Answer: y = 7cos(0.4π x) - 3
Step-by-step explanation:
The equation of a cosine function is: y = A cos(Bx - C) + D where
- Amplitude (A) is the distance from the midline to the max (or min)
- Period (P) is the length of one cosine wave --> P = 2π/B
- Phase Shift (C/B) is the horizontal distance shifted from the y-axis
- Midline (D) is the vertical shift. It is equal distance from the max and min
Midline (D) = -3
(-1.25, -3) is given as a point on the midline. We only need the y-value.
Horizontal stretch (B) = 0.4π
The max is located at (0,4) and also at (5, 4). Thus the period (length of one wave) is 5 units.
[tex]P=\dfrac{2\pi}{B}\qquad \rightarrow \qquad 5=\dfrac{2\pi}{B}\qquad \rightarrow \qquad B=\dfrac{2}{5}\pi[/tex] → B = 0.4π
Phase Shift (C) = 0
The max is on the y-axis so there is no horizontal shift.
Amplitude (A) = 7
The distance from the midline to the max is: A = 4 - (-3) = 7
Equation
Input A = 7, B = 0.4π, C = 0, and D = -3 into the cosine equation.
y = A cos(Bx - C) + D
y = 7cos(0.4π x - 0) - 3
y = 7cos(0.4π x) - 3
