Graphs of Functions §3.7
Fall 2013 - Math 1010
(Math 1010)
M 1010 §3.7
1 / 21
Roadmap
I
§3.7 Graphs of basic functions.
I
§3.7 Transformations of graphs of functions.
(Math 1010)
M 1010 §3.7
2 / 21
Recall Point-Plotting
Recall the Point-Plotting method from §3.2, used to graph equations:
1. Make a table with several coordinates (x, y ).
2. Plot the points on the rectangular coordinate frame.
3. Connect the points with a smooth curve. In some cases, use a line.
The graph of a function f is the same using the x-coordinate and f (x) as
the y -coordinate.
(Math 1010)
M 1010 §3.7
3 / 21
§3.7 Exercise # 2
Graph
f (x) = 1 − 3x
Determine its domain and range.
(Math 1010)
M 1010 §3.7
4 / 21
§3.7 Exercise # 4
Graph
1
f (x) = x 2 − 1
4
Determine its domain and range.
(Math 1010)
M 1010 §3.7
5 / 21
§3.7 - Basic Functions
A list of basic functions. From these we will make transformations, and
modify the basic graph.
I
A constant function, f (x) = c where c is a real number.
I
The identity function, f (x) = x.
I
I
The absolute value function, f (x) = |x|.
√
The square-root function, f (x) = x.
I
Power-of-x: The quadratic function: f (x) = x 2 .
I
Power-of-x: The cubic function: f (x) = x 3 .
Sample graphs follow, as well as the domains and ranges of each function.
(Math 1010)
M 1010 §3.7
6 / 21
§3.7 Constant f (x) = c
6 y
4
2
x
−1
1
2
3
4
Example: f (x) = 3, domain: −∞ < x < ∞, range: y = 3
(Math 1010)
M 1010 §3.7
7 / 21
§3.7 Identity f (x) = x
y
2
1
x
−2
−1
1
2
−1
−2
Example: f (x) = x, domain: −∞ < x < ∞, range: −∞ < y < ∞
(Math 1010)
M 1010 §3.7
8 / 21
§3.7 Absolute Value f (x) = |x|
y
2
1
x
−2
−1
1
2
−1
−2
Example: f (x) = |x|, domain: −∞ < x < ∞, range: y ≥ 0
(Math 1010)
M 1010 §3.7
9 / 21
§3.7 Square Root f (x) =
√
x
3 y
2
1
x
1
Example: f (x) =
(Math 1010)
√
2
3
4
5
x, domain: x ≥ 0, range: y ≥ 0
M 1010 §3.7
10 / 21
§3.7 Quadratic Function f (x) = x 2
y
4
3
2
1
x
−2
−1
1
2
Example: f (x) = x 2 , domain: −∞ < x < ∞, range: y ≥ 0
(Math 1010)
M 1010 §3.7
11 / 21
§3.7 Cubic Function f (x) = x 3
y
5
x
−8 −6 −4 −2
2
4
6
8
−5
Example: f (x) = x 3 , domain: −∞ < x < ∞, range: −∞ < y < ∞
(Math 1010)
M 1010 §3.7
12 / 21
§3.7 - Transformations
Transformations come in types: shifts, scales, and relections. We will give
examples of shifts and reflections. The basic functions described here will
undergo shifts and relfections.
(Math 1010)
M 1010 §3.7
13 / 21
§3.7 Shifts: Quadratic Function f (x) = x 2
y
4
3
2
1
x
−2
−1
1
2
g (x) = x 2 + 1 Domain: same as f , Range: y ≥ 1;
h(x) = (x + 1)2 Domain and Range: same f
(Math 1010)
M 1010 §3.7
14 / 21
§3.7 - Shifts
Adding or subtracting a number to every x value or f (x) value shifts the
graph in a direction. Let c be a positive real number, andlet y = f (x) be a
graph.
1. Upwards vertical shifts: g (x) = f (x) + c
2. Downwards vertical shifts: g (x) = f (x) − c
3. Left horizontal shifts: h(x) = f (x + c)
4. Right horizontal shifts: h(x) = f (x − c).
(Math 1010)
M 1010 §3.7
15 / 21
§3.7 - Reflections: Absolute value f (x) = |x|
y
2
1
x
−2
−1
1
2
−1
−2
Example: f (x) = |x|, domain: −∞ < x < ∞, range: y ≥ 0
g (x) = −|x|, domain: same as f , range: y ≤ 0.
(Math 1010)
M 1010 §3.7
16 / 21
§3.7 - Reflections: Cubic f (x) = x 3
y
5
x
−8 −6 −4 −2
2
4
6
8
−5
Example: f (x) = x 3 , domain: −∞ < x < ∞, range: −∞ < y < ∞
g (x) = −x 3 , domain, range: same as f
(Math 1010)
M 1010 §3.7
17 / 21
§3.7 - Reflections: Quadratic f (x) = x 2
4 y
2
x
−2
−1
1
2
−2
−4
Example: f (x) = x 2 , domain: −∞ < x < ∞, range: y ≥ 0
g (x) = −x 2 , domain: same as f , range: y ≤ 0
(Math 1010)
M 1010 §3.7
18 / 21
§3.7 - Relfections: Square Root f (x) =
√
x
y
2
x
−4
−2
2
4
−2
√
Example:
f
(x)
=
x, domain: x ≥ 0, range: y ≥ 0
√
g (x) = −x, domain: x ≤ 0 range: same as f
(Math 1010)
M 1010 §3.7
19 / 21
§3.7 - Reflections
Reflection across one of the coordinate axes occurs when multiplying a
variable by −1.
1. Reflection across the x-axis: g (x) = −f (x)
2. Reflection across the y -axis: g (x) = f (−x)
(Math 1010)
M 1010 §3.7
20 / 21
Assignment
Assignment:
1. Exercises from §3.7 due ????.
2. Exam # 1: Chapter 3 & Cumulative Chapters 1 - 2, October 2
(Math 1010)
M 1010 §3.7
21 / 21