Problem 2 - Algebra 2

Question

Graph each function. $$ y=3(10)^{x} $$

Step-by-Step Solution

Answer

The graph of the function is an exponential growth curve that crosses the y-axis at the point (0,3) and contains the point (1,30).

Step 1: Identify the Form of the Equation

In the given equation, \(y = 3(10)^x\), '3' is a scaling factor, '10' is the base, 'x' is the exponent. This is an exponential function and the base of the exponential is 10.

Step 2: Determine the y-intercept

The y-intercept of the function is the output value when \(x = 0\). Substituting \(x = 0\) in the equation, we get \(y = 3(10)^0 = 3\). This tells us that the graph crosses the y-axis at the point (0,3).

Step 3: Determine another point on the graph

Generally for exponential functions, it will be helpful to find the value of 'y' when \(x = 1\). Substituting \(x = 1\) in the equation, we get \(y = 3(10)^1 = 30\). So, the point (1,30) is on the graph.

Step 4: Plot the points and sketch the graph

Draw a graph with the x-axis and y-axis. Plot the y-intercept at (0,3) and the point at (1,30), and draw a curve passing through the points, rising rapidly from left to right - as is characteristic of exponential growth functions.

Key Concepts

graphing exponential functions

Exponential functions like the one described by the equation \( y = 3(10)^x \) are characterized by a constant base raised to a variable exponent. The graph of an exponential function can be understood by plotting points and observing the curve it forms. In our case, the function's base is 10, which indicates rapid growth. To graph this function:

• First, determine key points by substituting values for \( x \).


• For example, when \( x = 0 \), \( y = 3 \times 10^0 = 3 \).


• When \( x = 1 \), \( y = 3 \times 10^1 = 30 \).

Plotting these points (0,3) and (1,30) on a graph provides a starting framework. Exponential graphs like these tend to rise very quickly. Draw a smooth curve that passes through the points and continues to rise rapidly to the right. The distinct characteristic of an exponential graph is its dramatic upward curve, differentiating it from linear or quadratic graphs.

y-intercept of exponential functions

Determining the y-intercept of an exponential function is a vital step in graphing and understanding the function's behavior. The y-intercept is where the graph crosses the y-axis, which happens when \( x = 0 \).
In the function \( y = 3(10)^x \), substituting \( x = 0 \) gives:

• \( y = 3 \times 10^0 = 3 \)

Hence, the y-intercept is at the point (0,3). This provides a clear anchor point on the graph.
The y-intercept represents the starting output value of the function, and is especially useful when graphing because it acts as a reference point. Identifying the y-intercept assists in visualizing how the function begins on the graph before it rises steeply in exponential growth. This intercept is critical in sketching an accurate representation of the exponential curve.

characteristics of exponential growth

Exponential growth is a key characteristic of the function \( y = 3(10)^x \). Unique features of exponential growth differentiate it from other types of growth such as linear or polynomial.

• As the base is greater than 1 (in this case, 10), the function will show rapid increase as \( x \) increases.


• The rate of growth accelerates continuously, creating a curve that rises sharply.


• Every equal interval of \( x \) results in multiplying the preceding y-value by the base.

This growth pattern implies that for each step to the right on the x-axis, the corresponding y-value is significantly larger than before. In practical applications, exponential growth models things like population growth, financial investments, or even certain aspects of natural phenomena. Understanding these characteristics helps in predicting how the values will change and visualizing the graph's sharp upward trajectory as a distinctive exponential growth pattern.