Question

example: draw a rough sketch of the exponential function (y = 10(\frac{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 $b=\frac{1}{2}$, 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 a horizontal asymptote $y = 0$, the function will approach the x - axis as $x$ increases.

Answer:

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