Problem 47 - Algebra 1: Concepts and Skills

Question

Graph the exponential function. (Lesson 8.3) $$ y=3^{x} $$

Step-by-Step Solution

Answer

The graph of the function \( y = 3^x \) is an increasing exponential curve passing through the point (0, 1).

Step 1: Identify the type of the graph

The first step is to identify the type of graph. Here, the function to be graphed is \( y = 3^x \), which is an exponential function with base 3, so, the graph will be an increasing exponential curve passing through the point (0, 1).

Step 2: Create a table of values

To better plot the function, let's select values and calculate their corresponding \( y \) values. Let's take -2, -1, 0, 1, 2 as \( x \) values. Remember that any number to the power of zero equals one. So when \( x = 0 \), \( y = 3^0 = 1 \). Compute the rest of \( y \) values correspondingly.

Step 3: Plotting the points and drawing the curve

After having our table with pairs of (x, y), plot these coordinates on the graph. Connect the points to form the curve, make sure the graph is smooth and continuous. The graph should increase rapidly as we move to the right from the y-axis because of the exponential growth.

Key Concepts

Graphing Exponential Functions

When it comes to graphing functions, understanding exponential functions is key. An exponential function like \( y = 3^x \) possesses unique characteristics. Unlike linear functions that plot a straight line, exponential functions display a curve that either rises or falls sharply. This means the value of \( y \) will increase or decrease rapidly depending on the base.
Remember, for functions expressed as \( y = b^x \), the base \( b \) dictates the direction and steepness of the graph.

• If \( b > 1 \), the graph displays exponential growth.
• If \( 0 < b < 1 \), the graph demonstrates an exponential decay.

For the function \( y = 3^x \), because the base (3) is greater than 1, its graph will represent exponential growth. It will be an increasing curve that passes through the point (0, 1). This is because any number raised to the power of zero is always one.

Exponential Growth

Exponential growth occurs when the rate of growth of a mathematical function is proportional to its current value. In simpler terms, as the value of \( x \) increases, the function \( y = 3^x \) grows at a rate that becomes increasingly rapid.
To visualize this, you plot several points along the curve:

• When \( x = 0 \), \( y = 3^0 = 1 \).
• When \( x = 1 \), \( y = 3^1 = 3 \).
• When \( x = 2 \), \( y = 3^2 = 9 \).
• When \( x = 3 \), \( y = 3^3 = 27 \).

Notice how the y-values jump significantly as the x-values increase by just one step. This steepening curve is typical of exponential growth, highlighting how quick and substantial the changes can occur. By plotting these points and connecting them, one can appreciate how dramatically the function grows.

Base of an Exponential Function

The base of an exponential function is a fundamental component, determining the nature of the function’s graph. For the function \( y = 3^x \), the base is 3. The base tells us the factor by which the function changes for each one-unit increase in \( x \).
Here's a breakdown of what the base signifies:

• A base greater than 1 results in an increasing or growing curve, known as exponential growth.
• A base between 0 and 1 results in a decreasing or decaying curve, known as exponential decay.

For our function, the base 3 means that for every increment by 1 in \( x \), the value of \( y \) is multiplied by 3. This consistent multiplication factor is what creates the steep rise or fall in the graph. Understanding the base's role is crucial for anticipating how the function behaves as \( x \) changes.