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When sketching the graph of an exponential function like \(y = \frac{1}{2}e^{x}\), understanding the general shape of exponential curves is essential. Exponential functions have a characteristic smooth and continuously increasing curve. For a basic exponential function \(y = e^x\), the graph starts just above the x-axis and curves upward steeply to the right. When modifying this with a constant multiplier, such as \(\frac{1}{2}\) in our function, the shape still rises exponentially but at a slower rate due to the scaling effect of the constant.

To sketch this specific function, acknowledge the low y-intercept of 0.5, which pulls the starting point of the curve lower compared to the basic form. As x increases, you will see the curve gradually become steeper, growing towards positive infinity but never touching the x-axis. As x tends toward negative infinity, the curve flattens, getting closer to zero but not touching the x-axis. When sketching, it is useful to plot the y-intercept first, then draw the curve respecting these behaviors.

The y-intercept of a function is a critical point where the graph crosses the y-axis. It provides a reference for sketching the rest of the graph. For the function \(y = \frac{1}{2} e^x\), this occurs when \(x = 0\). Since the exponential term becomes \(e^0 = 1\), the calculation simplifies to \(y = \frac{1}{2} \times 1 = \frac{1}{2}\).

This means the graph crosses the y-axis at the point (0, 0.5). This point is a starting guide for drawing the curve, indicating that at \(x = 0\), the value of y will be 0.5. By plotting this point first, we establish a base reference for the ascending exponential curve that describes the function's shape in expected graphical behavior. Remember, the y-intercept helps to anchor the graph, adding clarity and orientation at its base starting line.

In graphing exponential functions, identifying horizontal asymptotes helps understand limits toward infinity. A horizontal asymptote is a horizontal line that a graph approaches but never actually reaches or crosses at extreme values of x. Here, for the function \(y = \frac{1}{2}e^{x}\), the horizontal asymptote is the x-axis, or \(y = 0\).

As x becomes very large positively (approaching positive infinity), the graph shoots upwards, distancing itself from the asymptote. When x becomes very negative (approaching negative infinity), the value of \(y\) in \(e^x\) gets closer to 0, making \(y = \frac{1}{2}\times 0 = 0\). Hence, the graph approaches the x-axis.

• This illustrates that no matter how large or small x becomes, the graph of the function will get infinitely close to the x-axis without ever touching it.
• The x-axis being the asymptote provides a boundary line, guiding the behavior of the function at extremes and ensuring continuity in how the graph is drawn.

Understanding horizontal asymptotes enhances comprehension of the function's end behavior and frames how the graph intersects with other components like y-intercepts and curve shape.