Graph each function. $\text{ƒ}\left(x\right)=(1/6{)}^{-x}$

Verified step by step guidance

1. Rewrite the function \( f(x) = \left( \frac{1}{6} \right)^{-x} \) using the property of exponents: \( a^{-b} = \frac{1}{a^b} \). This gives \( f(x) = 6^x \).
2. Identify the basic form of the function \( f(x) = 6^x \), which is an exponential function with base 6.
3. Determine the key characteristics of the exponential function: it passes through the point (0,1), has a horizontal asymptote at \( y = 0 \), and increases rapidly as \( x \) increases.
4. Plot the point (0,1) on the graph, as this is where the function intersects the y-axis.
5. Sketch the curve of the function, starting from the point (0,1) and moving upwards to the right, showing the rapid increase as \( x \) becomes larger, while approaching the x-axis as \( x \) becomes more negative.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In the given function f(x) = (1/6)^{-x}, the base is 1/6, and the exponent is negative, which indicates that the function will decrease as x increases. Understanding the behavior of exponential functions is crucial for graphing them accurately.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize the behavior of a function. For exponential functions, key points can be found by substituting values for 'x' and calculating 'f(x)'. Additionally, recognizing asymptotic behavior, where the graph approaches a horizontal line but never touches it, is essential for accurately representing the function's characteristics.

Transformations of Functions

Transformations of functions refer to changes made to the basic form of a function that affect its graph. In the case of f(x) = (1/6)^{-x}, the negative exponent indicates a reflection across the y-axis, which alters the typical growth of the function. Understanding these transformations helps in predicting how the graph will look compared to the parent function, allowing for more accurate graphing.