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Algebra Module 3 DBA
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Terms in this set (12)
3.01
Relation
•A relation describes a relationship and pairs input values with output values.
•Relations can be represented as ordered pairs, graphed on a coordinate plane, listed in an x/y table of values, or as a mapping.
Domain
•The domain of a relation is simply the input, or x-values of the relation.
Range
The range of a relation is simply the possible output, or y-values of the relation.
Function
•Some relations have a special relationship in that each input value is paired with exactly one output value. When this happens, the relation is called a function.
•You can test whether a relation is a function by comparing the x-values. The relation is not a function if any of the x-values repeat.
Vertical line test
•You can test whether the graph of a relation is a function using the vertical line test. If the graph crosses any vertical line at more than one point, the relation is not a function.
3.02
Function Notation
Function notation is used to write the equation of a function. Function notation simply sets an expression equal to f(x), read "f of x," which means "f is a function of x."
You may use any letter to represent a function: f(x), g(x), h(x), r(x), etc.
When given an equation in function notation (for example: f(x) = x + 1), you may be asked to do one of the following:
•Evaluate the function for a given x-value—substitute the value for x and simplify.
•Solve for x given a function value—substitute the function value and solve for x.
When given a table or a graph, you may be asked to do one of the following:
•Given an output value, identify the input value that corresponds.
•Given an input value, identify the output value that corresponds.
With any function, you can write inputs and outputs as ordered pairs: (input, output).
3.03
Slope
DON'T FORGET
•Slope is the "rise over run."
•Slope is the ratio of the vertical change to the horizontal change between two points.
•Given the line graphed on a coordinate plane, you find any two points on the line and count the rise (up or down) and the run (side to side) between the two points. Start with the point on the left. If you have to go up, your rise is positive. If you have to go down, your rise is negative. Since you always go to the right, your run will always be positive.
•To graph a line using its x- and y- intercepts: •Find the x-intercept: replace y with the number zero and solve for x.
•Find the y-intercept: replace x with the number zero and solve for y.
•Plot both intercepts on the coordinate plane, and then connect them to draw the graph.
Remember that the x-intercept can be written as an ordered pair where y is zero. The y-intercept can be written as an ordered pair where x is zero.
(2, 0) x-intercept
(0, −3) y-intercept
•To graph a line using the slope-intercept form: •Manipulate the equation into slope-intercept form, y = mx + b.
•Identify and plot the y-intercept of the line. Remember, the y-intercept is b. Don't forget: The sign goes with the number.
•Identify and use the slope of the line to find a second point. Remember, the slope is m. Don't forget: Starting at the y-intercept, the numerator tells you the rise (count up if positive and down if negative), the denominator tells you the run (count right.)
•Draw a straight line through the two points to complete the graph.
Linear Function
•A linear function is a function that is defined by a linear equation.
•To write a linear function with function notation, first write it in slope-intercept form, and then replace y with f(x).
•The graph of a linear function is all points (x, f(x)), where x is in the domain of the function.
3.05
Real World Situations
•Real-world situations can be analyzed with linear models. Tables, equations, and graphs show the complete picture.
•Describe a linear function with its key features: •What are the variables?
•What are the x- and y-intercepts?
•Is the function increasing, decreasing, or constant?
•What is the rate of change?
•What are the domain and range of the function?
Average Rate of Change
•The formula to find the average rate of change for any function, f(x), for any interval from a to b is: •Average rate of change = the quantity f of b minus f of a over the quantity b minus a
•The average rate of change in a linear function is the same as its slope.
•If the rate of change in a function is not constant, the function is not linear.
•Increasing the value of the y-intercept causes a vertical shift of a line up the y-axis; decreasing its value causes a vertical shift down. A function f(x) is shifted up or down the y-axis and becomes a new function g(x) depending on the value of k if g(x) = f(x) + k.
3.06
Point-Slope Form
•Point-slope form: y − y1 = m(x − x1) where (x1, y1) is any point on the line and m is the slope of the line.
•You can rearrange an equation from point-slope form to slope-intercept form or standard form.
•You can write any linear equation in function notation by replacing y with f(x).
•You can find the slope of a line if you know any two points on the line:
•You can graph a line from point-slope form by first changing it to slope-intercept form and then graphing it.
•To solve problems that have defined variables, such as a real-world problem, you need to analyze which variable is independent and which is dependent. Then, choose point-slope or slope-intercept form based on the information that is given.
Horizontal and Vertical Lines
Horizontal slope line: Zero (m=0) Equation Sign: Y= number
Vertical slope line: Undefined Equation Sign: X= number
Example: Given the point (4, −8):
1.The horizontal line through this point is y = −8 and has a slope of 0.
2.The vertical line through this point is x = 4 and has an undefined slope.
Horizontal relationship to lines: Parallel
Relation to vertical lines: Perpendicular
Vertical relationship to lines: Perpendicular
Relationship to Vertical lines: Parallel
Examples:
1.The lines y = 7 and y = −3 are parallel because they are both horizontal lines.
2.The lines y = 7 and x = 4 are perpendicular because the first line is horizontal and the second one is vertical.
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