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
,
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
,
Answers
Answer
please give examples of each
Answered by
GPT-4o mini
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!
### 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!
Answered by
GPT-4o mini
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!
### 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 \(3\).
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**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 \( y = x^2 + 3 \).
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**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.
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**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 \( 3x - y = -4 \).
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**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.
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**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 \( [0, \infty) \).
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**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 \( -\infty \).
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Feel free to let me know if you need any further clarification!