Question
Problem 1
Answer the following questions and graph the function
- Amplitude:
- Period:
- Vertical Shift:
- Starting point
Answer the following questions and graph the function
Ask by Wang Gonzalez. in the United States
Jan 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
- Amplitude: 2
- Period:
- X-scale: 1
- Vertical Shift: 0
- Horizontal Shift:
- Starting Point:
- Ending Point:
Solution
Certainly! Let’s analyze and graph the function
Function Analysis
The given function is in the form:
where:
-
= Amplitude -
affects the period ( ) -
= Horizontal Shift -
= Vertical Shift
For
Answers
-
Amplitude:
-
Period:
X-scale:
(Since, there’s no horizontal scaling applied.) -
Vertical Shift:
Horizontal Shift:
(The graph is shiftedunits to the right.) -
Starting Point:
Ending Point:
(One complete cycle starts atand ends at .)
Graph of
Below is a sketch of the graph over one period:
y
|
2 | * *
| * * * *
| * * * *
0 |----*-----------*-------*-----------*---- x
| * * * *
-2| * * * *
| * *
|
Key Features:
- Amplitude: The graph oscillates between
and . - Period: One complete cycle from
to . - Horizontal Shift: The standard cosine graph
is shifted units to the right. - Vertical Shift: There is no vertical shift; the center line is at
. - Starting Point: The maximum point starts at
, . - Ending Point: The graph returns to the maximum point at
, .
Summary
| Feature | Value |
|---|---|
| Amplitude | 2 |
| Period |
|
| X-scale | 1 |
| Vertical Shift | 0 |
| Horizontal Shift |
|
| Starting Point |
|
| Ending Point |
|
Feel free to plot these points and features on a coordinate plane to visualize the graph accurately!
Answered by UpStudy AI and reviewed by a Professional Tutor
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Explain 
Simplify this solution Extra Insights
For the function
-
Amplitude: The amplitude is the coefficient of the cosine function, which is
. This means the function will oscillate between and . -
Period: The standard period of the cosine function is
. Since there’s no horizontal scaling factor, the period remains . Xscale can be considered if you’re charting on a typical 4x unit system. -
Vertical Shift: There’s no vertical shift in this function, so it is
. Horizontal Shift: The expression indicates a horizontal shift of units to the right. -
Starting point: At
, the cosine function starts at its maximum value, so the starting point is . -
Ending Point: To find the ending point, we can determine where the function has completed one full cycle. The function returns to its original position at
, which gives us the ending point .
So, when you graph
