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Functions and Graphs Concept Map (Exponential Expressions (Power Rule for…
Functions and Graphs Concept Map
Slope
What is Slope? Slope is the measure of the steepness of a line, or possibly the section of a line, connecting two points.
Numeric Form- To find slope without a graph, you are going to use this formula---->
Graphical Form- To find the slope on a graph, you are going to use the equation of rise over run. To calculate the slope of a line you need only two points from that line, (x1, y1) and (x2, y2). The equation used to calculate the slope from two points is: On a graph, this can be represented as: ... Step One: Identify two points on the line. Step Two: Select one to be (x1, y1) and the other to be (x2, y2).
Real Life Problems
How much does Jenna make in one hour if she sells 5 items during that hour? Since x is the number of items Jenna sells during one hour, we've got x = 5.
Then y = 3(5) + 10 = 25, which means Jenna would be paid $25. This amount doesn't include tips. Yeah, she makes tips, too. What can we say, this girl knows how to turn a buck.
A plumber charges a fee of $50 to make a house call. He also charges $25.00 an hour for labor. Let x represent the number of hours for labor and y represent the total cost.
How much would it cost for a house call that requires 2.5 hours of labor? The equation that would be used to find the amount the plmuber charges is y = 25x + 50. You would substitute 2.5 in for the x.
A house call that was 2.5 hours would cost $112.50.
Critical Vocabulary
Equation: a statement that the values of two mathematical expressions are equal. Ex: if x = 2, 2x + 3 = 7
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. Ex: 2^3, in this the 3 is the exponent because the 2 is being multiplied by itself 3 times
Expression: a collection of symbols that jointly express a quantity. Ex: 2x + 3
Function: a relationship or expression involving one or more variables. Ex: 6x + 3y
Term: each of the quantities in a ratio, series, or mathematical expression. Ex: Slope intercept form
Exponential Expressions
Power Rule for Exponents:
Product Rule for Exponents: The bases have to be the same for the exponents to combine with addition
Negative Exponent Rules: negative exponents flip and become positive
Zero Exponent Rule: any base with an exponent of zero is equal to one.
Fractional Exponents:
Quotient Rule for Exponents: The exponents in the fractions combine by subtraction
problem #1 i started off by using the power rule, next i used the quotient rule, and lastly i used the negative exponent rule
problem #2 I started off by using the zero rule, so everything on the bottom easily turns to 1
Next, I used the fractional exponent rule to simplify
Lastly, I used the product rule to add the exponents of a together and then you get the answer
slope and point: slope 3 point (1,-3)
y + 3 = 3 (x-1)
y = 3x-3-3
y = mx + b --> y = 3x - 6
slope intercept
given m=2/3 b=5
y=mx + b --> y=2/3x + 5
two points on a line (2,-4) and (5,6)
find slope 6 + 4/5 - 2 m=10/3
y-6=10/3 (x-5)
point and the equation of a parallel line
through (1,3) parallel to y = -1/2x
y - 3 = -1/2 (x-1)
y = -1/2x + 3 1/2
point and equation of a perpendicular line
through (2,2) perp. to y = -4/5x
y = 5/4x - 18/4
graph of a line
y = 2/3x + 5
Function Notation
When defining an equation, the value of using function notation is that even though it is common to use the letter f when writing function notation, any letter can be used in either upper or lower case. We can also change the letter for the input variable, but it tends to stay in lower case. This flexibility provides valuable maneuverability in writing functions.
Function Notation: Function notation is the way a function is written. It is meant to be a precise way of giving information about the function without a long explanation. The most popular function notation is f (x) which is read "f of x".
Examples:
Critical features of A Function
Intervals of Increasing Value:
Intervals of increasing value are the domain of a function where its value is getting larger
Intervals of Decreasing Value:
Intervals of decreasing value are the domain of a function where its value is getting smaller
Zeroes:
A "zero" of a function in an input value that produces an output of a zero (0).
Maximums:
Largest value of the function, either within a given range or on the entire domain of a function
Range:
The range of a function is the complete set of all possible resulting values of the dependent variable, after we have substituted the domain
Minimums:
Smallest value of the function, either within a given range or on the entire domain 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
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 Functions
linear function y = x
The domain and range of the linear function consists of All Real Numbers or expressed in Interval Notation. Ex: y = m(4) + b
quadratic function y = x^2
Ex: y = 5x^2 + 3x + 8
D: (-∞, ∞) R: (Min,∞) U (-∞,Max) Vertex (-b/2a, Max, Min)
y = ax^2+bx+c Increasing (-b/2a, ∞) Decreasing (-∞, -b/2a)
Radical Function
Example
Rational function y = 1/x
Example
D: (-∞,-1) U (1, ∞) R: (-∞, ∞) Zero = 2,-2
absolute function
x if x>0
D: (-∞ , ∞) R: (4 , ∞) Min=4- In this case, there are no zeros becasue it never crosses the x-axis
Piecewise Functions
Piecewise Function: a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain.
Find the following: f(-2) and f(10)
https://docs.google.com/document/d/15j3WjLHJRO6GYoWofu8oROx9-uEcw4waaUfGexJYDc4/edit
