Problem 56 - Algebra 1

Question

Does the graph of the function contain the point \((0,1) ?\) $$y=60^{x}$$

Step-by-Step Solution

Answer

Yes, the point \((0,1)\) is on the graph of the function \(y = 60^x\).

Step 1: Understanding the problem

We need to determine whether or not the point \((0, 1)\) lies on the graph of the given function \(y = 60^x\). For this, we substitute the x-coordinate from the point into the equation and check if we get the corresponding y-coordinate as the output.

Step 2: Substituting the value of x

We substitute \(x = 0\) into the function \(y = 60^x\), which gives us \(y = 60^0\). According to the laws of exponents, any number (except 0) raised to the power of 0 is 1. So, \(y = 1\).

Step 3: Comparing the result with the y-coordinate

After substituting \(x = 0\) in the function, we got \(y = 1\) which is the same as the y-coordinate of the given point. Therefore, the point \((0,1)\) lies on the graph of the function \(y = 60^x\).

Key Concepts

Coordinate Plane

When graphing functions, the coordinate plane plays a foundational role. It consists of two axes: the horizontal axis (usually labeled as the x-axis) and the vertical axis (labeled as the y-axis). These axes intersect at a point called the origin, denoted as (0,0).

Every point on the plane can be represented by an ordered pair (x, y), where 'x' denotes the position along the horizontal axis and 'y' signifies the position along the vertical axis. In the context of exponential functions like \(y = 60^x\), graphing involves plotting points that represent the function's output for different input values along the x-axis and connecting these points to visualize the growth rate of the function.

Laws of Exponents

Exponential functions frequently require applying the laws of exponents to simplify and evaluate expressions. One of the foundational laws is that any nonzero number raised to the power of 0 equals 1, symbolically expressed as \(a^0 = 1\) provided that \(a eq 0\).

This rule is crucial when dealing with functions like \(y = 60^x\), especially at the point where \(x = 0\). Remembering this law allows for quick evaluation of the function at any point where the exponent is 0, significantly simplifying the function evaluation process.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. In the case of \(y = 60^x\), the function is defined for all real numbers 'x' and involves raising the base (60) to the power of 'x'.

To determine if a point lies on the graph of an exponential function, we substitute the x-coordinate into the equation and see if the resulting y-value matches the y-coordinate of the point in question. For the point (0,1), substituting 0 for x in accordance with the laws of exponents yields \(60^0 = 1\). The output matches the y-coordinate of the point, confirming that the point indeed lies on the graph of the function.