Unit 6 Functions

Question

Unit 6 Functions
Vocabulary

Key Terms
          Meaning
Example
U6 L1
What does it mean to be a Function?
Function:

Relation:

U6 L2
Independent variable/
Dependent variable
Independent variable:

Dependent variable:

U6 L2
Function notation

U6 L2
Ordered Pair (x, y)

U6 L2
Vertical Line Test

U6 L2
Linear/
NonLinear function
Linear:

NonLinear:

How to tell from a table:
How to tell from a graph:
How to tell from an equation:

U6 L3
Evaluating Function Notation

U6 L3
Interpreting piecewise, time-distance, and time-velocity
Graphs
Piecewise:

Time-Distance:

Time-velocity:

U6 L4
Meaning of the Slope in Context

U6 L4
Meaning of the y-intercept in context

U6 L4
How many points make up the graph of a Linear equation?

U6 L5
Increasing/
Decreasing Graphs
Increasing:

Decreasing:

U6 L5
4 possible slopes

U6 L5
Linear functions:  How do you find slope from a table?

U6 L6
Linear functions:  How do you know the y-intercept from a table?

U6 L7
Linear OR
NonLinear Functions
How to tell from a table:

How to tell from a graph:

How to tell from an equation:

U6 L7
Different forms of Linear Equations
Standard Form:

Point-Slope Form:
 
Slope-Intercept Form:

U7 L7
How do we change from one form to the other?

U6 L8
Compare Linear Functions
Rate of change:

y-intercept:

U6 L9
Linear Functions
Domain/Range
Domain:

Range:

Inequality Notation:

Interval Notation:

U6 L9
Linear Functions
End Behavior

,

GPT-4o mini · Answer

Certainly! Here's a complete vocabulary list based on your outline for Unit 6 on Functions.

Key Terms and Meanings

U6 L1

Function: A relation where each input (domain) has exactly one output (range).
Relation: A set of ordered pairs, where each pair consists of an input and its corresponding output.

U6 L2

Independent Variable: The variable that represents the input or cause; it can be manipulated (often denoted as x).
Dependent Variable: The variable that represents the output or effect; it depends on the independent variable (often denoted as y).

U6 L2

Function Notation: A way to represent a function using symbols, typically f(x), where f indicates the function and x is the input variable.

U6 L2

Ordered Pair (x, y): A pair of numbers used to represent a point on a graph, where x is the value of the independent variable and y is the value of the dependent variable.

U6 L2

Vertical Line Test: A method to determine if a graph represents a function; if a vertical line crosses the graph more than once, it is not a function.

U6 L2

Linear Function: A function whose graph is a straight line, described by a linear equation. The equation can be written in the form y = mx + b.
Nonlinear Function: A function that does not form a straight line when graphed (e.g., quadratic, exponential).

How to tell from a table: If the change in y divided by the change in x (slope) is constant, it is linear.
How to tell from a graph: If the graph is a straight line, it is linear; otherwise, it is nonlinear.
How to tell from an equation: If it can be written in the form y = mx + b, it is linear; otherwise, it is nonlinear.

U6 L3

Evaluating Function Notation: The process of substituting a specific value for the independent variable in a function to find the corresponding output value.

U6 L3

Interpreting piecewise, time-distance, and time-velocity graphs:
- Piecewise Function: A function that has different expressions for different intervals of the domain.
- Time-Distance Graph: A graph that shows the relationship between time and the distance traveled.
- Time-Velocity Graph: A graph that represents the velocity (rate of change of position) over time.

U6 L4

Meaning of the Slope in Context: The slope of a line in a functional relationship represents the rate of change; it shows how much the dependent variable changes for a unit change in the independent variable.

U6 L4

Meaning of the y-intercept in context: The y-intercept is the value of the dependent variable when the independent variable is zero; it represents the starting point of the function on the graph.

U6 L4

How many points make up the graph of a Linear equation? A linear equation is defined by two points, but technically consists of infinitely many points forming a straight line.

U6 L5

Increasing Graphs: A graph that rises from left to right; the dependent variable increases as the independent variable increases.
Decreasing Graphs: A graph that falls from left to right; the dependent variable decreases as the independent variable increases.

U6 L5

4 possible slopes:
1. Positive (increasing)
2. Negative (decreasing)
3. Zero (horizontal line)
4. Undefined (vertical line)

U6 L5

Linear functions: How do you find slope from a table? Calculate the change in y divided by the change in x (Δy/Δx) between two points.

U6 L6

Linear functions: How do you know the y-intercept from a table? Identify the output (y) when the input (x) is 0; this value is the y-intercept.

U6 L7

Linear OR NonLinear Functions
How to tell from a table: Look for a constant rate of change (linear) or varying rates of change (nonlinear).
How to tell from a graph: Identify if the graph forms a straight line (linear) or curves (nonlinear).
How to tell from an equation: Check for the form y = mx + b (linear) versus other forms like quadratic (nonlinear).

U6 L7

Different forms of Linear Equations
- Standard Form: Ax + By = C, where A, B, and C are constants.
- Point-Slope Form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
- Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.

U7 L7

How do we change from one form to the other? Use algebraic manipulation such as distributing, combining like terms, isolating y, or rearranging to transform between standard, point-slope, and slope-intercept forms.

U6 L8

Compare Linear Functions
- Rate of change: The slope of the function indicates how much y changes for a unit change in x.
- y-intercept: The value of the function when x is zero; indicates the starting value of the dependent variable.

U6 L9

Linear Functions Domain/Range
- Domain: The set of all possible input values (x-values).
- Range: The set of all possible output values (y-values).

Inequality Notation: Expresses restrictions on variables using inequalities.
Interval Notation: Uses intervals to denote the domain and range (e.g., [a, b] or (a, b)).

U6 L9

Linear Functions End Behavior: Describes the behavior of the function as the independent variable approaches positive and negative infinity, typically indicating whether the function goes up or down to extremes.

Feel free to adjust any definitions or examples as needed!

Answer

please give examples of each

GPT-4o mini · Answer

Certainly! Here are the key terms from Unit 6 on Functions, along with examples for each term.

Key Terms, Meanings, and Examples

U6 L1

Function: A relation where each input has exactly one output.
Example: \( f(x) = x^2 \) is a function because for each value of \( x \), there is one corresponding \( f(x) \).
Relation: A set of ordered pairs.
Example: The relation \( {(1, 2), (2, 3), (1, 4)} \) is not a function because the input \( 1 \) has two different outputs \( 2 \) and \( 4 \).

U6 L2

Independent Variable: The variable that represents the input.
Example: In the equation \( y = 2x + 3 \), \( x \) is the independent variable.
Dependent Variable: The variable that represents the output.
Example: In the same equation \( y = 2x + 3 \), \( y \) is the dependent variable because it depends on the value of \( x \).

U6 L2

Function Notation: A way of representing a function.
Example: \( f(x) = 3x + 5 \) is function notation for generating outputs based on the input \( x \).

U6 L2

Ordered Pair (x, y): A pair of numbers representing a point on a graph.
Example: \( (3, 7) \) is an ordered pair where \( 3 \) is the x-coordinate and \( 7 \) is the y-coordinate.

U6 L2

Vertical Line Test: A method to determine if a graph is a function.
Example: The graph of a circle fails the vertical line test because a vertical line can intersect it at two points.

U6 L2

Linear Function: A function whose graph is a straight line.
Example: \( f(x) = 2x + 1 \) is a linear function and its graph is a straight line.
Nonlinear Function: A function that does not form a straight line.
Example: \( g(x) = x^2 \) is a nonlinear function; its graph is a parabola.

How to tell from a table:

Linear: A table like \({(1, 2), (2, 4), (3, 6)}\) has a constant slope of 2 (each increment in \( x \) results in an increment of 2 in \( y \)).
Nonlinear: A table like \({(1, 1), (2, 4), (3, 9)}\) does not have a constant slope (slope goes from 3 to 5).

How to tell from a graph:

Linear: A straight line graph.
Nonlinear: A curved line (e.g., a parabola).

How to tell from an equation:

Linear: An equation like \( y = 3x + 4 \).
Nonlinear: An equation like \( y = x^3 - 2 \).

U6 L3

Evaluating Function Notation: Substituting a value into the function.
Example: For \( f(x) = x^2 \), to evaluate \( f(3) \), you substitute \( x = 3 \) to get \( f(3) = 3^2 = 9 \).

U6 L3

Interpreting piecewise, time-distance, and time-velocity graphs:
- Piecewise Function: A function with different expressions based on the input.
  Example:
  \[ f(x) = \begin{cases} x + 2 & \text{if } x < 0 \ x^2 & \text{if } x \geq 0 \end{cases} \]
- Time-Distance Graph: Represents distance traveled over time.
  Example: If a car moves at a constant speed, the graph would be a straight line going up (e.g., at 60 mph).
- Time-Velocity Graph: Shows how velocity changes over time.
  Example: If a car accelerates, the velocity might increase from 0 to 60 mph over 10 seconds, showing a curve in the graph.

U6 L4

Meaning of the Slope in Context: The slope indicates the rate of change.
Example: In the equation \( y = 5x + 2 \), the slope of 5 means that for each additional unit of \( x \), \( y \) increases by 5.

U6 L4

Meaning of the y-intercept in context: The y-intercept is where the line crosses the y-axis.
Example: In the equation \( y = 2x + 3 \), the y-intercept is \( 3 \); this is where the line crosses the y-axis (when \( x = 0 \)).

U6 L4

How many points make up the graph of a Linear equation? A linear equation is defined by two points but contains infinitely many points along the line.
Example: The points \( (0, 3) \) and \( (1, 5) \) define the line \( y = 2x + 3 \).

U6 L5

Increasing Graphs: A graph that rises from left to right.
Example: The line \( y = 2x + 1 \) is increasing.
Decreasing Graphs: A graph that falls from left to right.
Example: The line \( y = -3x + 4 \) is decreasing.

U6 L5

4 possible slopes:
- Positive: An upward slant, e.g., \( y = 2x + 1 \).
- Negative: A downward slant, e.g., \( y = -x + 2 \).
- Zero: A horizontal line, e.g., \( y = 3 \).
- Undefined: A vertical line, e.g., \( x = 2 \).

U6 L5

Linear functions: How do you find slope from a table?
Example: If the table shows \((1, 2)\) and \((2, 4)\), the slope is calculated as:
\[ \text{Slope} = \frac{4 - 2}{2 - 1} = 2 \]

U6 L6

Linear functions: How do you know the y-intercept from a table?
Example: If the table shows \((0, 3)\), then the y-intercept is \(3\).

U6 L7

Linear OR NonLinear Functions
How to tell from a table:
Linear: For example, the table \({(1, 1), (2, 2), (3, 3)}\) shows a constant change of 1 (linear).
Nonlinear: The table \({(1, 1), (2, 4), (3, 9)}\) shows increasing differences (nonlinear).

How to tell from a graph:

Linear: A straight line graph (e.g. \( y = mx + b \)).
Nonlinear: A curve, like a circle or a parabola.

How to tell from an equation:

Linear: An equation like \( y = 4x + 2 \).
Nonlinear: An equation like \( y = x^2 + 3 \).

U6 L7

Different forms of Linear Equations
- Standard Form: \( 3x + 4y = 12 \) is an example of standard form.
- Point-Slope Form: \( y - 2 = 5(x - 1) \) represents a line with a slope of 5 passing through the point \( (1, 2) \).
- Slope-Intercept Form: \( y = -2x + 5 \) shows that the slope is -2 and the y-intercept is 5.

U7 L7

How do we change from one form to the other?
Example 1: Converting from Point-Slope to Slope-Intercept:
From \( y - 3 = 2(x - 1) \), distribute to get \( y - 3 = 2x - 2 \), then add 3 to get \( y = 2x + 1 \) (which is the Slope-Intercept Form).

Example 2: Converting from Slope-Intercept to Standard:
From \( y = 3x + 4 \), rewrite as \( -3x + y = 4 \) or \( 3x - y = -4 \).

U6 L8

Compare Linear Functions
- Rate of change: In \( y = 4x - 1 \), the slope of 4 indicates that for each unit increase in \( x \), \( y \) increases by 4.
- y-intercept: In the function \( y = 1/2x + 3 \), the y-intercept is 3.

U6 L9

Linear Functions Domain/Range
- Domain: The set of x-values, e.g., for \( y = 2x + 1, \) the domain is all real numbers (\( \mathbb{R} \)).
- Range: The set of y-values, e.g., for \( y = 2x + 1, \) the range is also all real numbers (\( \mathbb{R} \)).

Inequality Notation: For a function defined on the domain \( x \geq 0 \), it's represented as \( { x | x \geq 0 } \).
Interval Notation: The domain can be expressed as \( [0, \infty) \).

U6 L9

Linear Functions End Behavior: Describes how the function behaves as \( x \) approaches \( \pm \infty \).
Example: For \( y = 2x + 3 \), as \( x \) approaches \( +\infty \), \( y \) also approaches \( +\infty \). As \( x \) approaches \( -\infty \), \( y \) approaches \( -\infty \).

Feel free to let me know if you need any further clarification!