Problem 52 - Algebra 1: Concepts and Skills

Question

Use a calculator to investigate the effects of a and b on the graph of \(y=a b^{x}\) In the same viewing rectangle, graph \(y=\left(\frac{1}{2}\right)^{x}, y=\left(\frac{1}{4}\right)^{x},\) and \(y=\left(\frac{1}{8}\right)^{x}\) How does a decrease in the value of \(b\) affect the graph of \(y=a b^{x}\) when \(0

Step-by-Step Solution

Answer

In the exponential function \(y=a b^{x}\), when \(0

Step 1: Graphing different functions

Start off by graphing the three functions: \(y=\left(\frac{1}{2}\right)^{x}, y=\left(\frac{1}{4}\right)^{x}\), and \(y=\left(\frac{1}{8}\right)^{x}\) using a calculator or a graphing software. Ensure that all three graphs are plotted on the same viewing rectangle for easy comparison.

Step 2: Observation of different graphs

Pay careful attention to the steepness of the curves as well as their horizontal asymptotes (if applicable). This will give you an idea about the pace of decrease in the function values with increasing x, which depends on the value of 'b'.

Step 3: Deciphering the relationship

The smaller the value of \(b\) in the form \(y=a b^{x}\) (where \(0

Key Concepts

Graph Transformations

When working with exponential functions like \( y = a b^x \), graph transformations play a pivotal role in understanding how the function behaves visually. A transformation refers to the changes made to the graph of a function, like shifting, stretching, compressing, or reflecting.

• **Vertical Stretch/Compression**: The value of \( a \) determines the vertical stretch or compression of the graph. If \( a > 1 \), the graph will stretch away from the x-axis, making it appear taller. If \( 0 < a < 1 \), it compresses, making it shorter.
• **Horizontal Stretch/Compression**: The base \( b \) influences the rate at which the graph either increases or decreases. In cases where \( 0 < b < 1 \), the function shows a decreasing trend as \( x \) increases.
• **Reflections**: Adjusting \( a \) to a negative value reflects the graph across the x-axis, thereby changing the direction in which it grows or decays.

Understanding these transformations will help you predict how graph changes with variations in \( a \) and \( b \). This is crucial as you investigate the behavior of functions in math and real-world applications.

Exponents

Exponents are at the heart of exponential functions. In \( y = a b^x \), \( x \) is the exponent and its role is essential as it determines the power to which the base \( b \) is raised. This power has significant implications:

• **Growth and Decay**: When \( b > 1 \), \( b^x \) represents growth, becoming larger as \( x \) increases. Conversely, \( 0 < b < 1 \) implies decay as the function decreases.
• **Exponential Increase Rate**: The larger the base \( b \), the faster the increase (if \( b > 1 \)), or the slower the decrease (if \( 0 < b < 1 \)).
• **Fractional Bases**: The exercise focuses on fractional bases \( \left( \frac{1}{2}, \frac{1}{4}, \frac{1}{8} \right) \). Here, smaller fractions result in slower decay, leading to gentler slopes closer to the x-axis, showcasing a prime example of exponential decay.

Mastering the role of exponents helps in anticipating how changes in \( x \) affect function values and graphically represent exponential functions.

Function Behavior

The behavior of exponential functions can reveal much about how they interact with different variables. Observing the function \( y = a b^x \), particularly when \( 0 < b < 1 \), provides insight into the following aspects:

• **Steepness and Asymptotic Nature**: As \( b \) decreases, the graph becomes less steep, tending more towards the x-axis as \( x \) increases. The curve never quite reaches the x-axis, demonstrating the asymptotic nature common in exponential decay.
• **Quadrant Occupation**: These functions primarily occupy the first quadrant (positive \( x \) and positive \( y \) values) when \( a > 0 \). They reflect to the third quadrant when \( a \) is negative.
• **Long-term Trends**: Over larger values of \( x \), smaller bases \( b \) lead to slower decreases, resulting in a lingering presence near, but above, the x-axis. This long-tailed behavior is significant in modeling scenarios that exhibit slow decay, such as radioactive decay or population declines in biology.

By understanding these characteristics of function behavior, you gain the ability to use exponential functions to model and analyze real-world phenomena effectively.

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Other exercises in this chapter

Problem 51

Copy and complete the statement using \(\). \((6 \cdot 3)^{3} ? 6 \cdot 3 \cdot 3\)

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Problem 52

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Problem 52

You buy 6 bagels and 8 donuts for a total of \(8.60. Then you decide to buy 3 extra bagels and 3 extra donuts for a total of \)3.75. How much did each bagel and

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Problem 52

Simplify the expression. Use only positive exponents. $$ \left(\frac{2 m^{3} n^{4}}{3 m n}\right)^{3} $$

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Key Concepts

Graph Transformations

When working with exponential functions like \( y = a b^x \), graph transformations play a pivotal role in understanding how the function behaves visually. A transformation refers to the changes made to the graph of a function, like shifting, stretching, compressing, or reflecting.

• **Vertical Stretch/Compression**: The value of \( a \) determines the vertical stretch or compression of the graph. If \( a > 1 \), the graph will stretch away from the x-axis, making it appear taller. If \( 0 < a < 1 \), it compresses, making it shorter.
• **Horizontal Stretch/Compression**: The base \( b \) influences the rate at which the graph either increases or decreases. In cases where \( 0 < b < 1 \), the function shows a decreasing trend as \( x \) increases.
• **Reflections**: Adjusting \( a \) to a negative value reflects the graph across the x-axis, thereby changing the direction in which it grows or decays.

Understanding these transformations will help you predict how graph changes with variations in \( a \) and \( b \). This is crucial as you investigate the behavior of functions in math and real-world applications.

Exponents

Exponents are at the heart of exponential functions. In \( y = a b^x \), \( x \) is the exponent and its role is essential as it determines the power to which the base \( b \) is raised. This power has significant implications:

• **Growth and Decay**: When \( b > 1 \), \( b^x \) represents growth, becoming larger as \( x \) increases. Conversely, \( 0 < b < 1 \) implies decay as the function decreases.
• **Exponential Increase Rate**: The larger the base \( b \), the faster the increase (if \( b > 1 \)), or the slower the decrease (if \( 0 < b < 1 \)).
• **Fractional Bases**: The exercise focuses on fractional bases \( \left( \frac{1}{2}, \frac{1}{4}, \frac{1}{8} \right) \). Here, smaller fractions result in slower decay, leading to gentler slopes closer to the x-axis, showcasing a prime example of exponential decay.

Mastering the role of exponents helps in anticipating how changes in \( x \) affect function values and graphically represent exponential functions.

Function Behavior

The behavior of exponential functions can reveal much about how they interact with different variables. Observing the function \( y = a b^x \), particularly when \( 0 < b < 1 \), provides insight into the following aspects:

• **Steepness and Asymptotic Nature**: As \( b \) decreases, the graph becomes less steep, tending more towards the x-axis as \( x \) increases. The curve never quite reaches the x-axis, demonstrating the asymptotic nature common in exponential decay.
• **Quadrant Occupation**: These functions primarily occupy the first quadrant (positive \( x \) and positive \( y \) values) when \( a > 0 \). They reflect to the third quadrant when \( a \) is negative.
• **Long-term Trends**: Over larger values of \( x \), smaller bases \( b \) lead to slower decreases, resulting in a lingering presence near, but above, the x-axis. This long-tailed behavior is significant in modeling scenarios that exhibit slow decay, such as radioactive decay or population declines in biology.

By understanding these characteristics of function behavior, you gain the ability to use exponential functions to model and analyze real-world phenomena effectively.