Functions and Graphs函数和图表
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substitute in 30 for x to find the function at 30. f (30)= 0.13(30)2 -0.21 (30) + 8.7
Now let’s try to evaluate using our calculators.
HOW DO WE EVALUATE A FUNCTION?
There are two ways to visually demonstrate if a relation is a function.
1. Mapping
2. Vertical Line Test
DETERMINE WHETHER THE RELATION IS A FUNCTION
a. {(1, 6), (2, 6), (3, 8), (4, 9)} b. {(6, 1), (6, 2), (8, 3), (9, 4)}
We can say that -2, 3, and 5 are the zeros of the function. The zeros of the function are the x- values that make f (x) = 0.
Therefore, the real zeros are the x-intercepts.
We use the special notation f(x) which reads as f of x and represents the function at the number x.
Example: f (x) = 0.13x2 -0.21x +8.7 If we are interested in finding f (30), we
HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION
x2 + y2 =4
Steps:
Solve the equation for y in terms of x.
Note:
If two or more y values are found then the equation is not a function.
PRACTICE:
Solve each equation for y and then determine whether the equation defines y as a function of x.
1. 2x + y = 6 2. x2 + y2 = 1
WHAT IS FUNCTION NOTATION?
HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION?
x2 + y = 4
Steps:
Solve the equation for y in terms of x.
Note:
If two or more y values are found then the equation is not a function.
interval, I if
f (x1) < f(x2) whenever x1<x2 for any x1 and
x2 in the interval.
A function is decreasing on an open interval, I, if f(x1) > f (x2) whenever x1 > x2
AIM #1.3: HOW DO WE IDENTIFY INTERVALS
ON WHICH A FUNCTION IS INCREASING OR DECREASING?
Increasing, Decreasing and Constant Functions
A function is increasing on a open
function? How can you determine if an equation in x and
y defines y as a function of x? Give an example.
AIM #1.2B: WHAT KIND OF INFORMATION CAN WE OBTAIN FROM GRAPHS OF FUNCTIONS?
HOW DO WE IDENTIFY DOMAIN AND RANGE FROM A FUNCTION’S GRAPH?
Domain: set of inputs
Found on x –axis
Range: set of outputs Found on y -axis
Using set builder notation it would look like this for the domain:
(Williams, 0.699%), (Brown, 0.621%)}
PRACTICE:
Find the domain and range of the following relation.
HOW DO WE DETERMINE IF A RELATION IS A FUNCTION?
A relation is a function if each domain only has ONE range value.
The graph of a function is the graph of the ordered pairs.
Let’s graph: a. f (x) = 2x b. g (x) = 2x + 4
USING THE VERTICAL LINE TEST
The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as
FUNCTIONS AS EQUATIONS
Functions are usually given in terms of equations instead of ordered pairs.
Example: y =0.13x2 -0.21x + 8.7
The variable x is known the independent variable and y is the dependent variable.
Note the closed dot indicates the graph does not extend from this point and its part of the graph.
Open dot indicates that the point does not extend and the point is not part of the graph.
Use Set Builder Notation. Domain:
Range:
IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH
IDENTIFYING INTERCEPTS FROM A FUNCTION’S GRAPH
FUNCTIONS AND GRAPHS
Aim #1.2: What are the basics of functions and their graphs?
Let’s Review: What is the Cartesian Plane or Rectangular
Coordinate Plans? How do we find the x and y-intercepts of any
Domain is the first number in the ordered pair.
Example:(4,-2)
Range is the second number in the ordered pair.
Example: : (4,30)
EXAMPLE 1:
Find the domain and range of the relation. {(Smith, 1.0006%), (Johnson, 0.810%),
F (x) = x2 + 3x + 5
Evaluate each of the following:
f (2) f (x + 3) f (-x)
Substitute the 2 for x and evaluate.
Then repeat.
GRAPHS OF FUNCTIONS
function Evaluate a function Graph functions by plotting points Use the vertical line to identify functions
WHAT IS A RELATION?
A relation is a set of ordered pairs. Example: (4,-2), (1, 2), (0, 1), (-2, 2)
for any x1 and x2 in the interval.
a functioቤተ መጻሕፍቲ ባይዱ of x.
Practice:
Use the vertical line test to identify graphs in which y is a function of x.
SUMMARY: ANSWER IN COMPLETE SENTENCES.
What is a relation? What is a function? How can you determine if a relation is a
function? How do we interpret the viewing rectangle
[-10,10, 1] by [-10, 10,1]?
IN THIS SECTION WE WILL LEARN:
How to find the domain and range? Determine whether a relation is a function Determine whether an equation represents a
Explain how to determine the domain and range of a function from its graph.
Does it make sense? Explain your reasoning.
I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.
{x 4 x 2}
Using Interval Notation: [-4, 2]
What would it look like for the range using both set builder and interval
IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH
A function can have more than one x-intercept, but at most one y-intercept.
SUMMARY: ANSWER IN COMPLETE SENTENCES.
Explain how the vertical line test is used to determine whether a graph is a function.
Now let’s try to evaluate using our calculators.
HOW DO WE EVALUATE A FUNCTION?
There are two ways to visually demonstrate if a relation is a function.
1. Mapping
2. Vertical Line Test
DETERMINE WHETHER THE RELATION IS A FUNCTION
a. {(1, 6), (2, 6), (3, 8), (4, 9)} b. {(6, 1), (6, 2), (8, 3), (9, 4)}
We can say that -2, 3, and 5 are the zeros of the function. The zeros of the function are the x- values that make f (x) = 0.
Therefore, the real zeros are the x-intercepts.
We use the special notation f(x) which reads as f of x and represents the function at the number x.
Example: f (x) = 0.13x2 -0.21x +8.7 If we are interested in finding f (30), we
HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION
x2 + y2 =4
Steps:
Solve the equation for y in terms of x.
Note:
If two or more y values are found then the equation is not a function.
PRACTICE:
Solve each equation for y and then determine whether the equation defines y as a function of x.
1. 2x + y = 6 2. x2 + y2 = 1
WHAT IS FUNCTION NOTATION?
HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION?
x2 + y = 4
Steps:
Solve the equation for y in terms of x.
Note:
If two or more y values are found then the equation is not a function.
interval, I if
f (x1) < f(x2) whenever x1<x2 for any x1 and
x2 in the interval.
A function is decreasing on an open interval, I, if f(x1) > f (x2) whenever x1 > x2
AIM #1.3: HOW DO WE IDENTIFY INTERVALS
ON WHICH A FUNCTION IS INCREASING OR DECREASING?
Increasing, Decreasing and Constant Functions
A function is increasing on a open
function? How can you determine if an equation in x and
y defines y as a function of x? Give an example.
AIM #1.2B: WHAT KIND OF INFORMATION CAN WE OBTAIN FROM GRAPHS OF FUNCTIONS?
HOW DO WE IDENTIFY DOMAIN AND RANGE FROM A FUNCTION’S GRAPH?
Domain: set of inputs
Found on x –axis
Range: set of outputs Found on y -axis
Using set builder notation it would look like this for the domain:
(Williams, 0.699%), (Brown, 0.621%)}
PRACTICE:
Find the domain and range of the following relation.
HOW DO WE DETERMINE IF A RELATION IS A FUNCTION?
A relation is a function if each domain only has ONE range value.
The graph of a function is the graph of the ordered pairs.
Let’s graph: a. f (x) = 2x b. g (x) = 2x + 4
USING THE VERTICAL LINE TEST
The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as
FUNCTIONS AS EQUATIONS
Functions are usually given in terms of equations instead of ordered pairs.
Example: y =0.13x2 -0.21x + 8.7
The variable x is known the independent variable and y is the dependent variable.
Note the closed dot indicates the graph does not extend from this point and its part of the graph.
Open dot indicates that the point does not extend and the point is not part of the graph.
Use Set Builder Notation. Domain:
Range:
IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH
IDENTIFYING INTERCEPTS FROM A FUNCTION’S GRAPH
FUNCTIONS AND GRAPHS
Aim #1.2: What are the basics of functions and their graphs?
Let’s Review: What is the Cartesian Plane or Rectangular
Coordinate Plans? How do we find the x and y-intercepts of any
Domain is the first number in the ordered pair.
Example:(4,-2)
Range is the second number in the ordered pair.
Example: : (4,30)
EXAMPLE 1:
Find the domain and range of the relation. {(Smith, 1.0006%), (Johnson, 0.810%),
F (x) = x2 + 3x + 5
Evaluate each of the following:
f (2) f (x + 3) f (-x)
Substitute the 2 for x and evaluate.
Then repeat.
GRAPHS OF FUNCTIONS
function Evaluate a function Graph functions by plotting points Use the vertical line to identify functions
WHAT IS A RELATION?
A relation is a set of ordered pairs. Example: (4,-2), (1, 2), (0, 1), (-2, 2)
for any x1 and x2 in the interval.
a functioቤተ መጻሕፍቲ ባይዱ of x.
Practice:
Use the vertical line test to identify graphs in which y is a function of x.
SUMMARY: ANSWER IN COMPLETE SENTENCES.
What is a relation? What is a function? How can you determine if a relation is a
function? How do we interpret the viewing rectangle
[-10,10, 1] by [-10, 10,1]?
IN THIS SECTION WE WILL LEARN:
How to find the domain and range? Determine whether a relation is a function Determine whether an equation represents a
Explain how to determine the domain and range of a function from its graph.
Does it make sense? Explain your reasoning.
I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.
{x 4 x 2}
Using Interval Notation: [-4, 2]
What would it look like for the range using both set builder and interval
IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH
A function can have more than one x-intercept, but at most one y-intercept.
SUMMARY: ANSWER IN COMPLETE SENTENCES.
Explain how the vertical line test is used to determine whether a graph is a function.