Unit 1: Functions
What is a Function? A function relates an input to an output.
It is like a machine that has an input and an output.
And the output is related somehow to the input.
It is like a machine that has an input and an output.
And the output is related somehow to the input.
Examples of Functions
![Picture](/uploads/3/7/8/6/37863677/164948229.jpg)
Linear Functions: Linear functions are lines
Can be as simple as: f(x) = x
Other Examples Include: f(x) = 3x + 2 and f(x) = -2x -7
The line in the picture is: f(x) = 1/2x + 5
Can be as simple as: f(x) = x
Other Examples Include: f(x) = 3x + 2 and f(x) = -2x -7
The line in the picture is: f(x) = 1/2x + 5
Non-linear Functions: Include Quadratic, Exponential, and Absolute Value
Absolute Value
![Picture](/uploads/3/7/8/6/37863677/741480000.png?250)
The absolute value |x| of a real number x is the non-negative value of x regardless of its sign.
In the picture: f(x) = |x|
In the picture: f(x) = |x|
Notes from Weds 8/20
What is a relation?
- A relation is any set of ordered pairs, (x, y).
- 3 forms of a relation:
- Table
- Set
- Graph
- All x values pair with 1 unique y value.
- THIS MEANS THAT X-COORDINATES CANNOT REPEAT WITH DIFFERENT Y-COORDINATES.
- Set {(-4, 6), (-3, 3), (1, 0), (4, 5), (10, 2)}
- Important Note: {(7, -2), (3, -4), (1, -6), (-2, -2)} IS a function. Tell me why.
- If you move your pencil vertically across a graph, it is only a function if at any time, the pencil only touches one point on the graph at a time.
- This is the quickest way to check to see if a function is a function.
- A function that is defined only for a set of numbers that can be listed, such as the set of whole numbers or the set of integers.
- Examples:
Continuous Functions
- A function is continuous when its graph is a single unbroken curve
Notes from Friday 8/22
Linear Functions and Relations
linear relation: relations that have straight line graphs
nonlinear relations: relations that are not linear
linear equation: has no operations other than addition, subtraction, and multiplication of a variable by a constant
Ex: Linear equations
3x - 5y = 16
x = 10
y = -23x - 1
y = 12x
Non-linear equations
2x + 6y2 = -25
y=x+2
x+xy=-58
y=1x
linear function: a function with ordered pairs that satisfy a linear equation.
Standard Form: a way of writing a linear equation in the form Ax + By = C, where A, B, and C are integers with a greatest common factor of 1.
Example:
Write -310x = 8y-15
A = ___ B = ___ C = ____
y-intercept: y-coordinate of the point at which a graph crosses the y-axis (0, y)
x-intercept: the x-coordinate of the point at which a graph crosses the x-axis (x, 0)
linear relation: relations that have straight line graphs
nonlinear relations: relations that are not linear
linear equation: has no operations other than addition, subtraction, and multiplication of a variable by a constant
Ex: Linear equations
3x - 5y = 16
x = 10
y = -23x - 1
y = 12x
Non-linear equations
2x + 6y2 = -25
y=x+2
x+xy=-58
y=1x
linear function: a function with ordered pairs that satisfy a linear equation.
Standard Form: a way of writing a linear equation in the form Ax + By = C, where A, B, and C are integers with a greatest common factor of 1.
Example:
Write -310x = 8y-15
A = ___ B = ___ C = ____
y-intercept: y-coordinate of the point at which a graph crosses the y-axis (0, y)
x-intercept: the x-coordinate of the point at which a graph crosses the x-axis (x, 0)
Notes on Rate of Change 8/25/14
Rate of change is a ratio that compares how much one quantity changes, on average, relative to the change in another quantity.
x is the independent variable
y is the dependent variable
rate of change= (change in y) / (change in x)
Point Slope Form
(y1 - y2) = m (x1 - x2)
Uses: When you know 2 points that are on the same line, plug them into the point slope form to find the slope.
Slope-Intercept Form
y = mx + b
x is the independent variable
y is the dependent variable
rate of change= (change in y) / (change in x)
Point Slope Form
(y1 - y2) = m (x1 - x2)
Uses: When you know 2 points that are on the same line, plug them into the point slope form to find the slope.
Slope-Intercept Form
y = mx + b
Parent Functions & Transformations 9/3/14
Parent Graphs
A family of graphs is a group of graphs that have one or more similar characteristics
Constant Function
The general equation of a constant function is f(x) = a, where a is any number. The domain is all real numbers, and the range consists of a single real number a.
Identity Function
The identity function f(x) = x passes through all points with coordinates (a, a). It is the parent function of most linear functions. Its domain and range are all real numbers.
Absolute Value Function
The parent function of absolute value functions is f(x) = |x|. The domain of f(x) = |x| is the set of all real numbers, and the range is the set of numbers greater than or equal to zero.
Quadratic Function
The parent function of a quadratic function is f(x) = x^2. The domain is the set of all real numbers, and the range is the set of numbers greater than or equal to zero.
Transformations of Functions
Translation
f(x + h), h>0 translates the graph h units left
f(x - h), h>0 translates the graph h units right
f(x) +k, k>0 translates the graph k units up
f(x) - k, k>0 translates the graph k units down
Reflection
-f(x) reflects the graph across the x-axis
f(-x) reflects the graph across the y-axis
Dilation
a•f(x), |a|>1, constricts the graph horizontally
a•f(x), 0<|a|<1 expands the graph horizontally
f(bx), |b|>1 stretches the graph vertically
f(bx), 0<|b|<1 shrinks the graph vertically
A family of graphs is a group of graphs that have one or more similar characteristics
Constant Function
The general equation of a constant function is f(x) = a, where a is any number. The domain is all real numbers, and the range consists of a single real number a.
Identity Function
The identity function f(x) = x passes through all points with coordinates (a, a). It is the parent function of most linear functions. Its domain and range are all real numbers.
Absolute Value Function
The parent function of absolute value functions is f(x) = |x|. The domain of f(x) = |x| is the set of all real numbers, and the range is the set of numbers greater than or equal to zero.
Quadratic Function
The parent function of a quadratic function is f(x) = x^2. The domain is the set of all real numbers, and the range is the set of numbers greater than or equal to zero.
Transformations of Functions
Translation
f(x + h), h>0 translates the graph h units left
f(x - h), h>0 translates the graph h units right
f(x) +k, k>0 translates the graph k units up
f(x) - k, k>0 translates the graph k units down
Reflection
-f(x) reflects the graph across the x-axis
f(-x) reflects the graph across the y-axis
Dilation
a•f(x), |a|>1, constricts the graph horizontally
a•f(x), 0<|a|<1 expands the graph horizontally
f(bx), |b|>1 stretches the graph vertically
f(bx), 0<|b|<1 shrinks the graph vertically