Problem 19 - Calculus of a Single Variable

Question

Sketching a Graph In Exercises \(17-22,\) sketch the graph of the function. $$ y=e^{x}+2 $$

Step-by-Step Solution

Answer

The graph of \(y=e^{x}+2\) is the graph of \(y=e^{x}\) shifted upwards by 2 units along the y-axis. The y-intercept is at (0,3).

Step 1: Understand the basic function

The basic function here is \(y=e^{x}\). The graph of \(y=e^{x}\) is a curve that increases rapidly for positive \(x\) and approaches zero for negative \(x\). The \(y\)-intercept is at (0,1).

Step 2: Understand the transformation

The function \(y=e^{x}+2\) is a vertical transformation of the basic function \(y=e^{x}\). The '+2' in the function translates the graph 2 units upward along the y-axis. So, instead of the y-intercept being at (0,1), it will now be at (0,3).

Step 3: Sketch the graph

To sketch the graph, first draw the basic function curve described in step 1. Then, move this curve upward by 2 units, as described in step 2. Label the main points and the y-intercept. Make sure the curve still approaches zero as \(x\) goes towards negative infinity and increases rapidly as \(x\) goes towards positive infinity. This is the final graph of \(y=e^{x}+2\)

Key Concepts

Exponential Function Transformations

When we talk about transforming an exponential function, we usually mean making changes to the graph's position. An exponential function like \( y = e^x \) has a characteristic curve that rises steeply as \( x \) moves to the right and flattens out as \( x \) goes to the left. Transformations can shift, stretch, or compress this curve.

Here are a few common forms of transformations:

• Vertical Shifts: Adding or subtracting a constant will move the graph up or down.
• Horizontal Shifts: Adding or subtracting a constant to the \( x \) within the exponent shifts the graph left or right.
• Reflections: Multiplying by a negative number can flip the graph across the x-axis or y-axis.
• Scaling: Multiplying the entire function by a number can stretch or compress the graph.

For \( y = e^x + 2 \), a vertical shift is used, moving the entire curve upwards by 2 units.

Y-Intercept

The y-intercept is a crucial part of understanding and sketching graphs. It is the point where the graph crosses the y-axis. At the y-intercept, the value of \( x \) is always zero.

For the function \( y = e^x \), we substitute \( x = 0 \) to find that the y-intercept is \( (0, 1) \). The calculation is simple: \[ y = e^0 = 1 \]
However, when there's a transformation involved, such as in the function \( y = e^x + 2 \), the y-intercept shifts. Substituting \( x = 0 \) again, we get:\[ y = e^0 + 2 = 3 \]So the new y-intercept is \( (0, 3) \). Understanding where the graph touches the y-axis helps in sketching and visualizing the function's changes.

Vertical Translation

Vertical translation is a straightforward but potent transformation. It involves moving the entire graph up or down on the coordinate plane without altering its shape.

In the expression \( y = e^x + 2 \), the "+2" indicates a vertical translation. This means every point on the graph of \( y = e^x \) is moved 2 units up. The overall shape remains the same, but the position changes.

To visualize this: imagine the graph of \( y = e^x \) as a piece of paper on your desk. Vertical translation is like sliding that piece of paper upward. The critical parts remain the same, like how the curve behaves as \( x \) moves positively or negatively, yet everything on the graph is now sitting higher up.
This movement can also alter other important graph features, such as the y-intercept, which shifts from \( (0, 1) \) to \( (0, 3) \) when you add 2. Vertical translation affects how the function looks in real-world scenarios, such as in growth models and decay processes, emphasizing its fundamental role in function graphing.