Download Functions and Their Graphs Part 1 (Section 1-3)

Graphs of Functions
(Section 1-3)
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Graph of a function f: the collection of ordered pairs (x, f(x)) such that x is in the
domain of f.
If you are asked to find f(x) for a specific x value…
If you have an equation, plug the number inside the parenthesis into the equation in
place of x. The answer is the value you get once you have evaluated the expression.
If you have a graph, look for the point on the graph where the number inside the
parenthesis is the x-coordinate. The answer is the y-coordinate at this point.
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use the graph of the function to find the domain and range of f. Then find f(0).
Example 1
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use the graph of the function to find the domain and range of f. Then find f(0).
Example 2
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use the graph of the function to find the domain and range of f. Then find f(0).
Example 3
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
To determine the domain and range algebraically:
In general, the domain, is all real numbers except if there is a number that you cannot
plug into the function. (ex: a number that will result in a negative number inside an even
root, or that will result in a zero in the denominator of a rational expression)
For the range, consider which y-values you can get by plugging in allowed x-values to the
equation (range is easiest to determine graphically).
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use a graphing utility to graph the function and estimate its domain and range. Then find
the domain and range algebraically.
Example 4
f ( x)   x 2  1
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use a graphing utility to graph the function and estimate its domain and range. Then find
the domain and range algebraically.
Example 5
h(t )  t  3
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use a graphing utility to graph the function and estimate its domain and range. Then find
the domain and range algebraically.
Example 6
f ( x)  
1
x 5
4
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use the given function to answer the questions.
Example 7 f(x) = x3 – 4x
(a) Determine the domain of the function
(b) Find the value(s) of x such that f(x) = 0.
(c) The values of x from part (b) are referred to as what graphically?
(d) Find f(0), if possible.
(e) The value from part (d) is referred to as what graphically?
(f) What is the value of f at x = 1? What are the coordinates of the point?
(g) What is the value of f at x = -1? What are the coordinates of the point?
(h) The coordinates of the point on the graph of f at which x = -3, can be labeled (-3, f(-3))
or (-3, _____).
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use the given function to answer the questions.
Example 8
(a) Determine the domain of the function
(b) Find the value(s) of x such that f(x) = 0.
(c) The values of x from part (b) are referred to as what graphically?
(d) Find f(0), if possible.
(e) The value from part (d) is referred to as what graphically?
(f) What is the value of f at x = 1? What are the coordinates of the point?
(g) What is the value of f at x = -1? What are the coordinates of the point?
(h) The coordinates of the point on the graph of f at which x = -3, can be labeled (-3, f(-3))
or (-3, _____).
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Vertical line test (pg 31): A set of points in a coordinate plane is the graph of y as a function
of x if and only if no vertical line intersects the graph at more than one point.
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use the Vertical Line Test to determine whether y is a function of x. Describe how you can
use the graphing utility to produce the given graph.
Example 9
1 2
y x
4
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use the Vertical Line Test to determine whether y is a function of x. Describe how you can
use the graphing utility to produce the given graph.
Example 10
x  y2  1
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Increasing: positive slope
Decreasing: negative slope
Constant: zero slope
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Determine the open intervals on which the function is increasing, decreasing or
constant.
Example 11
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
(A) Use a graphing utility to graph the function. (b) Determine the open intervals on
which the function is increasing, decreasing or constant.
Example 12 y = x3
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
(A) Use a graphing utility to graph the function. (b) Determine the open intervals on
which the function is increasing, decreasing or constant.
Example 13 y = x3 – 3x
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Section 1.3,
Definitions of Relative Minimum and Relative Maximum
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use a graphing utility to approximate any relative minimum or relative maximum values of
the function.
Example 14 f(x) = 3x2 – 4x – 2
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use a graphing utility to approximate any relative minimum or relative maximum values of
the function.
Example 15 f(x) = -x3 + x
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
HW #61 pg 38-39 (1-41 odd…37-41 only part b)
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Test for Even and Odd Functions
A function is even if, for each x in the domain of f, f(-x) = f(x).
A function is odd if, for each x in the domain of f, f(-x) = -f(x).
Essential Question: How do you find the domain and range of a function algebraically? How do you
**Note: Unlike a number, a function does not have to be even
or odd.**
determine
if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Algebraically determine whether the function is even, odd, or neither.
Example 16 f(x) = |x|
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Algebraically determine whether the function is even, odd, or neither.
Example 17 h(x) = x3 - 5
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use a graphing utility to determine whether the function is even, odd, or neither.
Example 18
f ( x)  9
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
Use a graphing utility to determine whether the function is even, odd, or neither.
Example 19 g (t )  3 t  1
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.
HW #62 pg 39-41 (31-49 odd, 59-65 odd, 73-85 odd, 95-100 all)
Essential Question: How do you find the domain and range of a function algebraically? How do you
determine if a function is even or odd?
Students will write a summary describing domain and range of a function and include how to determine if a function is
even or odd.