Characteristics of Graphs + Functions Flashcards
Function
a relation is a function if for every y-value (output) there is precisely one x-value (input)
- No x-value can reppeat in a table, list of ordered pairs or graph
- The graph must pass the vertical line test (VLT)
- In an equation, no y-value can be to an even power or in an absolute value symbol
Common notation and what it means in English:
f(x) > 0: ?
f(x) >/_ 0: ?
f(x) < 0: ?
f(x) </_ 0: ?
f(x) = g(x): ?
f(x) > 0: interval(s) where the graph is ABOVE the x-axis
f(x) >/_ 0: interval(s) where the graph is ABOVE or ON the x-axis
f(x) < 0: interval(s) where the graph is BELOW the x-axis
f(x) </_ 0: interval(s) where the graph is BELOW or ON the x-axis
f(x) = g(x): point(s) where the curves intersect
Domain
all possible x-coordinates; use interval notation; watch for discontinuities – holes, jumps, vertical asymptotes
Range
all possible y-coordinates; use interval notation; watch for horizontal asymptotes, holes and jumps
Increasing Interval
an interval over which the y-coordinates increase; use interval notationi, but ONLY use parenthesis (NEVER square brackets); record the x-coordinates
Decreasing Interval
an interval over which the y-coordinates decrease; use interval notation, but ONLY use parenthesis (NEVER square brackets); record the x-coordinates
Maximum
an increase DIRECTLY into a decrease; written as the y-coordinate of the ordered pair
- Relative/Local Maximum: all locations on the curve where there is an increase directly into a decrease; doesn’t have to be one or there can be more than one
- Absolute Maximum: the greatest maximum on a curve as long as no point of the range is higher
Minimum
a decrease DIRECTLY into an increase; written as the y-coordinate of the ordered pair
- Relative/Local Minimum: all locations on the curve where there is an decrease directly into an increase; doesn’t have to be one or there can be more than one
- Absolute Maximum: the lowest minimum on a curve as long as no point of the range is lower
End Behavior
Explains what the y-values are doing at the extreme right hand and left hand side of the graph
- Notation: as x –> -(infinity), y–> ___ and as x –> (infinity), y –> ___
Transformations of Graphs
The parent graph of any function can be moved around the Cartesian plane by manipulating its base equation. (Note: “main math idea” refers to the type of function that is presented (i.e. square root, linear, quadratic, etc…)
Vertical Shift
occurs when a value is added or subtracted outside the main math idea
f(x) + k, shifts every point on the “mother graph” up k units
f(x) - k, shifts every point on the “mother graph” down k units
Examples:
A. f(x) = 3^x - 2
B. g(x) = x^2 + 4
Horizontal Shift
occurs when a value is added or subtracted inside the main math idea
f(x - h), shifts every point on the “mother graph” to the right h units
f(x + h), shifts every point on the “mother graph” to the left h units
x-axis Reflection
occurs when a negative is multiplied outside the main math idea
-f(x), changes the sign on all the y-coordinates of the “mother graph”
Examples:
A. f(x) = -(square root of) x
B. g(x) = -x
y-axis Reflection
occurs when a negative is multiplied to the x inside the main math idea
f(-x), changes the sign on all the x-coordinates of the “mother graph”
Examples:
A. f(x) = (square root of) -x
B. h(x) = (1 - x)^2
Vertical Stretches and Compressions/Dilations
occurs when a number is multiplied outside the main math idea
af(x)
- If a > 1, then a vertical stretch by “a”
- If 0 < a < 1, then a vertical compression by “a”
Examples:
A. f(x) = 2x^2
B. h(x) = 2/3 (square root of) x
Horizontal Stretches and Compressions/Dilations
occurs when a number is multiplied to x inside the main math idea
f(ax)
- If a > 1, then a horizontal compression by “a” or a horizontal dilation by “a”
- If 0 < a < 1, then a horizontal stretch by “a” or a horizontal dilation by “a”
Examples:
A. f(x) = (square root of) 3x
B. h(x) = (absolute value of) 1/4x
Linear Functions
- All lines are functions except the vertical line. (It does not pass the vertical line test/x-coordinates repeat)
( - All lines have a constant rate of change called slope. Slope is the quotient of the change in y over the change in x
m = (y2-y1)/(x2-x1) ) - point-slope form: y - y1 = m(x - x1), m represents the slope and (x1,y1) represent a point on the line
( - Notive this format is the slope formula with (x2 - x1) multiplied to the opposite side. - This format is useful when given a point and a slope or two points )
- slope-intercept form: y = mx + b, m represents the slope, b represents the y-intercept or initial value
( - This format is useful when given the slope and y-intercept and for graphing purposes ) - standard form: Ax + By = C, A (e looking thing) Z^+ (A cannot be negative, A must be a whole number)
( - This form is useful when finding and/or graphing by x- and y- intercepts )
x-intercept: a point on the x-axis; written as (# , 0); found in an equation by replacing the y-value with zero and solving for x
y-intercept: a point on the y-axis; written as (0 , #); found in an equation by replacing the x-value(s) with zero and solving for y
Recognizing Linear Functions from an Equation and a Table
- An equation is linear if:
( - All variables have an exponent of precisely one - No variable in the denominator
- No variables in radicals )
- A table represents a line if the differences of consecutive x-
and - and y-values are the same
Square Root Function
f(x) = a(square root of) x - h (then) + k where (h , k) represents the terminal point
- Remember that a square root can be written as an exponent of one-half
- Has a restricted domain to prevent negatives under the radical
- Doesn’t always have an x- and y- intercept
- To determine if a table of values represents a square root function, sketch and look at the shape
Parent Function: f(x) = (square root of) x
Pattern on Graph: Up 1, Over 1, Up 2 over 4, Up 4 over 16, etc…
Evaluating Functions
- To evaluate a function means to subsitute the given value or expression into the location of all variables and simplify. “plug and chug”
Function Notation: f(x)
Read: f of x
Interpreted: a function named f written using x as the variable - The value in the parenthesis is what is to be subsiituted into the function’s variables
- The input (part in the parentheses) and the output (answer)
