The function given is:

$f(x) = \left(\frac{1}{2}\right)^x$

To sketch the graph of this function, let's analyze it step-by-step.

1. General Shape and Properties

The function $f(x) = \left(\frac{1}{2}\right)^x$ is an exponential function with a base less than 1. This means it represents exponential decay, where the value of $f(x)$ decreases as $x$ increases.

2. Key Points

Let's calculate a few key points to help with sketching the graph:

• When $x = 0$: $f(0) = \left(\frac{1}{2}\right)^0 = 1$
• When $x = 1$: $f(1) = \left(\frac{1}{2}\right)^1 = \frac{1}{2}$
• When $x = -1$: $f(-1) = \left(\frac{1}{2}\right)^{-1} = 2$
• When $x = 2$: $f(2) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$

These points provide a good guide for plotting.

3. Asymptote

Since the base is between 0 and 1, as $x \to \infty$, $f(x)$ approaches 0 but never actually reaches it. Thus, the x-axis (or $y = 0$) is a horizontal asymptote.

4. Sketching the Graph

The graph should:

• Pass through the points $(0, 1)$, $(1, \frac{1}{2})$, $(-1, 2)$, etc.
• Approach the x-axis as $x \to \infty$ (exponential decay).
• Show rapid growth for negative values of $x$, where $f(x)$ increases as $x$ becomes more negative.

Would you like me to provide a visual sketch of this graph?

Further Questions to Explore:

1. How does changing the base from $\frac{1}{2}$ to another fraction (e.g., $\frac{1}{3}$) affect the graph?
2. What happens to the graph if the base is changed to a value greater than 1 (e.g., $f(x) = 2^x$)?
3. How would the graph change if the function were multiplied by a constant, like $f(x) = 3 \left(\frac{1}{2}\right)^x$?
4. What would the graph look like if we added a constant to the function, like $f(x) = \left(\frac{1}{2}\right)^x + 1$?
5. Can you determine the equation of the line tangent to the graph at $x = 0$?

Tip: When sketching exponential decay functions, remember that the graph always gets closer to the x-axis but never actually touches it. This helps in visualizing the asymptotic behavior.