Find an equation for the cosine graph, f(x) = A cos(Bx + C) + D.
Check the final answer first, then review the worked steps.
Problem
Find an equation for the cosine graph, f(x) = A cos(Bx + C) + D.
Answer
\(f(x) = 3 cos(x) + 0\)
Step-by-step solution
- Identify the Amplitude (A): The amplitude is half the distance between the maximum and minimum values of the function. The maximum value is 3 and the minimum value is -3. So, $A = \frac{3 - (-3)}{2} = \frac{6}{2} = 3$.
- Identify the Vertical Shift (D): The vertical shift is the midline of the graph. The midline is halfway between the maximum and minimum values. So, $D = \frac{3 + (-3)}{2} = \frac{0}{2} = 0$.
- Identify the Period: The period is the horizontal distance over which the graph completes one full cycle. We can observe that the graph completes one full cycle from $x=0$ to $x=2\pi$. Thus, the period is $2\pi$.
- Identify the Angular Frequency (B): The relationship between the period ($P$) and the angular frequency ($B$) is $P = \frac{2\pi}{|B|}$. Since the period is $2\pi$, we have $2\pi = \frac{2\pi}{|B|}$. This implies $|B| = 1$. We can choose $B=1$ for simplicity, as the graph appears to be a standard cosine wave in terms of its horizontal compression.
- Identify the Phase Shift (C): The standard cosine function $y = \cos(x)$ starts at its maximum value at $x=0$. This graph also starts at its maximum value (3) at $x=0$. Therefore, there is no horizontal shift, meaning $C=0$. The general form for phase shift is $-\frac{C}{B}$. Since there is no shift, $-\frac{C}{1} = 0$, which means $C=0$.
- Write the Equation: Substitute the values of A, B, C, and D into the form $f(x) = A \cos(Bx + C) + D$. We have $A=3$, $B=1$, $C=0$, and $D=0$. Therefore, the equation is $f(x) = 3 \cos(1x + 0) + 0$, which simplifies to $f(x) = 3 \cos(x)$.