Y-Axis Reflection: Finding Points On G(x)

Hey guys! Today, we're diving into a fun little problem involving function transformations, specifically a reflection across the y-axis. We'll start with a given function, reflect it, and then figure out which point lies on the new function. Buckle up, it's gonna be a math-tastic ride!

Understanding the Problem

So, the question is this: We have a function ${f(x) = \frac{1}{6}(\frac{2}{5})^x}$. This function is reflected across the y-axis, creating a new function called ${g(x)}$. Our mission is to find which of the given ordered pairs is actually on ${g(x)}$.

Before we dive into the answer choices, let's make sure we understand what reflecting across the y-axis really means. When you reflect a function across the y-axis, you're essentially flipping it horizontally. Mathematically, this transformation is achieved by replacing ${x}$ with ${-x}$ in the original function. So, if ${f(x)}$ is our original function, then ${g(x)}$, the reflected function, is simply ${f(-x)}$.

Let's break down the concept of y-axis reflection. Imagine you have a graph of a function drawn on a piece of paper. Now, picture folding that paper along the y-axis. The image you see on the other side of the fold is the reflected function. Each point ${(x, y)}$ on the original function becomes ${(-x, y)}$ on the reflected function. The y-coordinate stays the same, but the x-coordinate changes its sign. This is crucial for understanding how to find points on ${g(x)}$.

Now, let's explicitly determine ${g(x)}$. Given ${f(x) = \frac{1}{6}(\frac{2}{5})^x}$, we find ${g(x)}$ by substituting ${-x}$ for ${x}$ in ${f(x)}$. This gives us:

${g(x) = f(-x) = \frac{1}{6}(\frac{2}{5})^{-x} = \frac{1}{6}(\frac{5}{2})^x}$

So, ${g(x) = \frac{1}{6}(\frac{5}{2})^x}$. Now that we have the equation for ${g(x)}$, we can test the provided ordered pairs to see which one satisfies this equation. Essentially, we'll plug in the x-coordinate of each ordered pair into ${g(x)}$ and see if we get the corresponding y-coordinate.

Checking the Options

Let's examine the given options and see which one lies on ${g(x) = \frac{1}{6}(\frac{5}{2})^x}$.

Option A: (-3, 4/375)

We need to check if ${g(-3) = \frac{4}{375}}$.

${g(-3) = \frac{1}{6}(\frac{5}{2})^{-3} = \frac{1}{6}(\frac{2}{5})^3 = \frac{1}{6} \cdot \frac{8}{125} = \frac{8}{750} = \frac{4}{375}}$

So, ${g(-3) = \frac{4}{375}}$. This means the point ${(-3, \frac{4}{375})}$ does lie on ${g(x)}$.

Since we found a correct answer, we could technically stop here. But for the sake of thoroughness (and because we're awesome!), let's check the other options too. This will reinforce our understanding and make sure we haven't made any sneaky mistakes.

Option B: (-2, 35/24)

We need to check if ${g(-2) = \frac{35}{24}}$.

${g(-2) = \frac{1}{6}(\frac{5}{2})^{-2} = \frac{1}{6}(\frac{2}{5})^2 = \frac{1}{6} \cdot \frac{4}{25} = \frac{4}{150} = \frac{2}{75}}$

So, ${g(-2) = \frac{2}{75}}$. This is NOT equal to ${\frac{35}{24}}$, so the point ${(-2, \frac{35}{24})}$ does not lie on ${g(x)}$.

Option C: (2, 35/24)

Remember, to check if this point is on the graph, we're looking to see if the y-value matches the result of plugging the x-value into our equation, ${g(x) = \frac{1}{6}(\frac{5}{2})^x}$. So, we need to check if ${g(2) = \frac{35}{24}}$.

${g(2) = \frac{1}{6}(\frac{5}{2})^2 = \frac{1}{6} \cdot \frac{25}{4} = \frac{25}{24}}$

Is ${\frac{25}{24}}$ equal to ${\frac{35}{24}}$? Nope! So, the point ${(2, \frac{35}{24})}$ is not on the graph of ${g(x)}$.

The Answer

After checking all the options, we found that only option A, ${(-3, \frac{4}{375})}$, satisfies the equation for ${g(x)}$. Therefore, the ordered pair that lies on ${g(x)}$ is ${(-3, \frac{4}{375})}$.

Key Takeaways

• Reflection Across the Y-Axis: Reflecting a function across the y-axis means replacing ${x}$ with ${-x}$ in the function's equation.
• Finding Points on a Function: To check if a point ${(x, y)}$ lies on a function, plug the x-coordinate into the function and see if the result is the y-coordinate.
• Be Thorough: Always double-check your work and, if possible, test all the options to ensure you've found the correct answer.

Wrapping Up

So, there you have it! We successfully reflected a function across the y-axis and identified a point that lies on the new function. Hopefully, this explanation has helped clarify the concept of y-axis reflections and how to work with them. Keep practicing, and you'll become a master of function transformations in no time!

Keep rocking those math problems, and I'll catch you in the next one! Peace out!

This was a fun problem demonstrating a reflection across the y-axis. Remember, the key to these problems is understanding what the transformation does to the function and then carefully applying that transformation to find the new equation. From there, it's just a matter of testing the points!

Function transformations can seem tricky at first, but with practice, they become second nature. Always remember to take it step by step, and don't be afraid to draw diagrams or use visual aids to help you understand what's going on. Math is all about building a solid foundation of understanding, and with that foundation, you can tackle even the toughest problems.

Understanding transformations such as reflections, translations, stretches, and compressions is fundamental in mathematics. These transformations allow us to manipulate and analyze functions in various ways, providing valuable insights into their behavior. So keep honing those skills, and you'll be well on your way to mathematical mastery! And remember, math is not about memorizing formulas; it's about understanding concepts. Once you truly understand the underlying principles, the formulas will naturally fall into place.