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Functions and Graphs Concept Map (Critical Features of a function (Domain:…
Functions and Graphs Concept Map
Exponential Expressions
Zero Exponent Rule: If a base has an exponent that is zero then the equation would be equal to One .
Ex:
Ex:
Product Rule for Exponents: If the exponents are the same in the equation one is able to combine the exponents with addition.
Ex:
Ex:
Quotient Rule for Exponents: When exponents are over each other they subtract.
Ex:
Ex:
Power Rule for Exponents
Negative Exponent Rule
Negative exponents flip and become pos;(t)
Fractional Exponents
Vocabulary
Term: a term is either a single number or variable, or numbers and variables multiplied together. Ex. 4b X 3F
Expression: An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add,subtract,multiply, and divide). Ex. A-11
Equation:a statement that the values of two mathematical expressions are equal Ex. Y=Mx+b
Exponent
a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression (e.g., 3 in 23 = 2 × 2 × 2).
Function
a relationship or expression involving one or more variables
2 Problems using exponent rules
Zero rule affected the problem by changing it to a 1
Power Rule affected the equation by multiplying by 2
Negative exponent rule affected the problem by dropping the a below the equation
Slope
What is Slope? Slope is a measure of the steepness of a line, or a section of a line, connecting two points on a graph.It is rise over run which is the change in Y over the change in X.
The slope of the graph is 2/3 this means that the line will go up 2 and over 3.
Real World Problems
Brian planted a patch of trees a few years ago. He wants to know when the trees will be 20 feet high . The trees one year ago were 8 feet and this year they are 10 feet. When will the tree be 20 ft? The tree grows at rate of 2 feet per year. That means the slope is 2/1.
Jims price to rake your lawn is 50 dollars plus 6 dollars for every hour he works.
Jim raked his neighbor Tims lawn for for 4 hours. How much does Tim owe Jim. Jim’s rate is y=6(x)+50. The slope of Jim’s cost is 6$ multiplied by X. X represents the number of hours Jim works.
Tim owes Jim y=6(4)+50
2nd Problem: Involves Product, Quotient, and Fractional rule.
Fractional Rule
Product Rule
Quotient Rule
Linear Equations Different Knowns
Slope and an intercept
Given m =4/6 b=3
Y=mx+b --> Y=4/6+3
Slope and a point
Given through (3,4), Slope 2
Y=mx+b --> 4=2(3)+b --> Y=2x -2
Two points on a line
Given (-3,3) (3,-1)
Equation: Point slope form -1 - 3 / 3 - - 3 or y2 - y1 / x2 - x1
Equation: point slope equals - 2/3 for a slope, now substitution 3 = -2/3 * -3 + b --> 3 = 2 + b --> b = 1
Y=mx+b --> Y=-2/3x+1
Point and the equation of a parallel line
Given parallel line Y=3x+1 (2,-3)
Y=mx+b --> Y=3x-9
Point and the equation of a perpendicular
Given x+3y=6 through (1,5)
Y=mx+b --> Y= -1/3x + 2 m1*m2= -1 --> 5=3(1) =2 --> Y=3x+2
Graph of a line
Y=mx+b --> b=3 mx=2 --> Y=2x+3
Function Notation
Function Notation is the way a function is written. It is written usually as f of x or f(x)= it is almost the same as y except any value can be put in to x.
Function Notation Advantages
Names have different letters, such as f (x) and g (x).
2.
Identifies the independent variable in a problem. f (x) = x + 2b + c, where the variable is "x".
3.
it quickly states which element of the function is to be examined. Find f (2) when f (x) = 3x, is the same as saying, "Find y when x = 2, for y = 3x."
Evaluating Functions
Evaluating Values
Evaluating Expressions
Critical Features of a function
Domain
: The domain is the set of all possible x-values which will make the function "work", and will output real y-values.
Range
: The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain.
Zeroes
: a "zero" of a function is an input value that produces an output of zero (0).
Intervals of increasing values
: Intervals of increase are the domain of a function where its value is getting larger.
Intervals of decreasing values
: Intervals of decreasing are the domain of a function where its value is getting smaller.
Maximums
: Largest value of the function, either within a given range or on the entire domain of a function.
Minimums
: smallest value of the function, either within a given range or on the entire domain of a function.
Asymptotes
: A horizontal asymptote is a horizontal line that the graph of a function approaches, but never touches as x approaches negative or positive infinity.
Six parent function
Linear
Equation: Y=X
Critical Feature: Domain ( -∞, ∞ ) Range: ( -∞, ∞)
Example:
Quadratic
Equation y= x^2
Critical Feature: Domain ( -∞, ∞ ) Range [min, ∞) or (- ∞, max]
Example:
Radical
Equation: y=√x
Critical Feature: Domain ( -∞, # ] or [#, ∞)
Example:
Rational
Equation: y= 1/x
Critical Feature: Domain all others of x except those that gives you a zero a denominator
Example:
Examples of parent functions
Linear equation
4x + 3 = y is the function. The domain (-∞,∞) The range (-∞,∞) Intervals of increasing value because the slope is positive and going upward.
Quadratic equation
Y= x^2 +2 is the function
The domain is (-∞,∞) The range [2,∞) Intervals of increasing value because the slope is positive and going upward. Min (0,2)
Radical
Y= √3-x is the function
The domain (-∞,3] The range [0,∞) Intervals of decreasing value because the slope is going in a negative direction.
Rational function
Y=1/x-4
Y=1/x-4 is the function
The domain (-∞,-4) U (-4,4) U (4,∞) The range is (-∞,∞) The asymptotes are y=4 x=0
Piecewise Functions
is a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain (a sub-domain).
Finding the Domain of a piecewise function. Domain is the is the set of possible x-values of a function. To find the domain of a function you need to look at the function graphed. Determine how far left or negative the function goes, wherever the function stops is first value of the domain. If the domain goes on forever it is infinity ∞. Repeat this process for the right side or positive side of the graph.
