Problem 7 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

Yes, the graph of the function \(y=2^{x}\) contains the point \((0,1)\), as substituting \(x=0\) into the function yields \(y=1\) which is the y coordinate of the given point.

Step 1: Substitute the x coordinate

Take the x coordinate of the point \((0,1)\) and substitute it into the equation. It becomes \(y=2^{0}\).

Step 2: Simplify to find y

In accordance with the rule of exponents that anything raise to power 0 is 1, \(2^{0}\) simplifies to 1. This gives us \(y = 1\).

Step 3: Compare the y coordinate

After simplification, the y value gotten is 1 which happens to be the y coordinate of the given point. This indicates that the point is on the graph.

Key Concepts

Coordinate Plane

The coordinate plane is the canvas on which we can visually represent mathematical concepts and functions. Think of it as a map that mathematicians use to find the location of points, lines, and curves. It's composed of two perpendicular number lines: the horizontal axis (commonly called the x-axis) and the vertical axis (the y-axis). These axes intersect at a point known as the origin, which has the coordinates \(0,0\).

When dealing with functions, such as exponential functions, the coordinate plane allows us to plot points that represent the output of the function for specific inputs, thus creating a graph. For example, if we were to plot the function \(y=2^x\), points like \(0,1\), which is the focus of our exercise, help us analyze the behavior of the function and check for correctness in our calculations.

Exponential Functions

Exponential functions are characterized by the presence of an exponent in the function's formula where the variable is in the exponent's position. The general form is \(y=ab^x\), where \(a\) is not zero (it's the initial value), \(b\) is a positive constant (the base of the exponential function), and \(x\) is the exponent. In our exercise, \(2\) is the base and \(x\) is the variable exponent.

What's fascinating about exponential functions is their growth rate. Depending on the base, they can illustrate rapid increases or decreases (if the function defines exponential decay). This is why you often see exponential functions used to model phenomena such as population growth or radioactive decay. Our function \(y=2^x\) is a classic example, where values of \(x\) increase, and thus \(y\) grows quite rapidly.

Exponents Rules

Exponents rules are the guidelines that make dealing with exponential expressions more manageable. One of the fundamental rules is that any nonzero number raised to the power of 0 equals 1. This is why in the exercise, \(2^0=1\). Let's break down a few more rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\), meaning if you multiply two powers of the same base, you add the exponents.
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\), indicating that when you divide two powers of the same base, you subtract the exponents.
• Power of a Power: \(\left(a^m\right)^n = a^{m \ast n}\), which tells us that when you raise a power to another power, you multiply the exponents.

These rules make the simplification of exponential expressions efficient and permit the connection between algebraic manipulation and the representation on a coordinate plane.