Question

Sketch the graph of the function.\(g(x)=4^{x}\)

Short answer

The graph of \(g(x) = 4^{x}\) is an upward-sloping curve that passes through the y-intercept (0,1), starts near the x-axis for negative x values and rises steeply as x becomes larger. This is typical behavior for an exponential function with a base greater than 1, showing exponential growth.

Step-by-step solution

Recognize the Function Type

The function \(g(x)=4^{x}\) is an exponential function, which has a base of 4. Exponential functions have the form \(f(x) = a^{x}\), where 'a' is a positive real number, not equal to 1.

Identify Key Attributes of the Function

1) Y-intercept: The y-intercept of any exponential function \(f(x) = a^{x}\) is always 1, as substituting x=0 into the function will always result in a value of 1 (since any number to the power of 0 equals 1). Hence, for this function, the y-intercept is (0,1).\n 2) As x tends towards positive infinity: The value of the function \(4^{x}\) tends to infinity.\n 3) As x tends towards negative infinity: The value of the function \(4^{x}\) tends to 0.

Sketch the Graph

1) Start by marking the y-intercept (0,1) on the graph.\n 2) Draw a curve that starts near the x-axis (but never touches it) on the left side of the graph, goes through the y-intercept, and then rises steeply on the right-hand side.\n 3) Note the key characteristic of exponential growth here, which is that as x increases, the function's value is growing at a faster and faster rate.

Key concepts

Exponential Function

The term 'exponential function' refers to a powerful mathematical concept utilized to describe growth or decay processes. When we talk about an exponential function, we typically mean an equation of the form \( f(x) = a^{x} \), where \( a \) is a positive real number, and \( x \) is the exponent. In our example, \( g(x)=4^{x} \) is an exponential function with a base of 4. This base number is crucial because it dictates how the function will behave—whether it will increase or decrease, and at what rate. For instance, if the base is greater than 1, the function will exhibit exponential growth as it is in our case with the base 4.

Understanding the nature of exponential functions is vital in many fields, from finance, where it can represent compound interest, to biology, for modeling population growth. The exponential function is a hallmark of continuously consistent growth or decay and can dramatically illustrate how quickly values can rise or fall as the variable \( x \) changes.

Y-Intercept

The y-intercept of a graph is a point where the curve intersects the y-axis. This is an essential attribute because it provides a starting point for drawing the graph and understanding its behavior. For exponential functions, finding the y-intercept is surprisingly uncomplicated. Regardless of the base \( a \) of the exponential function \( f(x) = a^{x} \), as long as \( a \) is positive and not equal to 1, the y-intercept will always be at \( f(0) = a^{0} \), which simplifies to 1 since any non-zero number raised to the power of 0 equals 1. This is why the y-intercept of our example function \( g(x) = 4^{x} \) is at the point (0,1).

Knowing this characteristic allows students to quickly plot an initial point on the graph and provides a reliable reference for further understanding the graph's behavior and shape. The y-intercept is a crucial stepping stone in the journey of plotting any function's graph.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input (or 'x' value) gets very large in the positive or negative direction. For exponential functions, this characteristic is particularly notable. In our function \( g(x) = 4^{x} \), as \( x \) approaches positive infinity, the value of the function increases without bound, ‘exploding’ to infinity. Conversely, as \( x \) approaches negative infinity, the function doesn’t continue downward forever. Instead, it asymptotically approaches the x-axis, never actually touching it, but getting infinitely close – heading towards 0.

This behavior gives the graph of an exponential function its distinctive 'J' shape when the base is greater than one. Visualizing this concept is vital as it illustrates the unbounded growth to the right and the leveling off to the left. A practical interpretation of asymptotic behavior in real-world applications, such as physics or finance, helps explain situations with limiting factors – for example, the maximum capacity that a population can reach given its environment.