Question

Precalculus please help

Answer 1

Answer:

amplitude = 2

period = 4

midline = y = -1

equation = either $2sin(\frac{pi}{2}x+\frac{pi}{2} )-1$ or $2sin(\frac{pi}{2}(x+1) )-1$ depending on what format you're using

Step-by-step explanation:

the amplitude is the distance from the bottommost point to the topmost point

so if we use the points (0,1) and (2,-3), the min y value is -3 and the max y value is 1

so amplitude = ymax - ymin = 1 - - 3 = 4

amplitude = 4

the period is the distance it takes to complete one cycle

we have (0,1) and (4,1) which are two crest points

the distance between these is x2 - x1 = 4 - 0 = 4

period = 4

the midline is a horizontal line bisecting the sin line, which would be in the middle of its min and max y values

we already found that ymax = 1 and ymin = -3

so the midline would be y = (ymax + ymin) / 2 = (1 -3) /2 = -1

midline = y = -1

to find the sin equation, we will fill out the equation

asin(b(x-d))+c

a is the amplitude = 2

b = 2π/period = 2π/4 = π/2

c is the vertical shift. the midline of sine is originally at y = 0

since it is at y = -1 here, the vertical shift is -1 - 0 = -1

d is the horizontal shift. the middle of the upwards portion of the sine graph is originally at x = 0, but in the picture we see that it is at x = -1 so the shift is -1 - 0 = -1

a = 2, b = π/2, c = -1, d = -1

plug these values into the equation to get

asin(b(x-d))+c

2sin(π/2(x--1))+-1

=2sin(π/2(x+1))-1

=2sin(πx/2+π/2)-1

equation = either $2sin(\frac{pi}{2}x+\frac{pi}{2} )-1$ or $2sin(\frac{pi}{2}(x+1) )-1$ depending on what format you're using