Key Terms
Meaning
Example
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What does it mean to be a Function?
Function: A function is a relation where each input (x-value) has exactly one output (y-value). It can also be described as a rule or an equation that assigns each input with a unique output
Relation:A relation is a set of ordered pairs. It describes the relationship between two sets of values or variables.
A function can be represented by the equation y = 2x + 3, where for every input value of x, there is a unique output value of y.
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Independent variable/
Dependent variable
Independent variable:The independent variable is the variable that is input or controlled by the person or experiment. It is also known as the "x-variable" in a function.
Dependent variable:The dependent variable is the variable that is output or dependent on the independent variable. It is also known as the "y-variable" in a function.:
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Function notation
Function notation is a way of representing a function using symbols. It is typically written as f(x), where f represents the function and x represents the input value.
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Ordered Pair (x, y)
An ordered pair is a pair of numbers in a specific order. In the context of functions, an ordered pair represents a pair of values where the first value is the input (x-value) and the second value is the output (y-value).
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Vertical Line Test
The vertical line test is a test used to determine whether a graph represents a function. If a vertical line can intersect the graph at more than one point, then the graph does not represent a function.
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Linear/
NonLinear function
Linear:
NonLinear: A nonlinear function is a function that does not have a constant rate of change. The change in the output value is not directly proportional to the change in the input value. It cannot be represented by a straight line on a graph.
How to tell from a table:if the change in the output (y-value) is constant for each change in the input (x-value), then the function is linear. If the change in the output is not constant, then the function is nonlinear.
How to tell from a graph:On a graph, if the points form a straight line, the function is linear. If the points do not form a straight line, the function is nonlinear.
How to tell from an equation:In an equation, if the highest exponent of the variable is 1, the function is linear. If the highest exponent is greater than 1 or if there are terms with variables multiplied together or raised to a power, the function is nonlinear.
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Evaluating Function Notation
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Interpreting piecewise, time-distance, and time-velocity
Graphs
Piecewise: In a piecewise function, different rules or equations are used to define the function for different intervals or regions.
Time-Distance:Time-Distance graphs show the relationship between the time it takes to travel a certain distance. The x-axis represents time, and the y-axis represents distance
Time-velocity:Time-Velocity graphs show the relationship between the time and velocity of an object. The x-axis represents time, and the y-axis represents velocity.
For example, a piecewise function f(x) could be defined as f(x) = x^2 if x is less than or equal to 0, and f(x) = 2x + 1 if x is greater than 0
A linear Time-Distance graph would have a constant slope, indicating a constant speed.
The slope of a Time-Velocity graph represents acceleration, with a positive slope indicating a increasing velocity and negative slope indicating decreasing velocity.
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Meaning of the Slope in Context
In the context of a linear equation, the slope represents the rate of change or the steepness of a line
For example, if a linear equation represents the relationship between a person's age and their height, the slope would represent the amount their height changes for each year of age.
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Meaning of the y-intercept in context
In the context of a linear equation, the y-intercept represents the value of the dependent variable (y) when the independent variable (x) is zero.
For example, in the equation y = mx + b, the y-intercept, represented as (0, b), would give the starting value or initial condition of the scenario being modeled.
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How many points make up the graph of a Linear equation?
. A linear equation represents a straight line on a graph.
A linear equation will have exactly two points that make up the graph, as it only takes two points to determine a line.
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Increasing/
Decreasing Graphs
Increasing:An increasing graph shows a positive slope, where the y-values increase as the x-values increase.
Decreasing:A decreasing graph shows a negative slope, where the y-values decrease as the x-values increase
The line or curve on the graph generally moves upward from left to right.
The line or curve on the graph generally moves downward from left to right.
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4 possible slopes
The four possible slopes of a linear equation are positive slope, negative slope, zero slope (horizontal line), and undefined slope (vertical line).
- Positive slope: y = 2x + 1, where the line increases as x increases.
- Negative slope: y = -3x + 2, where the line decreases as x increases.
- Zero slope: y = 4, where the line is horizontal and does not change as x increases.
- Undefined slope: x = 5, where the line is vertical and does not change as y increases.
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Linear functions: How do you find slope from a table?
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Linear functions: How do you know the y-intercept from a table?
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Linear OR
NonLinear Functions
How to tell from a table:
How to tell from a graph:
How to tell from an equation:
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Different forms of Linear Equations
Standard Form:
Point-Slope Form:
Slope-Intercept Form:
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How do we change from one form to the other?
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Compare Linear Functions
Rate of change:
y-intercept:
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Linear Functions
Domain/Range
Domain:
Range:
Inequality Notation:
Interval Notation:
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Linear Functions
End Behavior
,
give a example to all of them
Example for Function: The equation y = 2x + 3 is a function because for every input value of x, there is a unique output value of y. For example, if we input x = 1, the function gives us an output of y = 5.
Example for Relation: A relation can be represented by a set of ordered pairs such as {(1, 2), (3, 4), (5, 6)}. This set of ordered pairs describes the relationship between the sets of values or variables.
Example for Independent variable: In the equation y = 2x + 3, x is the independent variable because it is input or controlled by the person or experiment. The value of x can be chosen or manipulated.
Example for Dependent variable: In the equation y = 2x + 3, y is the dependent variable because it is the output or dependent on the value of the independent variable x. The value of y depends on the value of x chosen or manipulated.
Example for Function notation: The function notation f(x) is used to represent a function. For example, f(x) = 2x + 3 is the function notation for the equation y = 2x + 3.
Example for Ordered Pair (x, y): An example of an ordered pair is (2, 5), where 2 represents the input (x-value) and 5 represents the output (y-value). This ordered pair represents a point on the graph of a function.
Example for Vertical Line Test: If we have a graph of a relation, we can use the vertical line test to determine if the graph represents a function. If a vertical line can intersect the graph at more than one point, then the graph does not represent a function. For example, a graph of a circle does not represent a function because a vertical line can intersect it at multiple points.
Example for Linear: An example of a linear function is y = 2x + 3. The output (y-value) changes at a constant rate of 2 for every change in the input (x-value) of 1. The graph of a linear function is a straight line.
Example for Nonlinear: An example of a nonlinear function is y = x^2. The output (y-value) does not change at a constant rate for every change in the input (x-value). The graph of a nonlinear function is not a straight line.
Example for Evaluating Function Notation: To evaluate the function f(x) = 2x + 3 at x = 4, we substitute x = 4 into the equation and find the value of y. So, f(4) = 2(4) + 3 = 11.
Example for Interpreting piecewise, time-distance, and time-velocity Graphs: A piecewise function could be defined as f(x) = x^2 if x is less than or equal to 0, and f(x) = 2x + 1 if x is greater than 0. This function has different rules for different intervals of x.
Example for Meaning of the Slope in Context: In the equation y = mx + b, the slope (m) represents the rate of change or the steepness of a line. For example, if a linear equation represents the relationship between the amount of rainfall and the number of days, the slope would represent the amount of rainfall that increases or decreases for each additional day.
Example for Meaning of the y-intercept in context: In the equation y = mx + b, the y-intercept (b) represents the value of the dependent variable (y) when the independent variable (x) is zero. For example, in the equation y = 2x + 3, the y-intercept is 3, which represents the initial value of y when x is 0.
Example for How many points make up the graph of a Linear equation: A linear equation will have exactly two points that make up the graph, as it only takes two points to determine a line. For example, the equation y = 2x + 3 represents a linear equation with a graph that consists of two points (0, 3) and (1, 5).
Example for Increasing/Decreasing Graphs: An increasing graph shows a positive slope, where the y-values increase as the x-values increase. For example, y = 2x + 3 represents an increasing graph. A decreasing graph shows a negative slope, where the y-values decrease as the x-values increase. For example, y = -2x + 3 represents a decreasing graph.
Example for 4 possible slopes:
- Positive slope: y = 2x + 1, where the line increases as x increases.
- Negative slope: y = -3x + 2, where the line decreases as x increases.
- Zero slope: y = 4, where the line is horizontal and does not change as x increases.
- Undefined slope: x = 5, where the line is vertical and does not change as y increases.
Example for Linear functions: How do you find slope from a table? To find the slope from a table, you can choose any two points on the table and use the formula: slope = (change in y-values)/(change in x-values). For example, if the table has the points (2, 4) and (5, 10), the slope would be (10 - 4)/(5 - 2) = 2.
Example for Linear functions: How do you know the y-intercept from a table? To find the y-intercept from a table, you can look for the value of y when x is 0. For example, if the table has the point (0, 3), the y-intercept would be 3.
Example for Linear OR NonLinear Functions: To determine if a table represents a linear or nonlinear function, you can look for a constant rate of change in the output (y-values) for each change in the input (x-values). If the change in output is constant, the function is linear. If the change in output is not constant, the function is nonlinear.