Math Problem Statement
Find a function of the form y = A sin(kx) + C or y = A cos(kx) + C whose graph matches the function shown.
Solution
From the image, the problem is to determine a function of the form:
that matches the graph shown. Here's how we solve this:
Step-by-step Analysis:
-
Determine the amplitude :
- The amplitude is the distance from the centerline to the peak (or trough).
- From the graph, the peak is at and the trough is at . Hence, .
-
Determine the vertical shift :
- The centerline is midway between the peak and the trough. This value is , as the centerline is along .
-
Determine the period and :
- The period of a sine or cosine function is given by:
- From the graph, one complete cycle occurs from to . Thus, the period is , and:
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Choose sine or cosine:
- At , the graph starts at and rises, which matches the sine function. Therefore, we use:
Final Answer:
The equation of the function is:
Would you like a detailed explanation of any step? 😊
Related Questions:
- What is the difference between sine and cosine graphs?
- How do you calculate the period of a sine or cosine function?
- What is the significance of the vertical shift ?
- How do you graph a sine function from its equation?
- How can we verify if a function matches a given graph?
Tip:
To easily identify , check how many cycles fit into and use .
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Graphing Sine and Cosine Functions
Amplitude
Period
Vertical Shift
Formulas
y = A sin(kx) + C
y = A cos(kx) + C
Period = 2π / k
Theorems
Properties of Sine and Cosine Graphs
Suitable Grade Level
Grades 10-12