Sketching Exponential Functions And Finding Intersections

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In this comprehensive guide, we will delve into the process of sketching the graphs of exponential functions, specifically focusing on the function f(x) = 3^(-2-x) and the constant function g(x) = 9. Furthermore, we will explore how to determine the exact value of x where these two functions intersect, i.e., where f(x) = g(x). This exploration will not only enhance your understanding of exponential functions but also provide valuable insights into graphical analysis and equation solving.

Understanding Exponential Functions

Exponential functions are a fundamental concept in mathematics, characterized by their rapid growth or decay. The general form of an exponential function is f(x) = a^x, where a is a constant base and x is the exponent. The behavior of the function is heavily influenced by the value of the base a. If a is greater than 1, the function exhibits exponential growth, while if a is between 0 and 1, it demonstrates exponential decay.

In our case, we have the function f(x) = 3^(-2-x). This function can be rewritten as f(x) = 3^-(x+2), which further simplifies to f(x) = (1/3)^(x+2). This transformation reveals that the function is an exponential decay function with a base of 1/3. The term (x+2) in the exponent indicates a horizontal shift of the graph. Understanding these transformations is crucial for accurately sketching the graph.

On the other hand, we have the function g(x) = 9, which is a constant function. Its graph is a horizontal line at y = 9. Constant functions are straightforward to graph and serve as a good contrast to the exponential function in our analysis.

Sketching the Graph of f(x) = 3^(-2-x)

Sketching the graph of an exponential function involves identifying key features and plotting a few points to establish the curve's shape. For f(x) = 3^(-2-x), we can follow these steps:

  1. Identify the base: The base of the exponential function is 3, but due to the negative exponent and the transformation, we can rewrite it as (1/3)^(x+2). This tells us it's an exponential decay function.

  2. Determine the horizontal shift: The (x+2) term in the exponent indicates a horizontal shift of 2 units to the left. This means the graph will be shifted 2 units to the left compared to the basic exponential decay function f(x) = (1/3)^x.

  3. Find key points: To sketch the graph, we can find a few key points by plugging in different values of x. Some useful points to consider are:

    • x = -2: f(-2) = 3^(-2-(-2)) = 3^0 = 1
    • x = -3: f(-3) = 3^(-2-(-3)) = 3^1 = 3
    • x = -1: f(-1) = 3^(-2-(-1)) = 3^(-1) = 1/3
    • x = 0: f(0) = 3^(-2-0) = 3^(-2) = 1/9

  4. Plot the points and sketch the curve: Plot the points we calculated and draw a smooth curve through them. Remember that since it's an exponential decay function, the graph will approach the x-axis (y = 0) as x increases and rise sharply as x decreases. The horizontal shift we identified earlier will also be visible in the graph.

  5. Asymptote: Identify the horizontal asymptote. For exponential functions of the form f(x) = a^(x+b), the horizontal asymptote is y = 0. This line is approached but never touched by the graph as x approaches infinity.

Sketching the Graph of g(x) = 9

Sketching the graph of g(x) = 9 is much simpler. This is a constant function, meaning the value of g(x) is always 9, regardless of the value of x. The graph is a horizontal line at y = 9. To sketch it, simply draw a horizontal line across the coordinate plane that intersects the y-axis at the point (0, 9).

Finding the Intersection Point

Finding the intersection point of the two graphs means determining the x-value where f(x) = g(x). In other words, we need to solve the equation 3^(-2-x) = 9.

To solve this equation, we can follow these steps:

  1. Express both sides with the same base: Since 9 can be written as 3^2, we can rewrite the equation as 3^(-2-x) = 3^2.

  2. Equate the exponents: When the bases are the same, we can equate the exponents. This gives us the equation -2 - x = 2.

  3. Solve for x: Solve the linear equation for x:

    • -x = 2 + 2
    • -x = 4
    • x = -4

Therefore, the exact value of x where f(x) = g(x) is x = -4. This is the x-coordinate of the point where the graphs of the two functions intersect.

Verifying the Solution

Verifying the solution is a crucial step to ensure the accuracy of our answer. We can do this by substituting x = -4 back into both f(x) and g(x) and checking if the values are equal.

For f(x):

f(-4) = 3^(-2-(-4)) = 3^(-2+4) = 3^2 = 9

For g(x):

g(-4) = 9 (since it's a constant function)

Since f(-4) = g(-4) = 9, our solution x = -4 is verified. This confirms that the graphs of f(x) and g(x) intersect at the point (-4, 9).

Implications and Applications

Understanding the intersection of functions has significant implications in various fields, including:

  1. Mathematics: Solving equations and inequalities, analyzing the behavior of functions, and understanding graphical representations.

  2. Science and Engineering: Modeling physical phenomena, such as population growth, radioactive decay, and circuit analysis.

  3. Economics and Finance: Analyzing supply and demand curves, predicting market trends, and modeling financial investments.

  4. Computer Science: Algorithm design, data analysis, and machine learning.

By mastering the concepts of sketching graphs and finding intersections, you gain a powerful tool for analyzing and solving problems in a wide range of disciplines.

Conclusion

In this guide, we have explored the process of sketching the graphs of the exponential function f(x) = 3^(-2-x) and the constant function g(x) = 9. We learned how to identify key features of exponential functions, such as the base, horizontal shifts, and asymptotes, to accurately sketch their graphs. Furthermore, we determined the exact value of x where the two functions intersect by solving the equation f(x) = g(x). This involved expressing both sides of the equation with the same base, equating the exponents, and solving for x. Finally, we verified our solution by substituting it back into the original functions.

This exploration not only enhances your understanding of exponential functions and their graphs but also provides valuable skills in graphical analysis, equation solving, and problem-solving in various fields. By mastering these concepts, you can confidently tackle more complex mathematical problems and apply them to real-world applications.

Key takeaways from this guide:

  • Exponential functions have the form f(x) = a^x and exhibit rapid growth or decay depending on the base a.
  • Sketching exponential functions involves identifying key features, plotting points, and drawing a smooth curve.
  • Constant functions have the form g(x) = c and their graphs are horizontal lines.
  • Finding the intersection of two functions means solving the equation f(x) = g(x).
  • Verifying solutions is crucial to ensure accuracy.
  • Understanding the intersection of functions has significant implications in various fields.

By practicing these techniques and applying them to different problems, you can further solidify your understanding of exponential functions and their applications.