Question

draw a rough sketch of the exponential function y = 10(1/2)^x. growth or decay? y - intercept: asymptote?

Explanation:

Step1: Determine growth/decay

The general form of an exponential function is $y = a(b)^x$. When $0 < b< 1$, it is a decay - function. Here $b=\frac{1}{2}$ and $0<\frac{1}{2}<1$, so it is a decay function.

Step2: Find y - intercept

To find the y - intercept, set $x = 0$. Substitute $x = 0$ into $y = 10(\frac{1}{2})^x$. We know that any non - zero number to the power of 0 is 1, so $y=10(\frac{1}{2})^0=10\times1 = 10$. The y - intercept is 10.

Step3: Find asymptote

As $x
ightarrow+\infty$, $(\frac{1}{2})^x
ightarrow0$. So $y = 10(\frac{1}{2})^x
ightarrow0$. The horizontal asymptote is $y = 0$.

Step4: Sketch the function

Plot the y - intercept at the point $(0,10)$. Since it is a decay function with an asymptote $y = 0$, the graph will approach the x - axis (the line $y = 0$) as $x$ increases. The graph will be above the x - axis for all real values of $x$ and will decrease as $x$ moves from left to right.

Answer:

Growth or Decay: Decay
Y - intercept: 10
Asymptote: $y = 0$
For the sketch: Plot the point $(0,10)$ and draw a curve that decreases and approaches the x - axis as $x$ increases.