answer of the equation. Here it would be (1,7) To graph, just replace f(x) with y.
Types of advanced functions
y = x², y = √x, y = |x| (they can be added or multiplied by sometimes as well, as seen in the previous card)
Distance formula
√(x₂-x₁)²+(y₂-y₁)²