Question

Sketch the graph of the function.\(y=e^{0.2 x}\)

Short answer

The graph is a curve that starts from the bottom left, passes through (0,1), and then rises to the top right. As \(x\) increases, \(y\) also increases exponentially.

Step-by-step solution

Identify the Base and the Exponent

The equation given is \(y=e^{0.2 x}\). This is an exponential growth function where \(e\) is the base (approximately equal to \(2.71828\)) and \(0.2 x\) is the exponent. This means the function grows exponentially as \(x\) increases.

Create a Table of Values

To sketch the graph, we will need some points. Let's choose some \(x\)-values and calculate the corresponding \(y\)-values. For \(x = -2, -1, 0, 1, 2\), we can plug these into the function to get \(y = e^{(0.2*-2)}, e^{(0.2*-1)}, e^{0}, e^{0.2*1}, e^{0.2*2}\) respectively.

Plot the Values

With these points, we can now start plotting our graph. We mark the x-y pairs from the table on the coordinate system.

Draw the Curve

The final step is to sketch the curve of the function that passes through these points. As x increases, y will also increase, so we can see the typical curve of the exponential function.

Key concepts

Exponential Growth

Exponential growth is a concept fundamental to understanding certain functions in mathematics and applications in the real world, such as population growth, compound interest, and even certain types of technological advancements. An exponential growth function is characterized by the variable being in the exponent, leading to the value of the function increasing at a rate proportional to its current value.

For the function in our exercise, we consider the formula of the form (y = e^{kx}), where e is a mathematical constant approximately equal to 2.71828, k is a constant that determines the growth rate, and x represents the independent variable. In the example (y=e^{0.2x}), the positive coefficient '0.2' indicates that as the value of x increases, the function grows at an exponential rate.

Table of Values

A table of values is a systematic way to calculate and organize pairs of numbers that represent the input and output of a function. It is particularly useful for sketching graphs of functions. To create a table of values for the function (y=e^{0.2x}), we typically select a range of x-values, sometimes including negative numbers, zero, and positive numbers to get a sense of the function's behavior across its domain.

For each selected x-value, we compute the corresponding y-value by substituting x into the function. For instance, when x=0, we get y=e^0, which simplifies to 1 as any number raised to the zeroth power is 1. By calculating and plotting multiple points, we get an adequate representation of the function's overall shape and direction of growth.

Sketching Graphs

Sketching the graph of an exponential function helps us visualize its behavior. Once we have our table of values, we plot the corresponding points on a coordinate system. It's crucial to pay attention to the scale, particularly for exponential functions, because they can increase rapidly.

After plotting the points, we draw a smooth curve that passes through them. This curve will reveal the general trend of the exponential function. For exponential growth functions, the graph typically rises steeply as it moves to the right (toward positive x-values). The left side of the graph tends to approach the x-axis but never touches it, illustrating the principle that the output of an exponential growth function never becomes zero but can get infinitely close to it.

Exponentials with Base e

In mathematics, exponentials with base e are so common that they are given their own name: natural exponentials. The number e is an irrational number known as Euler's number, and it's one of the most important constants in mathematics, playing a crucial role in calculus, complex numbers, and many areas of applied mathematics.

The function (y=e^{0.2x}) is an example of a natural exponential function. Because e is the base, the function models situations where growth occurs continuously rather than at discrete intervals. This is why natural exponentials are often used to model natural phenomena such as radioactive decay and continuous compounding of interest. The exponent, '0.2x' in this case, dictates how quickly the function grows or decays over time, with larger exponents corresponding to faster changes.