Problem 9 - Algebra 1: Concepts and Skills

Question

Tell whether the graph of the function contains the point \((0,1) .\) Explain your answer. $$y=2(3)^{x}$$

Step-by-Step Solution

Answer

No, the graph of the function \(y=2(3)^{x}\) does not contain the point (0,1) because \(y\) equals 2 when \(x\) equals 0, not 1.

Step 1: Substituting x-coordinate into the function

Using the given function \(y=2(3)^{x}\), replace \(x\) with 0 to get \(y=2(3)^{0}\).

Step 2: Simplifying the equation

We know that any number (except 0) raised to the power of 0 equals 1. So, \(3^{0} = 1\). Hence, the equation simplifies to \(y=2*1\).

Step 3: Calculate y-coordinate

Continue to compute the value of \(y\). The calculation \(2*1 = 2\) gives the result \(y=2\).

Key Concepts

Exponential Functions

Exponential functions represent one of the fundamental concepts in mathematics and have a wide range of applications in various fields such as finance, physics, and biology. In essence, an exponential function is written in the form of f(x) = ab^x, where a is a constant term, b is the base and must be a positive real number different from 1, and x is the exponent, which can take any real value.

An important characteristic of exponential functions is their rate of growth. When the base b is greater than 1, the function grows exponentially as x increases, which is known as exponential growth. Conversely, if 0 < b < 1, the function exhibits exponential decay. This behavior of exponential functions is crucial to understanding phenomena like population growth or radioactive decay.

In the function y = 2(3)^x, the constant 2 is the initial value when x is zero, and 3 is the base of the exponential function. As x increases, the value of y will rise rapidly due to the nature of exponential growth.

Coordinates in a Graph

Graphing is a valuable skill in mathematics, helping visualize functions and understand their behaviors. When it comes to plotting, coordinates play a crucial role. A coordinate is composed of an ordered pair of numbers, typically written as (x, y), where x represents the position along the horizontal axis, and y indicates the position along the vertical axis.

Graphing the function y = 2(3)^x involves plotting several points on the graph to reveal the shape of the curve. Each point corresponds to a specific input-output pair, where the input (or the x-value) is plugged into the function, and the corresponding output (or the y-value) is calculated. If a point lies on the graph, its coordinates satisfy the function. For instance, if we were examining whether the point (0, 1) is on the graph of y = 2(3)^x, we would need to determine whether a y-value of 1 corresponds to an x-value of 0.

Function Evaluation

Evaluating functions is a key skill in mathematics, as it allows us to determine the output of a function given a specific input. This process often requires substituting a number for the variable in the function and simplifying. The function in our exercise is evaluated at x = 0. To perform the evaluation, one substitutes 0 for x in the expression to find the corresponding y-value.

For the exponential function y = 2(3)^x, evaluating the function at x = 0 is straightforward due to the exponent property that any non-zero base raised to the power of 0 equals 1. After substituting x with 0, we simplify to find that y equals 2, indicating that the point (0, 2) is on the graph. If a different x-value results in the y-value of 1, then the point (0, 1) would not be on the graph of this particular function. It's important to accurately perform function evaluation for precise graphing and understanding functions.