write a cosine function that has a midline of ( y = 3 ), an amplitude of 5 and a period of ( \frac{pi}{2}…
write a cosine function that has a midline of ( y = 3 ), an amplitude of 5 and a period of ( \frac{pi}{2} ).\nanswer\n( f(x) = )
Answer
Answer:
$f(x)=5\cos(4x)+3$
Explanation:
Step1: Recall the general form of cosine function
The general form of a cosine function is $y = A\cos(Bx - C)+D$.
Step2: Determine the values of parameters
- Amplitude $|A|$: Given amplitude is $5$, so $A = 5$.
- Mid - line $y = D$: Given mid - line $y = 3$, so $D=3$.
- Period formula $T=\frac{2\pi}{|B|}$: Given $T = \frac{\pi}{2}$. Using the formula $\frac{\pi}{2}=\frac{2\pi}{|B|}$, solve for $B$. Cross - multiply gives $\pi|B|=4\pi$, so $|B| = 4$. We can take $B = 4$ (assuming no phase shift, so $C = 0$).
Substitute $A = 5$, $B = 4$, $C = 0$, $D = 3$ into the general form $y = A\cos(Bx - C)+D$, we get $f(x)=5\cos(4x)+3$.