An exponential function is given by the equation y=3x . Using the points x
and x+1
, show that the y-values increase by a factor of 3 between any two points separated by x2−x1=1 .(4 points)
Given the exponential function y=3x , we can find the y-values at points x
and x+1
as follows:
At point x
: y1=3x
At point x+1
: y2=3(x+1)=3x+3
To show that the y-values increase by a factor of 3 between these two points, we need to compare the ratio of y2 to y1:
y2/y1 = (3x+3)/(3x)
= (3x/3) + (3/3)
= x + 1
Since x2-x1=1 , the ratio y2/y1 simplifies to:
y2/y1 = x + 1
This shows that the y-values increase by a factor of 3 between any two points separated by x2−x1=1 .