Question

Sketch the graph of the exponential equation. $$y=2^{x}$$

Short answer

The graph of the function \(y = 2^{x}\) is a curve that increases rapidly from the point (0,0), passing through the points (0,1), (1,2) and (-1,0.5).

Step-by-step solution

- Understand the exponential function

The general form of an exponential function is \(y = a^x\), where \(a\) is a positive constant. For the equation \(y = 2^x\), our basis \(a\) is 2. When \(x\) is a number, the value of \(y\) is found by raising 2 to the power of that number.

- Plot important points

There are a few important values we can calculate to draw the graph. For \(x = 0\), the output is 1, since \(2^0=1\). This gives the point (0,1). When \(x = 1\), \(2^1=2\) so the point is (1,2). And when \(x = -1\), \(2^{-1}=0.5\), so the point is (-1,0.5). It's also important to note that as \(x\) approaches negative infinity, \(y = 2^x\) approaches 0. This is because any non-zero number to the power of negative infinity is zero.

- Sketching the graph

On your graph paper, you can plot the points calculated in step 2. Now, for every time \(x\) increases by 1, \(y\) doubles. This is because we are continuously multiplying by 2 for each increase in \(x\). Hence, as \(x\) goes to infinity \(y\) becomes larger and larger (rises rapidly), and as \(x\) goes to negative infinity, \(y\) approaches 0 (but never reaches it). Thus, the graph is a curve that rises rapidly from (0,0), passing through the points (0,1), (1,2) and (-1,0.5).

- Further graph characteristics

Additional characteristics to consider when sketching your graph are that it has no x-intercepts, one y-intercept at (0,1), domain is all real numbers and range is \(y>0\).

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of the given exercise, we consider the function y = 2^x. This type of function is known for its rapid growth or decay depending on the exponent's value.

The key property of an exponential function like y = 2^x is that it increases or decreases at a rate proportional to its current value. That's why as x increases, the value of y does not simply add up linearly but doubles in this specific case. Conversely, as x becomes more negative, y halves, approaching zero without ever reaching it. This distinctive characteristic distinguishes exponential growth from other types of increase, such as linear.

Understanding the exponential function is crucial for interpreting its behavior. It governs many real-world phenomena like population growth, radioactive decay, and interest compounding, making it a vital concept in mathematics and various science fields.

Plotting Points

Plotting points is a fundamental skill in graphing that involves marking specific locations on a coordinate system. These points represent ordered pairs that correlate to the independent variable (x) and the dependent variable (y).

When plotting an exponential function like y = 2^x, choosing a range of x-values to cover various parts of the graph is vital. Typically, starting with x = 0 gives us the y-intercept. As demonstrated in our exercise, when x = 0, y = 1 since any non-zero number to the power of zero equals one. The points (1,2) and (-1,0.5) demonstrate the exponential growth and decay.

Plotting these important points on graph paper gives us a scaffold for our exponential curve. To ensure accuracy, it is helpful to plot several points and then connect them with a smooth curve that reflects the function's nature—arapid upswing for positive x and a slow descent towards zero for negative x values.

Graph Characteristics

Key Features of Exponential Graphs

Exponential graphs like the one for y = 2^x have distinct traits that define their shape and position on a coordinate grid. Some of the main characteristics you want to look for include:

• Y-intercept: The point where the graph crosses the y-axis. For y = 2^x, this is at (0,1).
• Domain: The set of all possible x-values. For an exponential function, the domain is all real numbers since we can raise 2 to any real power.
• Range: The set of all possible y-values. Here, the range includes all positive real numbers (> 0) since an exponential function with a positive base never reaches zero or becomes negative.
• Asymptote: A line that the graph approaches but never touches. In this case, the x-axis (y = 0) is a horizontal asymptote.
• End behavior: Describes what happens to the y-values as x goes to positive or negative infinity. As x approaches negative infinity, y approaches the asymptote; as x approaches positive infinity, y increases without bound.

These characteristics are crucial for understanding and sketching the complete graph of an exponential function. They help in visualizing the function's behavior across all x-values. Recognizing these features in exponential functions can also form a basis for more complex understanding, such as transformations and compound interest calculations.