use the graphing calculator to graph these functions: (y_1 = 2x); (y_2 = 2x^{2}); (y_3 = 2^{x}). after what…
use the graphing calculator to graph these functions: (y_1 = 2x); (y_2 = 2x^{2}); (y_3 = 2^{x}). after what (y)-value does the exponential function appear to surpass the linear function?
Answer
Explanation:
Step1: Set the two functions equal
Set $y_1 = 2x$ and $y_3=2^x$. We want to find when $2^x>2x$. First, set $2^x = 2x$.
Step2: Test integer - values
For $x = 1$, $y_1=2\times1 = 2$ and $y_3=2^1 = 2$, so $2^1=2\times1$. For $x = 2$, $y_1=2\times2 = 4$ and $y_3=2^2 = 4$, so $2^2=2\times2$. For $x = 3$, $y_1=2\times3 = 6$ and $y_3=2^3 = 8$, so $2^3>2\times3$. When $x = 3$, $y_1 = 6$ and $y_3 = 8$. The value of the linear function $y_1 = 2x$ and the exponential function $y_3 = 2^x$ are equal at $x = 1$ and $x = 2$, and the exponential function $y_3 = 2^x$ surpasses the linear function $y_1 = 2x$ when $x = 3$. Substituting $x = 3$ into the linear function $y_1=2x$, we get $y = 6$.
Answer:
6