Problem 54 - Algebra 1

Question

Does the graph of the function contain the point \((0,1) ?\) $$y=4^{x}$$

Step-by-Step Solution

Answer

Yes, the graph of the function \(y = 4^x\) does contain the point \((0,1)\).

Step 1: Identify and Understand the Given Function and Point

The given function is exponential of base 4, denoted by \(y = 4^x\). An exponential function has the property that for any base \(b > 0\) and \(b \neq 1\), \(b^0 = 1\). Our goal is to determine if the point \((0,1)\) lies on the graph of the function \(y = 4^x\). The x-coordinate of the given point is 0, and the y-coordinate is 1.

Step 2: Evaluate the Function at the X-coordinate of the Given Point

We need to substitute \(x = 0\) into the given function to find \(y=4^0\). According to the property of exponential functions, \(4^0 = 1\). Therefore, \(y(0) = 1\).

Step 3: Compare Y-value with Given Y-coordinate

The y-value obtained from the function by substituting \(x = 0\) is 1, which is the same as the y-coordinate of the given point (0,1). This means the point \((0,1)\) is indeed on the graph of the function \(y = 4^x\).

Key Concepts

Evaluating Exponential Functions

Exponential functions are essential in representing growth or decay processes, such as population growth or radioactive decay. The process of evaluating exponential functions involves substituting the given x-values into the equation and simplifying to find the corresponding y-values. For instance, to evaluate the exponential function given by the equation y = 4^x at x = 0, we use the exponent rule that states any non-zero number raised to the power of zero equals one. Thus, we get y = 4^0 = 1. This rule is crucial in understanding why the point \text{\((0,1)\)} is always on the graph of non-trivial exponential functions. When students face difficulties in evaluating exponential functions, remembering the exponent rules and ensuring they correctly substitute values can greatly improve their understanding and accuracy.

Evaluating exponential functions accurately is critical to understanding their behavior and leads seamlessly into the next concept: graphing these functions.

Graphing Exponential Functions

Graphing is a powerful tool to visualize the behavior of exponential functions. To graph an exponential function, such as y = 4^x, we can plot several points by choosing different x-values and computing the corresponding y-values. Typically, we'd start with integral values of x such as -2, -1, 0, 1, and 2. Graphing the point \text{\((0,1)\)} is particularly important, as this represents the y-intercept of all exponential functions with the form y = b^x where b is the base and x is the exponent. The point \text{\((0,1)\)} signifies that no matter what value of x we choose, when we come back to zero, the y-value will always be one, assuming the base b is positive and not equal to one. The graph typically shows a rapid increase or decrease away from the y-intercept depending on whether the base is greater than one (growth) or between zero and one (decay).

Understanding the shape and characteristics of the graph helps students to predict the function's growth rate and to find key features such as intercepts and asymptotes.

Exponential Function Properties

Exponential functions come with a set of distinct properties that set them apart from other types of functions. Recognizing these properties is critical for a proper understanding and manipulation of such functions in various mathematical contexts. Here are a few fundamental properties:

• Non-Zero Base: The base of an exponential function must be a positive number other than one for it to be defined as exponential.
• Positive Growth: If the base is greater than one, the function will represent growth as we move from left to right on the graph.
• Decay: If the base is between zero and one (excluding zero itself), the function will represent decay.
• Horizontal Asymptote: Exponential functions have a horizontal asymptote, typically the x-axis (y=0), which the graph approaches but never crosses or touches.
• Domain and Range: An exponential function has a domain of all real numbers, and its range is all positive real numbers.
• Continuous and Smooth: The graph of an exponential function is continuous (no breaks) and smooth (no sharp turns or corners).

For students working with exponential functions, grasping these properties can aid in solving different types of problems more effectively, such as understanding long-term behavior of the function or applying transformations, like reflections and shifts, to the graph.