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How to Find the Midline of a Sinusoidal Function from a Graph
Mathematics
Grade 10 (High School)
Question Content
What is the equation of the midline of the sinusoidal function? Enter your answer in the box.
Correct Answer
y = 1
Detailed Solution Steps
1
Step 1: Recall that the midline of a sinusoidal function is the horizontal line that lies exactly halfway between the maximum and minimum y-values of the wave.
2
Step 2: Identify the maximum y-value from the graph: the peaks of the wave reach y = 4.
3
Step 3: Identify the minimum y-value from the graph: the troughs of the wave reach y = -2.
4
Step 4: Calculate the midline by finding the average of the maximum and minimum values: Midline = (Maximum + Minimum) / 2 = (4 + (-2)) / 2 = 2 / 2 = 1.
5
Step 5: Write the equation of the horizontal midline, which is y = 1.
Knowledge Points Involved
1
Sinusoidal Function Midline
The midline of a sinusoidal (sine or cosine) function is a horizontal line that represents the central axis of the oscillating wave. It is calculated as the average of the maximum and minimum output values of the function, and its equation is in the form y = k where k is the midline value. It is a key component of the general sinusoidal function form y = A sin(B(x - C)) + k or y = A cos(B(x - C)) + k, where k directly defines the midline.
2
Reading Key Features of Graphs
When analyzing graphs of functions, identifying critical points like maximums (peaks) and minimums (troughs) is essential for determining key properties. For sinusoidal graphs, these points are evenly spaced around the midline, and their values are used to calculate amplitude, midline, and vertical shifts.
3
Average of Two Values
The average of two numbers is found by adding the numbers together and dividing the sum by 2. This calculation is used to find the midpoint between two values on a number line, which applies directly to finding the midline between the highest and lowest points of a sinusoidal wave.
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