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Unit 1 Functions and Graphs Concept Map (Vocab: Term, Expression, Equation…
Unit 1 Functions and Graphs Concept Map
Vocab: Term, Expression, Equation, Exponent, Funtions
Term: A term is a single number or variable, it can also be a single number with a variable.
Expression: An expression is a phrase that contains numbers, variables (x,y), and operations (add, subtract, multiply, division)
Equation: A function that says two things are equal. It consists of two expressions, one on each side of an 'equals' sign.
Exponent: An exponent is a superscript 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.
Function: a function is a relation between a set of inputs and a set of outputs, it's important to remember that each input has only one output
Simplifying Exponential Expressions:
https://docs.google.com/document/d/1p0wpM5-T7B3gpXJKfALL44fAglJ-rcwLwdXc2Pmu8Hc/edit
Example 1:
Example 2:
Negative Exponent Rule: If an exponent is negative they flip to become positive.
Zero Exponent Rule: Anything raised to the zero power is 1.
Power Exponent Rule: If you raise an exponent to another exponent you have to multiple the powers.
Product Rule for Exponents: When multiplying two exponents with the same base, add the powers.
Quotient Rule for Exponents: when you are dividing two exponents with the same base, you must subtract the exponents (
https://study.com/academy/lesson/the-quotient-rule-for-exponents.html
)
Fractional Exponent Rule:
Slope can be defined as a rate of change.
Formula:
Duxbury Bridge has a zero slope
Example of slope
Points: (-6,5) and (2,4)
The cost of a school banquet is $95 plus $15 for each person attending. Write an equation that gives total cost as a function of the number of people attending. What is the cost for 77 people?
Roller coasters have a positive slope
How to write a linear equation: 6 ways
Slope and Intercept Known
Slope and Point Known
Two Points on a line Known
Point and Parallel Line Given
Point and Perpendicular Line Given
Graph of a line Known
Evaluating Functions
Function Notation: Function notation helps us write functions without the use of y, making them easier to compute and understand.We can then rewrite it with function notation: f(x) = 3x + 7. All we did is replace y with f(x), so y = f(x).
Using Function Notation to Define an Equation: Using function notation helps to make equations more clear picture of how a function needs to be evaluated. For example take f(x)= 4x + 2, when f(x)=2 , x=2, f(2)=4(2)+2 = 10. We can also try equations like f(x(y))= 2x+3y, when x=4 and y=3. This would be near impossible to do with our standard y=mx+b notation.
Evaluating Expressions
Evaluating Values
Analyzing Functions
Parent Functions
Linear Parent Function:
Example:
Graph:
Domain: (-∞, ∞) Range: (-∞, ∞) Zeros: (-1.5,0) Intervals: Are increasing
Y-Intercept: (0,3)
Quadratic Parent Function
Example:
Graph
Zeros: (3.4,0) and (.59,0) Intervals: Begin decreasing, then increase. Min: -2, Max: infinity
Logarithmic Parent Function:
Example
Domain: (-∞, ∞) Range:
Absolute Value Parent Function
Example
Graph
Domain: (-∞, ∞) Range: [0,∞)
Zeros: (1.75,0)
Increasing and decreasing
Min=0
Y-intercept=(0,7)
Rational Parent Function:
Example
Graph
Asymptotes: x=0, y=5 Zero = (-.05,0)
Radical Parent Function:
Example
Graph
Y intercept: (0,-2.59) Zeros: (14,0) Intervals are increasing. Min =
(-2,-4) Max = infinity
Critical Features of Function
Maximums and Minimums: you can find the minimum or maximum value algebraically by using the formula x = -b / 2a, then solve for y
Intervals of increasing and decreasing values: If f(x) > 0, then f is increasing on the interval, and if f(x) < 0, then f is decreasing on the interval
Zeroes: its zeros are the x-coordinates of the points where its graph meets the x-axis, also known as the x intercept or roots
Domain & Range: The range is the resulting y-values we get after substituting all the possible x-values. The domain is the resulting x values we get after substituting all the possible y values.
Asymptotes: a line that continually approaches a given curve but does not meet it at any finite distance.
Piecewise Functions: A piecewise function is a function that passes through different intervals. It excludes certain values and is defined at different points.
Domain of 2x+8, when x is less than but equal to -2 (-∞, -2)
Domain of x^2, when x is greater than [-2, ∞)
