Reflecting Exponential Functions: Finding Points On G(x)

Hey math enthusiasts! Let's dive into the fascinating world of exponential functions and transformations. Today, we're going to explore what happens when we reflect a function over the y-axis. Specifically, we'll be looking at the function $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$ and figuring out how its reflection, $g(x)$, behaves. The goal? To identify which points lie on the graph of $g(x)$. Buckle up; this is going to be fun!

Understanding the Basics: Reflections and Exponential Functions

First things first: what does it mean to reflect a function over the y-axis? Imagine the y-axis as a mirror. When you reflect a function, you're essentially flipping it over this mirror. The y-axis remains unchanged, but the x-values change their sign. So, if a point $(x, y)$ is on the original function $f(x)$, then the point $(-x, y)$ will be on the reflected function $g(x)$. This is the key concept to remember. Now, let's talk about exponential functions. These are functions of the form $f(x) = a \cdot b^x$, where 'a' and 'b' are constants, and 'x' is the exponent. The base 'b' determines the growth or decay of the function. In our case, we have $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$. Here, $-\frac{2}{7}$ is the coefficient and $\frac{5}{3}$ is the base. Because the base is greater than 1, the function exhibits exponential growth, but due to the negative coefficient, the function is reflected over the x-axis, meaning it starts from negative values and approaches the x-axis from below as x increases. When we reflect this function over the y-axis, the x-values are negated, while the y-values are preserved. Therefore, we'll need to figure out the rule for $g(x)$ first. Then, we can check if the given points satisfy it. This requires us to substitute the x values of the coordinates given into the function $g(x)$ and check whether the y values we obtain match the coordinates provided.

Transforming the Function: Finding g(x)

To find $g(x)$, we need to replace $x$ with $-x$ in the original function $f(x)$. So, if $f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x$, then $g(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x}$. We can also rewrite this as $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$. This new function, $g(x)$, represents the reflection of $f(x)$ over the y-axis. Now that we have the equation for $g(x)$, we can test the given points to see if they lie on the graph of the function. The process involves substituting the x-coordinate of each point into $g(x)$ and verifying whether the result matches the y-coordinate. If it matches, the point is on the graph; if it doesn't, it's not. Remember, the negative sign in the original function is critical: it reflects the function across the x-axis. Therefore, the reflection over the y-axis is a bit more intricate, as it changes the input of the exponential part while the initial negative sign is maintained. To be successful at this type of problem, it is vital to know that if the base of the exponential term is greater than 1, as x increases, the exponential function also increases, and if the base of the exponential term is between 0 and 1, as x increases, the exponential function decreases. In the function $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$, the base is $\frac{3}{5}$, so as x increases, the exponential function decreases. Since this exponential term is multiplied by a negative number, the exponential function always yields a negative value, which is then added to 0, which means that the entire function will always have a negative value. Given the importance of precision in mathematical problem-solving, each step should be performed with extra care. Let's see how this plays out in our next steps!

Checking the Points: Which Ones Fit?

Now, let's go through the points one by one and see if they belong on $g(x)$.

Analyzing Point A: $(-7, -10.206)$

For point A, let's plug $x = -7$ into $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$:

$g(-7) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-7} = -\frac{2}{7}\left(\frac{5}{3}\right)^7 \approx -\frac{2}{7} \cdot 17.476 = -4.993$

Since $g(-7) \approx -4.993$, and the y-coordinate of point A is $-10.206$, this point does not lie on the graph of $g(x)$.

Analyzing Point B: $(-2.5, -4.474)$

For point B, we plug in $x = -2.5$:

$g(-2.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-2.5} = -\frac{2}{7}\left(\frac{5}{3}\right)^{2.5} \approx -\frac{2}{7} \cdot 6.094 = -1.741$

Here, $g(-2.5) \approx -1.741$. Since this doesn't match the y-coordinate of $-4.474$, this point is also not on the graph.

Analyzing Point C: $(0, -1.773)$

Let's check point C with $x = 0$:

$g(0) = -\frac{2}{7}\left(\frac{3}{5}\right)^0 = -\frac{2}{7} \cdot 1 = -\frac{2}{7} \approx -0.286$

The y-coordinate here is $-0.286$, and it is not equivalent to the y coordinate of point C, so $(0, -1.773)$ is not on the graph.

Analyzing Point D: $(0.5, ?)$ (Note: The y-coordinate is missing in the question)

Unfortunately, the y-coordinate is missing for point D. To properly evaluate this, we would need the y-value. Let's calculate what $g(0.5)$ should be:

$g(0.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{0.5} \approx -\frac{2}{7} \cdot 0.775 = -0.221$

Without the y-coordinate, we can't definitively say if this point is on the graph, but if the given point had been (0.5, -0.221), it would have been on the graph of g(x). Based on the context of the question, we can assume that we need to calculate the y-coordinate to see if any of the other coordinates is equivalent to this y-coordinate, but since the question did not specify the y-coordinate, we cannot choose any of the answer. Remember to check all the coordinates given in your math problems.

Conclusion: Identifying Points on the Reflected Function

In this problem, we've learned how to reflect an exponential function over the y-axis and how to check if specific points lie on the reflected graph. The key takeaway? When reflecting over the y-axis, replace $x$ with $-x$ in the function's equation. Then, carefully evaluate the function at the given x-values to determine the corresponding y-values and confirm whether the given points lie on the transformed function. Precision in calculating and a solid grasp of the transformation rules will help you ace these problems. Always remember to check your work and double-check your calculations, especially when exponents and negative signs are involved!

This exercise highlights the importance of understanding function transformations. By correctly applying the rules of reflection, we can accurately determine the new function's equation and find points on its graph. Practice these types of problems to become more adept at working with function transformations!

For further exploration on exponential functions and transformations, you might find the following resource helpful:

• Khan Academy's lessons on transformations of exponential functions: This is a great place to start! Khan Academy - Transformations of Exponential Functions

I hope this explanation has been helpful! Let me know if you have any questions. Keep practicing, and you'll become a pro at function transformations in no time!