Problem 62 - Algebra A Combined Function

Question

Explain why an exponential function \(y=b^{x}\) has a \(y\) -intercept of (0,1)

Step-by-Step Solution

Answer

The y-intercept is (0,1) because when \( x=0 \), \( y=b^0=1 \).

Step 1: Definition of the Exponential Function

An exponential function has the form \( y = b^x \), where \( b \) is the base and \( x \) is the exponent. For any base \( b > 0 \), such as \( b = 2, 3, \ldots \), the function describes the exponential growth or decay depending on the size of \( b \).

Step 2: Determine the Y-Intercept

The *y-intercept* of a function is the point where the graph of the function crosses the y-axis. At this point, the x-coordinate is 0.

Step 3: Substitute X=0 in the Function

Substitute \( x = 0 \) into the exponential function: \( y = b^0 \).

Step 4: Evaluate the Exponential Expression

Evaluate \( b^0 \). Any non-zero number raised to the power of 0 is 1: \( b^0 = 1 \).

Step 5: Conclude the Y-Intercept

Since \( b^0 = 1 \), when \( x = 0 \), \( y = 1 \). Thus, the y-intercept of the function \( y = b^x \) is at the point (0,1).

Key Concepts

Exponential Growth

Exponential growth is a concept where things increase rapidly over time. You can see it in nature, technology, and finance. The basic idea is that as something grows, it keeps adding on more and more, creating a curve that rises steeply as you move along the x-axis.
There are a few key points to remember about exponential growth:

• The growth rate of the function is proportional to its current value.
• In mathematical terms, this is represented as \( y = b^x \), where each increase in \( x \) results in a multiplication of \( y \) by \( b \).
• This type of growth can look like a curve that starts flat and then rises steeply.

In real life, an example could be the spread of a virus. At first, the increase in cases is slow, but soon it becomes very rapid as more and more people get infected. Similarly, compounded interest on money in a bank account shows exponential growth as the interest is added to the principal over time.

Y-Intercept

The y-intercept is an essential concept in understanding graphs. It's the point where a graph crosses the y-axis. At this crossing point, the x-value is always zero. Finding the y-intercept of a function helps you understand where your function begins on the y-axis.
In the case of exponential functions like \( y = b^x \), the y-intercept is straightforward. When you substitute \( x = 0 \) into the expression, you're effectively asking what \( y \) is when the function starts. Calculating it gives:

• \( y = b^0 \)
• And because \( b^0 = 1 \), it shows that the y-intercept is at \( (0, 1) \)

This calculation is based on the rule that any non-zero number raised to the power of zero is 1. So for the exponential function \( y = b^x \), the graph will always start with a y-intercept at \( (0, 1) \).

Mathematical Expressions

Mathematical expressions are a way of representing numbers and operations in a concise form using symbols and numbers. They help us capture mathematical ideas in a way that can be easily manipulated and solved.
Within expressions, you'll often find:

• Variables, like \( x \), which stand for unknown values.
• Numbers and constants, such as the base \( b \) in exponential functions.
• Operators, such as addition (+), subtraction (-), multiplication (\( \times \)), and division (\( \div \)).

In the context of exponential functions, mathematical expressions describe how something grows or decays exponentially. The expression \( y = b^x \) shows the relationship between the base \( b \) and the variable \( x \). It indicates that the value of \( y \) is found by raising \( b \) to the power of \( x \), allowing us to model real-world phenomena where quantities change at exponential rates. Understanding these expressions is key to analyzing and predicting patterns and behaviors.