Problem 37 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=6^{x} $$

Step-by-Step Solution

Answer

Graph of \( f(x)=6^{x} \) rises steeply after \( x=0 \), passing through (0,1) with \( y \)-axis as asymptote.

Step 1: Understand the Exponential Function

The function we have is an exponential function, given by \( f(x) = 6^x \). Here, the base is 6, and the exponent is the variable \( x \). This means as \( x \) increases, \( 6^x \) will increase rapidly. The graph of this function will show continuous growth and will pass through the point (0,1) since any positive number to the power of 0 equals 1.

Step 2: Create a Table of Values

Select a set of \( x \) values to compute \( f(x) = 6^x \) for each. Starting with basic values, calculate:- \( x = -2, f(x) = 6^{-2} = \frac{1}{36} \)- \( x = -1, f(x) = 6^{-1} = \frac{1}{6} \)- \( x = 0, f(x) = 6^0 = 1 \)- \( x = 1, f(x) = 6^1 = 6 \)- \( x = 2, f(x) = 6^2 = 36 \)This table enables us to plot specific points on a graph.

Step 3: Plot the Points on the Graph

Using the table from Step 2, plot the points on a coordinate graph: - Plot \((-2, \frac{1}{36})\), \((-1, \frac{1}{6})\), \((0, 1)\), \((1, 6)\), \((2, 36)\).These points represent the function at various \( x \) values, showing the rapid increase of \( 6^x \) as \( x \) becomes positive and its decrease as \( x \) is negative.

Step 4: Draw the Exponential Curve

Connect the plotted points with a smooth curve. Make sure the curve passes through all points plotted and keeps tending towards the x-axis for large negative \( x \) values, illustrating the asymptotic behavior.The curve will rise sharply as \( x \) increases past zero.

Step 5: Check Graph Characteristics

Ensure the graph reflects the characteristics of the exponential function:- It passes through (0,1), confirming the y-intercept.- It has an asymptote along the x-axis, as the function approaches zero but never touches it for negative \( x \).- The graph increases rapidly as \( x \) becomes positive.

Key Concepts

Graphing Techniques

Graphing exponential functions, like the one we have in this example with the equation \( f(x) = 6^x \), involves several useful strategies. Start by recognizing the core features of the function. The base, 6 in this case, indicates that the function will grow rapidly as \( x \) increases. Knowing this helps in planning your graph layout.

Begin by creating a table of values, selecting both negative and positive \( x \) values to cover various sections of the graph and gain a complete picture of the function’s behavior.

• For \( x = -2, -1, 0, 1, 2 \), compute \( f(x) = 6^x \) as \( \frac{1}{36}, \frac{1}{6}, 1, 6, 36 \)
• This gives you precise points: \((-2, \frac{1}{36}), (-1, \frac{1}{6}), (0, 1), (1, 6), (2, 36)\)

Plot these points on a coordinate plane carefully. Ensure that you connect them with a smooth curve. Unlike linear functions, exponential curves will bend and demonstrate rapid changes, especially as \( x \) becomes larger.

Asymptotic Behavior

Asymptotic behavior refers to the way a graph approaches a line but never quite touches it. For the exponential function \( f(x) = 6^x \), its asymptote is the x-axis, meaning the curve will approach y = 0 but never actually intersect it. This feature becomes evident as \( x \) takes on large negative values.

Consider when \( x = -2 \). You get \( f(x) = \frac{1}{36} \), a very small positive number, showing the curve getting closer to zero. As \( x \) becomes even more negative, \( 6^x \) becomes smaller yet remains positive. This is a signature behavior of exponential functions, contrasting with polynomial graphs that may cross their asymptotes.

It's useful to remember the visual behavior:

• The curve never touches or crosses the x-axis.
• This helps define the "limit" as \( x \) approaches negative infinity: \( \lim\limits_{x \to -\infty} 6^x = 0 \).

Function Properties

Understanding function properties of \( f(x) = 6^x \) illuminates how exponential functions operate. One primary feature is continuous and rapid growth. The base of 6 indicates that each increase by 1 in \( x \) multiplies \( f(x) \) by 6, showcasing exponential growth.

Other key properties include:

• **Domain**: All real numbers, since you can substitute any real \( x \).
• **Range**: All positive real numbers, because \( 6^x \) can never be negative or zero.
• **Intercepts**: The y-intercept occurs at (0, 1), fitting with the identity that bases raised to the zero power are 1.
• **Monotonicity**: The function is always increasing, meaning \( f(a) < f(b) \) for \( a < b \).

The clarity in recognizing these properties allows you to predict the behavior and features of any exponential function quickly and accurately.