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Chapter 12: Problem 63

Explain why the graph of an exponential function \(y=b^{x}\) contains the point \((1, b)\)

Short Answer

Expert verified
The graph of \( y = b^x \) passes through \((1, b)\) because substituting \( x = 1 \) yields \( y = b \).

Step by step solution

01

Identify the Function

We are given the exponential function \( y = b^x \). This function has a base \( b \) raised to the power of \( x \).
02

Substitute the x-value in the Function

To find the y-coordinate when \( x = 1 \), we substitute \( x = 1 \) into the function: \( y = b^1 \).
03

Simplify the Exponent

Since \( b^1 = b \), the equation simplifies to \( y = b \).
04

Determine the Point

From the simplification, when \( x = 1 \), \( y = b \). Therefore, the graph of the function passes through the point \((1, b)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph of a Function
To understand the graph of a function, especially an exponential one, it's important to know what a graph represents. A graph of a function visually depicts all the possible outputs, or
y-values
, for every possible input or
x-value
. It allows you to see how the function behaves as x changes. Exponential functions, like
y=b^x
, have a distinctive curve that increases quickly if the base
b
is greater than 1.
The shape of the graph depends heavily on the value of
b
.
  • If
    b > 1
    , the graph rises steeply.
  • If
    0 < b < 1
    , the graph falls steeply, indicating a decaying process.
This behavior is crucial for interpreting the graph and understanding what happens as
x
changes, particularly when analyzing points like
(1, b)
, which tells us the value of the base itself.
Base of an Exponential Function
The base of an exponential function is a fundamental component that shapes the entire character of the curve. In the equation
y = b^x
,
b
is the base. The choice of base impacts the steepness and direction of the curve.
This function implies that for any x-value, the output is the base
b
raised to the power of that
x-value
.
  • A greater base (>1) means the curve rises quickly, representing exponential growth, such as population growth or compound interest.
  • A smaller base (<1) indicates decay, helpful in scenarios like radioactive decay or cooling.
When looking at the point
(1, b)
, by substituting
x=1
, the value of the function is simply
b
. This confirms the importance of the base as it's directly the y-coordinate at that specific x-value. It shows how substitution in the function gives us meaningful insights about the graph.
Substitution in Equations
Substitution in equations is a pivotal concept, especially in a mathematical function analysis. In the context of the exponential function
y = b^x
, substituting a specific value of
x
helps find the corresponding
y-value
.
For instance, by substituting
x = 1
into the function, you replace
x
with the given number, which means solving for
y = b^1
.
  • The calculation becomes straightforward because any number raised to the power of one is simply the number itself, hence
    y = b
    .
  • This simplification through substitution helps find specific points on the graph easily.
Understanding substitution is critical for solving equations quickly and analyzing graph behavior. It's this technique that clarifies why certain points, such as
(1, b)
, are part of the exponential function graph. This straightforward approach allows simplification in the understanding of complex mathematical graphs.

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