QuestionFind an equation for the graph shown to the right.
Type the equation in the form or
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression.)
Studdy Solution
STEP 1
What is this asking?
We need to find the equation of a wave-like graph that's shown in a picture, and the equation should look like or .
Watch out!
Don't mix up the *amplitude*, *frequency*, and *vertical shift*.
Also, remember that cosine starts at the **peak** and sine starts at the **midline**.
STEP 2
1. Find the vertical shift.
2. Find the amplitude.
3. Find the angular frequency.
4. Choose sine or cosine and write the equation.
STEP 3
The **vertical shift** is the middle height of the graph.
It's like the average height of the highest and lowest points.
STEP 4
The highest point is and the lowest point is .
Let's find the middle!
STEP 5
So, the **midline** is at .
This means our vertical shift is **zero**!
STEP 6
The **amplitude** is the distance from the **midline** to the highest point (or the midline to the lowest point--it's the same distance).
STEP 7
We found that the **midline** is at , and the highest point is .
STEP 8
The distance between and is .
So, our amplitude is !
STEP 9
The graph repeats itself every **3** units along the -axis.
This is called the **period**, sometimes written as .
So, .
STEP 10
The **angular frequency** is related to the period by the formula:
STEP 11
Let's plug in our **period** : So, our angular frequency is !
STEP 12
The graph starts at a **maximum** value when .
This is just like the cosine function!
If it started at the **midline**, we'd use sine.
STEP 13
We found: * * *
STEP 14
Putting it all together, our equation is: We can simplify because adding zero doesn't change anything!
STEP 15
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