Problem 18 - Algebra 2

Question

Sketch the graph of each function. Then state the function's domain and range. $$ y=2(3)^{x} $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((0, \infty)\).

Step 1: Identify the type of function

The function given is of the form \(y = a \cdot b^x\), which is called an exponential function. Here, the base is \(b = 3\) and the coefficient is \(a = 2\). Exponential functions have characteristic shapes and are crucial to understand before graphing.

Step 2: determine exponential function properties

Exponential functions have specific properties: they grow steadily by a consistent rate and never touch the x-axis. Since \(a = 2\) is positive and \(b > 1\), this function will have rapid growth as \(x\) increases.

Step 3: Calculate key points

Select a few values of \(x\) to calculate corresponding \(y\) values. For example, when \(x = 0\), \(y = 2(3)^0 = 2\). When \(x = 1\), \(y = 2(3)^1 = 6\). When \(x = -1\), \(y = 2(3)^{-1} = \frac{2}{3}\), and so on.

Step 4: Plot key points on Cartesian plane

Plot the calculated \(x, y\) pairs on graph paper or a graphing utility. For this function, you might plot points such as (0, 2), (1, 6), and (-1, 2/3) to help visualize.

Step 5: Draw the exponential curve

Connect the plotted points with a smooth, continuous curve. Ensure the curve reflects the exponential growth: rising steeply to the right and flattening out (approaching the x-axis but never touching) to the left.

Step 6: Determine the Domain

The domain of any exponential function \(y = a \cdot b^x\) is all real numbers, written as \((-\infty, \infty)\) because \(x\) can take any real value.

Step 7: Determine the Range

For \(y = 2(3)^x\), the range is all positive numbers \((0, \infty)\) because the function outputs are always positive, regardless of the input.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions might seem tricky but becomes intuitive with practice. They follow a rather unique shape on a graph: one that climbs steeply and doesn't flatten out but instead approaches an axis asymptotically. An exponential function, such as the one given, follows the general form \( y = a \cdot b^x \), where "a" determines the vertical stretch or compression, and "b" is the base of the exponential curve.

To plot an exponential graph like \( y = 2(3)^x \), start by selecting some values for "x," like \(-1, 0,\) and \(1\). These restrained choices help understand the general behavior. Evaluate the function for these integers:

• For \(x = 0\), \(y = 2(3)^0 = 2\).
• For \(x = 1\), \(y = 2(3)^1 = 6\).
• For \(x = -1\), \(y = 2(3)^{-1}=\frac{2}{3}\).

With these points, plot them onto a graph, ensuring that you mark the coordinate axes. Connect them with a smooth curve rising to the right and approaching the x-axis on the left, indicating rapid growth. The plotted curve should never touch the x-axis, as exponential functions only approach it asymptotically.

Domain and Range

Understanding the domain and range is vital for interpreting the behavior of exponential functions. The domain of an exponential function like \( y = 2(3)^x \) includes all real numbers. In simpler terms, the x-value can be any real number without restriction, thus giving the domain

• Domain: \((-\infty, \infty)\).

For the range, things differ slightly. An exponential function with a positive base and positive a, like \( y = 2(3)^x \), always yields positive outputs. As the x-value moves to negative infinity, the function's output heads toward zero but never hits it.

• Range: \( (0, \infty) \).

This indicates that exponential functions will output positive values only, regardless of what x-value is substituted into the function.

Properties of Exponential Functions

Exponential functions carry distinctive properties worth noting. They describe situations with consistent multiplicative growth or decay. For example, the function \( y = 2(3)^x \) represents **exponential growth**. This is because the base of the exponent, "3," exceeds one.

Some properties of exponential functions you should remember include:

• They never touch the x-axis—exponential curves display asymptotic behavior, becoming infinitely close to the axis but never intersecting it.
• Functions like \( y = a \cdot b^x \) grow rapidly if \( b > 1 \); if \( 0 < b < 1 \), they decay.
• Values of "a" influence the amplitude, stretching, or compressing the curve vertically.

These attributes are essential as they help predict an exponential function's behavior visually and numerically.