What is the equation of the graph described below in the discussion?

Answer 1

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A sin curve varies evenly either side of the x-axis; the graph varies between -3.5 and +5.5; the mid-point of -3.5 and +5.5 is (5.5 - 3.5) ÷ 2 = 1

→ it has been shifted up the y-axis by 1 unit

→ k = 1

As the curve has been shifted up the y-axis by 1 unit, it varies from (-3.5 -1) = -4.5 to (5.5 - 1) = +4.5

The sin curve varies between -1 and +1

→ it has been stretched by 4.5

→ a = 4.5

Using the points with y-coordinate of 1, the curve is negative between the first two and positive between the second two; thus half the sine curve, (ie between 0 and π radians) is between two of them.

Between the first two of (-5.495, 1) and (0.785, 1) is 0.785 - -5.495 = 6.28 = 2 × 3.14 = 2π

Similarly between the second two of (0.785, 1) and (7.065, 1) is 7.065 - 0.785 = 6.28 = 2 × 3.14 = 2π

→ the x-axis has been stretched by 2π ÷ π = 2

→ b = 2

The sine curve starts increasing from zero, thus (0.785, 1) was originally at the origin (0, 0)

→ the curve has been shifted right by 0.785 = π/4

→ h = π/4

Thus the curve has equation:

y = 4.5 sin (2(x - π/4))