Problem 28 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$h(x)=-5(3)^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(h(x)=-5 \cdot 3^x\) starts at the y-intercept point (0,-5). Since the function is a negative exponential, the graph decreases rapidly from (0, -5) and then approaches the x-axis as 'x' increases.

Step 1: Identify Basic Form

Recognize that the equation \(h(x)=-5 \cdot 3^x\) is in the form of \(y=ab^x\). Here, 'a' is -5 and 'b' is 3.

Step 2: Determine Y-Intercept

The y-intercept is found when 'x' equals 0. Plugging 'x' as 0 into the function equation, \(h(0) = -5 \cdot 3^0 = -5\), we find that the y-intercept is -5.

Step 3: Identify Horizontal Asymptote

The graph never touches the horizontal line \(y = 0\), but it gets arbitrarily close as 'x' tends to infinity. This line is called a horizontal asymptote.

Step 4: Determine the Behavior and Sketch the Graph

Since the 'a' value is negative, the usual growth pattern of an exponential function is reflected across the x-axis. We have a decreasing type exponential function. Start at the intercept and draw a curve that approaches the line \( y = 0\) as \( x \rightarrow \infty \) and the line \( y = -5 \) becomes an upper bound as \( x \rightarrow -\infty\).

Key Concepts

Graph Sketching

Graph sketching involves drawing a rough graph of a function by understanding its key features. To sketch the graph of an exponential function like \( h(x) = -5 \, (3)^x \), we start by identifying important points and behaviors.

• Identify the Equation Form: Recognize the function is in the form \( y = ab^x \). Here, \( a = -5 \) and \( b = 3 \).
• Determine the Y-Intercept: Plugging \( x = 0 \) into the equation gives \( h(0) = -5 \times 3^0 = -5 \). So, the graph crosses the y-axis at -5.

Graph sketching is a crucial skill because it helps visualize how values of \( x \) affect the function. When sketching, note that this graph reflects across the x-axis because of the negative sign in front of 5. Always label key points like the y-intercept and mark the direction they approach horizontal asymptotes.

Horizontal Asymptote

Horizontal asymptotes offer insight into the behavior of a graph at extreme values of \( x \). With the function \( h(x) = -5 (3)^x \), this graph has a horizontal asymptote at \( y = 0 \).

• Understanding Asymptotic Behavior: As \( x \) becomes very large, \( 3^x \) increases exponentially, but the negative sign before 5 causes \( h(x) \) to decrease and get very close to zero.
• Graph Never Touches Asymptote: Despite approaching \( y = 0 \), the graph will never actually touch this line.

This concept is key in exponential functions, as it helps predict the end behavior of a graph. Identifying horizontal asymptotes is essential for defining the sketch boundaries visually.

Exponential Decay

Exponential decay describes a process where quantities reduce at a consistent percentage rate over time. The function \( h(x) = -5 \times 3^x \) demonstrates exponential decay when reflected due to the negative coefficient.

• Reflecting Across the X-axis: Typically, an exponential function like \( b = 3 \) would imply growth. However, the negative sign flips the growth to decay.
• Visualizing Decay: As \( x \) increases, \( h(x) \) decreases towards the horizontal asymptote at \( y = 0 \).

Exponential decay is widely observed in real-life scenarios such as radioactive decay or depreciation of value. It's important to distinguish when a function is increasing versus decreasing to correctly interpret associated applications.