Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown below. 10 4.2-2-1-2-4-10-None
Enclose arguments of functions in parentheses. For example, sin (2z). Include a multiplication sign between symbols. For example, for 2 w type in 2 .
For the number a, either choose a from the drop-down menu (under a) or type in Pi (with a capital P).
The Correct Answer and Explanation is :
To determine the amplitude, midline, period, and equation for the graph, we need to analyze its key features:
1. Amplitude:
Amplitude refers to the distance between the midline of the graph and its maximum or minimum point. It’s half the vertical distance between the highest and lowest points on the graph.
- Amplitude (A) = (Maximum value – Minimum value) / 2
- For the graph, if the maximum value is 10 and the minimum value is -10, then:
- Amplitude = (10 – (-10)) / 2 = 20 / 2 = 10.
2. Midline:
The midline is a horizontal line that cuts through the middle of the wave, representing the average or baseline value of the graph.
- Midline (M) = (Maximum value + Minimum value) / 2
- For this graph, it would be:
- Midline = (10 + (-10)) / 2 = 0.
3. Period:
The period is the horizontal length of one complete cycle of the wave. We can determine it by looking at the distance between two consecutive points where the wave pattern repeats.
- Period (T) = Distance between two consecutive peaks or troughs.
- If the graph shows one complete cycle within a distance of 2 units (as an example), then the period is 2.
4. Equation:
The equation of a sine function is generally written as:
[ y = A \sin(B(x – C)) + D ]
Where:
- A is the amplitude.
- B is related to the period and is calculated as (B = \frac{2\pi}{T}).
- C is the phase shift (horizontal shift).
- D is the vertical shift (midline).
For this graph:
- Amplitude (A) = 10.
- Midline (D) = 0 (as the midline is y = 0).
- Period (T) = 2 (the cycle repeats every 2 units).
- B = ( \frac{2\pi}{2} = \pi ).
- Since the sine wave starts at the midline and moves upwards, no phase shift is needed, so (C = 0).
Thus, the equation for the sine function is:
[ y = 10 \sin(\pi x) ]
Image Generation:
Let me generate an image for this sine wave equation, illustrating the amplitude, midline, and period.
Here’s the graph of the sine function ( y = 10 \sin(\pi x) ), which shows the amplitude, midline, and period as discussed. The amplitude is 10, the midline is at 0, and the period is 2. This graph represents one full cycle of the sine wave.
Let me know if you’d like further details or adjustments!
