Find equations of the line, the exponential curve, and the power curve that each pass through the points (1, 3) and (2, 12).

Question

1.  Find equations of the line, the exponential curve, and the power curve that each pass through the points (1, 3) and (2, 12).

1. Graph the equations in the same coordinate plane.
2. Describe what happens when the equations are used as models to predict ​y​-values for ​x​-values greater than 2.

show all work please

Answer 1

Line: y-3=(12-3)/(2-1)(x-1)=9(x-1); y=9x-9+3=9x-6 (red)

Exponential: y=abˣ; 3=ab; 12=ab²; 12/3=ab²/ab=b, b=4; 3=4a or 12=16a, a=¾; y=¾(4ˣ)=3(4ˣ⁻¹) or 3(4^(x-1)) (blue)

Power: y=ax²; 3=a, 12=4a; y=3x² (green)

(a)

(b) For x>2 the exponential curve models to the highest value, the power curve coming next highest and the line coming lowest.