Graphing Reflections Across The Y Axis Exploring F(x) = 1.5(0.5)^x

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Understanding Reflections Across the Y-Axis

A reflection across the y-axis is a transformation that creates a mirror image of a function with respect to the y-axis. In simpler terms, for every point (x, y) on the original function, there will be a corresponding point (-x, y) on the reflected function. This transformation essentially flips the graph horizontally.

Key Concepts and Principles

Before we dive into the specifics of graphing the reflection of f(x) = 1.5(0.5)^x, let's recap some essential concepts:

  1. Function Notation: A function, denoted as f(x), represents a relationship between an input x and an output f(x). In our case, f(x) = 1.5(0.5)^x is an exponential function.
  2. Exponential Functions: Exponential functions have the general form f(x) = ab^x, where a is the initial value and b is the base. The base b determines whether the function represents exponential growth (b > 1) or exponential decay (0 < b < 1).
  3. Transformations: Transformations alter the graph of a function. Common transformations include translations (shifts), reflections, stretches, and compressions. A reflection across the y-axis is a specific type of transformation.
  4. Initial Value: The initial value of a function is the y-value when x = 0. It represents the point where the graph intersects the y-axis.

Step-by-Step Guide to Graphing the Reflection

Step 1: Identify the Original Function

Our starting point is the function f(x) = 1.5(0.5)^x. This is an exponential decay function because the base (0.5) is between 0 and 1. The coefficient 1.5 represents the initial value of the function.

Step 2: Calculate the Initial Value of the Reflected Function

To find the reflection of f(x) across the y-axis, we need to determine the function g(x) such that g(x) = f(-x). This substitution effectively mirrors the graph across the y-axis. To calculate the initial value of the reflected function, we evaluate g(0):

g(0) = f(-0) = f(0)

Since f(x) = 1.5(0.5)^x, we have:

f(0) = 1.5(0.5)^0

Any number raised to the power of 0 is 1, so:

f(0) = 1.5 * 1 = 1.5

Therefore, the initial value of the reflected function g(0) is also 1.5. This means the reflected graph will intersect the y-axis at the point (0, 1.5).

Step 3: Plotting the Initial Value of the Function

Now, let's plot the initial value on the coordinate plane. The initial value of the reflected function g(x) is 1.5, which corresponds to the point (0, 1.5) on the graph. Mark this point on your graph.

Step 4: Determine the General Form of the Reflected Function

To find the general form of the reflected function g(x), we substitute -x for x in the original function f(x):

g(x) = f(-x) = 1.5(0.5)^{-x}

Using the property a^{-b} = (1/a)^b, we can rewrite the expression:

g(x) = 1.5((0.5){-1})x

Since (0.5)^{-1} = 1 / 0.5 = 2, the reflected function becomes:

g(x) = 1.5(2)^x

Notice that the base of the exponential function has changed from 0.5 to 2. This transformation has converted the exponential decay function into an exponential growth function. The reflected function g(x) = 1.5(2)^x now represents exponential growth, which is a mirror image of the original decay function across the y-axis.

Step 5: Choose Additional Points to Plot

To accurately graph the reflected function, we need to plot a few additional points. Choose some values for x and calculate the corresponding values of g(x). Here are some suggested values:

  • x = 1: g(1) = 1.5(2)^1 = 1.5 * 2 = 3
  • x = 2: g(2) = 1.5(2)^2 = 1.5 * 4 = 6
  • x = -1: g(-1) = 1.5(2)^{-1} = 1.5 * 0.5 = 0.75
  • x = -2: g(-2) = 1.5(2)^{-2} = 1.5 * 0.25 = 0.375

Now we have the following points to plot: (0, 1.5), (1, 3), (2, 6), (-1, 0.75), and (-2, 0.375).

Step 6: Plot the Points and Draw the Graph

Plot the points you calculated in Step 5 on the coordinate plane. These points will help you visualize the shape of the reflected function. Connect the points with a smooth curve to create the graph of g(x) = 1.5(2)^x. The graph should exhibit exponential growth, starting from the initial value of 1.5 and increasing rapidly as x increases.

Step 7: Compare the Original and Reflected Graphs

To further solidify your understanding, compare the graph of the reflected function g(x) = 1.5(2)^x with the original function f(x) = 1.5(0.5)^x. You should observe that:

  • The graphs are mirror images of each other across the y-axis.
  • The original function f(x) represents exponential decay, while the reflected function g(x) represents exponential growth.
  • Both functions share the same y-intercept (initial value) of 1.5.

This comparison will highlight the effect of the reflection transformation and reinforce your understanding of exponential functions.

Additional Tips and Considerations

  1. Use Graphing Tools: Utilize graphing calculators or online graphing tools to visualize the functions and verify your manual graphing.
  2. Practice with Variations: Explore reflections of other exponential functions with different initial values and bases to broaden your understanding.
  3. Understand the Impact of Transformations: Pay attention to how different transformations (e.g., translations, reflections, stretches) affect the graph of a function.
  4. Apply Real-World Contexts: Consider how exponential functions and their transformations are used in real-world applications such as population growth, radioactive decay, and financial modeling.

Conclusion

Graphing the reflection of f(x) = 1.5(0.5)^x across the y-axis involves transforming an exponential decay function into an exponential growth function. By following the step-by-step guide outlined in this article, you can accurately graph the reflected function g(x) = 1.5(2)^x and gain a deeper understanding of transformations in mathematics. This skill is crucial for further studies in calculus, algebra, and various scientific fields where functions play a central role.

Remember, the key to mastering transformations is practice and visualization. Continue to explore different functions and transformations to hone your skills and build a strong foundation in mathematical concepts.