Problem 30 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$h(x)=4\left(\frac{2}{3}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of \(h(x)=4(\frac{2}{3})^{x}\) is an exponential decay curve that intercepts the y-axis at (0,4), falls to the right and rises to the left. It has no x-intercept.

Step 1: Identify domain and range

An exponential function's domain is all real numbers, so the domain of \(h(x)=4(\frac{2}{3})^{x}\) is all real numbers (i.e., \(-\infty\) to \(+\infty\)). The range of an exponential function is always \(y>0\), so in this case, the range is \(y > 0\).

Step 2: Identify intercepts

The y-intercept is the value of the function when \(x=0\). So to find y-intercept, set \(x=0\). We get \(h(x)=4(\frac{2}{3})^{0}\). As any number to the power of 0 is 1, the y-intercept is at \(y = 4(1) = 4\). For x-intercept, set \(h(x) = 0\). In the case of \(h(x)=4(\frac{2}{3})^{x}\), there is no x-intercept, because \(h(x)=0\) has no solution for \(x\) when \(y > 0\).

Step 3: Determine end behavior

As \(x\) goes to \(+\infty\), \((\frac{2}{3})^{x}\) approaches 0, so \(h(x)=4(\frac{2}{3})^{x}\) also approaches 0. As \(x\) goes to \(-\infty\), \((\frac{2}{3})^{x}\) goes to \(+\infty\), so the function approaches \(+\infty\). The graph of our function will fall to the right and rise to the left.

Step 4: Draw the graph

Upon all the points acquired, draw the graph using the y-intercept (0,4) and the asymptotic behavior as \(x\) approaches \(-\infty\) and \(+\infty\).

Step 5: Label the graph

All graphs should be labeled with its function equation and main points. Thus, label the y-intercept and write the function equation near the graph.

Key Concepts

Exponential Growth and Decay

Exponential functions are mathematical expressions where variables appear as exponents. These functions showcase exponential growth or decay depending on the base of the exponential term. For the function \(h(x) = 4\left(\frac{2}{3}\right)^{x}\), the base \(\frac{2}{3}\) is less than 1, indicating exponential decay. This means as \(x\) increases, \(h(x)\) decreases towards zero, but never touches it.
Exponential growth occurs with bases greater than 1, where the function values increase rapidly as \(x\) increases. Here, since the base is \(\frac{2}{3}\), each step to the right on the x-axis results in smaller values of \(h(x)\), showcasing the function's decay pattern.

Domain and Range

In the context of exponential functions such as \(h(x) = 4\left(\frac{2}{3}\right)^{x}\), understanding the domain and range is crucial. The domain of any exponential function is all real numbers. This means you can plug any real number into \(x\) and get a result for \(h(x)\).

• The domain for the function \(h(x)\) is \( (-\infty, +\infty) \).
• The range, however, is limited to positive values.
• This is because the function moves closer to zero but never becomes zero, resulting in a range of \(h(x) > 0\).

Even as the base of the function is a fraction, it cannot produce zero or negative results, maintaining the range solely in positive territory.

Intercepts

Intercepts in a graph give us important points where the graph crosses the axes. For \(h(x) = 4\left(\frac{2}{3}\right)^{x}\), let's explore its y-intercept and x-intercept.

• Y-intercept: This is found by setting \(x = 0\). At \(x = 0\), \(h(x) = 4\cdot 1 = 4\). So, the y-intercept is at the point \((0, 4)\).
• X-intercept: This is found by setting \(h(x) = 0\). However, because exponential functions with positive bases never actually reach zero, \(h(x)\) does not cross the x-axis, and hence has no x-intercept.

These intercepts help define the graph's starting point and behavior concerning the axes.

End Behavior

Understanding the end behavior of \(h(x) = 4\left(\frac{2}{3}\right)^{x}\) helps visualize how the function behaves as \(x\) moves towards the extremities of the number line.
As \(x\) approaches \( +\infty \), the term \(\left(\frac{2}{3}\right)^{x}\) diminishes towards zero. Therefore, \(h(x)\) also approaches 0 from above. This means the function's graph gradually falls towards the x-axis but never actually touches it, due to the property of exponential decay.
Conversely, as \(x\) goes towards \(-\infty\), \(\left(\frac{2}{3}\right)^{x}\) increases significantly, and subsequently, \(h(x)\) approaches \(+\infty\).
This tells us that the graph will rise to the left, signifying a steady increase without bound.