Reflecting Functions Over The Y-Axis: Finding Points On G(x)

Hey guys, let's dive into the wild world of functions and transformations! Today, we're tackling a cool problem from our mathematics category. We've got this function, $f(x)=-\frac{2}{7}\left(\frac{5}{3}\right)^x$, and we're going to reflect it over the $y$-axis to create a brand new function, $g(x)$. Our mission, should we choose to accept it, is to figure out which of the given points actually lie on this new function $g(x)$. This is a fantastic exercise in understanding how reflections affect the coordinates of points on a graph, and it's super useful for visualizing and analyzing function behavior. So, grab your calculators, maybe a snack, and let's get this done!

Understanding the Reflection

First things first, what does it mean to reflect a function over the $y$-axis? Think about it like looking in a mirror. If you have a point $(x, y)$ on the original function $f(x)$, when you reflect it over the $y$-axis, the $y$-coordinate stays the same, but the $x$-coordinate flips its sign. So, a point $(x, y)$ on $f(x)$ becomes $(-x, y)$ on the reflected function. This means that if our original function is $f(x)$, the reflected function $g(x)$ will have the relationship $g(x) = f(-x)$. Let's apply this to our specific function: $f(x)=-\frac{2}{7}\left(\frac{5}{3}\right)^x$. To find $g(x)$, we simply replace every instance of $x$ with $-x$. So, $g(x) = f(-x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x}$. Now, we can simplify this expression further using exponent rules. Remember that $\left(\frac{a}{b}\right)^{-x} = \left(\frac{b}{a}\right)^x$. Applying this to our function, we get $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$. This is our new function, $g(x)$, after the reflection. The key takeaway here is that reflecting over the $y$-axis transforms $x$ to $-x$. This transformation is crucial for understanding how the graph of a function changes. When we reflect a graph over the $y$-axis, we are essentially looking at the function's behavior in terms of its negative input values. This can reveal interesting symmetries and patterns. For example, if a function is even, meaning $f(x) = f(-x)$, then reflecting it over the $y$-axis results in the exact same function. Our function isn't even, so we expect a noticeable change. The process of finding $g(x)$ involves careful substitution and manipulation of exponents, which are fundamental skills in algebra. Understanding these concepts is not just about solving problems; it's about building a solid foundation for more advanced mathematical concepts you'll encounter later on. So, make sure you're comfortable with these steps before we move on to checking the points!

Evaluating Points on g(x)

Alright, now that we've got our function $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$, it's time to test those points. We need to see which of the given ordered pairs $(x, y)$ satisfy the equation $y = g(x)$. This means we'll plug the $x$-value of each point into our $g(x)$ function and see if the result matches the $y$-value of the point. Let's go through them one by one. Remember, we're checking against $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$.

Point A: $(-7, -10.206)$ We need to calculate $g(-7)$. $g(-7) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-7} = -\frac{2}{7}\left(\frac{5}{3}\right)^{7}$ Let's crunch these numbers. $\left(\frac{5}{3}\right)^{7} \approx (1.6667)^7 \approx 152.937$ So, $g(-7) \approx -\frac{2}{7} \times 152.937 \approx -0.2857 \times 152.937 \approx -43.70$. This does not match $-10.206$. So, Point A is out.

Point B: $(-2.5, -4.474)$ We need to calculate $g(-2.5)$. $g(-2.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-2.5} = -\frac{2}{7}\left(\frac{5}{3}\right)^{2.5}$ Let's calculate $\left(\frac{5}{3}\right)^{2.5} \approx (1.6667)^{2.5} \approx 4.305$. So, $g(-2.5) \approx -\frac{2}{7} \times 4.305 \approx -0.2857 \times 4.305 \approx -1.23$. This does not match $-4.474$. So, Point B is out.

Point C: $(0, -1.773)$ We need to calculate $g(0)$. $g(0) = -\frac{2}{7}\left(\frac{3}{5}\right)^{0}$ Anything raised to the power of 0 is 1 (as long as the base isn't 0). So, $\left(\frac{3}{5}\right)^{0} = 1$. $g(0) = -\frac{2}{7} \times 1 = -\frac{2}{7}$ Now, let's convert $-\frac{2}{7}$ to a decimal. $-\frac{2}{7} \approx -0.2857$. This does not match $-1.773$. So, Point C is out. Wait a minute, did I make a mistake in the initial calculation? Let's re-evaluate $f(x)$ for this point. If the point $(0, -1.773)$ was on $f(x)$, then $f(0) = -\frac{2}{7}(\frac{5}{3})^0 = -\frac{2}{7} \approx -0.2857$. This also doesn't match. This means that maybe my understanding of the question or the points provided needs a second look. Let's double-check the original function and the reflection process. The reflection over the y-axis means $g(x) = f(-x)$. So $g(x) = -\frac{2}{7}(\frac{5}{3})^{-x} = -\frac{2}{7}(\frac{3}{5})^{x}$. This is what I used. Hmm.

Let's reconsider the possibility that the points given might be for the original function $f(x)$, and we are supposed to find the corresponding points on $g(x)$ after reflection. If a point $(x_0, y_0)$ is on $f(x)$, then $y_0 = f(x_0)$. When reflected over the y-axis, the new point on $g(x)$ will be $(-x_0, y_0)$. Let's test this idea with the provided points as if they were on $f(x)$ and see if their reflected versions land on $g(x)$.

Let's re-evaluate Point C: $(0, -1.773)$. If this point were on $f(x)$, then the point on $g(x)$ would be $(-0, -1.773)$, which is just $(0, -1.773)$. We already calculated $g(0) \approx -0.2857$. So, $(0, -1.773)$ is not on $g(x)$. This is confusing. Let me re-read the question carefully: "Which points represent ordered pairs on $g(x)$?" This means we must plug the given $x$ values into $g(x)$ and see if we get the given $y$ values. My calculations for $g(x)$ seem correct: $g(x) = -\frac{2}{7}(\frac{3}{5})^{x}$. Let me re-calculate the decimal approximations more precisely.

Let's try the points again, being very careful with the calculator. $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$

Point C Revisited: $(0, -1.773)$ $g(0) = -\frac{2}{7}\left(\frac{3}{5}\right)^0 = -\frac{2}{7} \times 1 = -\frac{2}{7}$. As a decimal, $-\frac{2}{7} \approx -0.285714$. This is definitely not $-1.773$. There might be a typo in the question or the options provided, or perhaps the function $f(x)$ was different. Let's assume for a moment that the question implied that the original $f(x)$ passes through some points, and we need to find the reflected points on $g(x)$. If $(x,y)$ is on $f(x)$, then $(-x,y)$ is on $g(x)$. The points given are supposed to be on $g(x)$. So, we must test them directly in $g(x)$.

Let's check the options again and be very meticulous. It's possible that the number $-1.773$ is related to $f(x)$ in some way, but not directly $g(0)$. For $f(x)=-\frac{2}{7}\rac{5}{3}\ ext{ ) }^x$, if we evaluate $f(1) = -\frac{2}{7}(\frac{5}{3})^1 = -\frac{10}{21} \approx -0.476$. If we evaluate $f(2) = -\frac{2}{7}(\frac{5}{3})^2 = -\frac{2}{7}(\frac{25}{9}) = -\frac{50}{63} \approx -0.793$. The values of $f(x)$ become more negative as $x$ increases. The values of $g(x)$ will become more negative as $x$ decreases since $g(x) = -\frac{2}{7}(\frac{3}{5})^{x}$.

Let's re-examine the provided options and my calculations for $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$. It's crucial to get these calculations spot on.

Point A: $(-7, -10.206)$ $g(-7) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-7} = -\frac{2}{7}\left(\frac{5}{3}\right)^{7} \approx -\frac{2}{7}(152.937) \approx -43.70$. Still not matching.

Point B: $(-2.5, -4.474)$ $g(-2.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-2.5} = -\frac{2}{7}\left(\frac{5}{3}\right)^{2.5} \approx -\frac{2}{7}(4.305) \approx -1.23$. Still not matching.

Point C: $(0, -1.773)$ $g(0) = -\frac{2}{7}\left(\frac{3}{5}\right)^0 = -\frac{2}{7}(1) = -0.2857$. Still not matching.

Point D: $(0.5, -0.221)$ $g(0.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{0.5} = -\frac{2}{7}\sqrt{\frac{3}{5}} \approx -0.2857 \times \sqrt{0.6} \approx -0.2857 \times 0.7746 \approx -0.2213$. Bingo! This one matches very closely. So, Point D is a valid point on $g(x)$. The slight difference is likely due to rounding in the provided option.

Let's consider the possibility that there might be another point that works. Given the structure of these problems, sometimes there are multiple correct answers.

Wait, let's rethink. If Point D works, does that mean my initial calculation for Point C was wrong, or is there something else going on? The y-intercept of $g(x)$ is always $g(0)$, which we calculated as $-2/7 \\\approx -0.2857$. So $(0, -1.773)$ can never be on $g(x)$. This strongly suggests a typo in the question or options if C is supposed to be correct.

Let me re-read the prompt. "Which points represent ordered pairs on $g(x)$? Check all that apply." This implies there could be more than one. Since I found one that works perfectly (Point D), let me re-verify my calculations for the others very carefully, as it's possible I made a calculation error or the given values are rounded approximations. It is also possible that the question expects us to recognize the relationship between $f(x)$ and $g(x)$ and deduce points without direct calculation if the numbers are tricky.

Let's re-evaluate $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$. A point $(x,y)$ is on $g(x)$ if $y = -\frac{2}{7}\left(\frac{3}{5}\right)^x$.

Point A: $(-7, -10.206)$ $x = -7$. $g(-7) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-7} = -\frac{2}{7}\left(\frac{5}{3}\right)^{7}$. Calculate $(\frac{5}{3})^7 \approx 152.93735$. $g(-7) \approx -\frac{2}{7} \times 152.93735 \approx -43.70$. Definitely not $-10.206$.

Point B: $(-2.5, -4.474)$ $x = -2.5$. $g(-2.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-2.5} = -\frac{2}{7}\left(\frac{5}{3}\right)^{2.5}$. Calculate $(\frac{5}{3})^{2.5} \approx 4.3053$. $g(-2.5) \approx -\frac{2}{7} \times 4.3053 \approx -1.2301$. Not $-4.474$.

Point C: $(0, -1.773)$ $x=0$. $g(0) = -\frac{2}{7}\left(\frac{3}{5}\right)^0 = -\frac{2}{7} \times 1 = -0.2857$. Not $-1.773$.

Point D: $(0.5, -0.221)$ $x=0.5$. $g(0.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{0.5} = -\frac{2}{7}\sqrt{0.6}$. Calculate $\sqrt{0.6} \approx 0.7745966$. $g(0.5) \approx -\frac{2}{7} \times 0.7745966 \approx -0.221313$. This is extremely close to $-0.221$. So, Point D is confirmed.

Now, let's think about potential errors in the question or my interpretation. What if the question meant that the points given were on $f(x)$ and we had to find the corresponding points on $g(x)$? If $(-7, -10.206)$ was on $f(x)$, then $(-(-7), -10.206) = (7, -10.206)$ would be on $g(x)$. Let's test $g(7)$: $g(7) = -\frac{2}{7}(\frac{3}{5})^7 \approx -0.0014$. Not $-10.206$. This confirms we need to test the points directly on $g(x)$.

Is it possible that the original function $f(x)$ had a different structure? For example, if $f(x) = -2/7 * (5/3)^x$, then $g(x) = f(-x) = -2/7 * (5/3)^{-x} = -2/7 * (3/5)^x$. This is consistent. My $g(x)$ calculation is correct. My evaluation of $g(0.5)$ is correct and matches option D.

Let's consider the possibility that one of the other options might also be correct due to rounding, or if there's a mathematical property I'm overlooking. However, the calculations for A, B, and C are quite far off from the given y-values. For instance, $g(0)$ is definitively $-2/7$, which is approximately $-0.2857$, not $-1.773$. The difference is too large to be just rounding.

Could there be a typo in $f(x)$ itself? Suppose $f(x) = -C imes b^x$. Then $g(x) = f(-x) = -C imes b^{-x} = -C imes (1/b)^x$. Here, $C = 2/7$ and $b = 5/3$. So $g(x) = -2/7 imes (3/5)^x$. This formula is solid.

Let me re-check the value $-1.773$ for option C. If $g(x) = -1.773$ when $x=0$, this implies $-2/7 = -1.773$, which is false. Could $-1.773$ be related to the original $f(x)$? $f(0) = -2/7 \approx -0.2857$. $f(1) \approx -0.476$. $f(2) \approx -0.793$. $f(3) \approx -1.32$. $f(4) \approx -2.20$. So, maybe there was a point $(x, -1.773)$ on $f(x)$ where $x$ is between 3 and 4. If that were the case, the corresponding point on $g(x)$ would be $(-x, -1.773)$ where $x$ is between 3 and 4. So the $x$-coordinate would be between $-3$ and $-4$. This doesn't match option C's $x=0$.

Let's re-examine the numbers provided in the options. They are given to three decimal places. This level of precision suggests that we should expect our calculations to yield results very close to these values if they are correct. We found that $g(0.5) \approx -0.2213$, which rounds to $-0.221$. This is a strong match.

What if there was a misunderstanding of the base? Is it possible that the base is negative? The base of an exponential function is typically positive. $(\frac{5}{3})$ is positive. So that's fine.

Let's consider the structure of the points. We have negative and positive $x$ values. $g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x$. Since $(\frac{3}{5})$ is less than 1, $(\frac{3}{5})^x$ will increase as $x$ increases. Therefore, $g(x)$ will become less negative (closer to 0) as $x$ increases. Conversely, as $x$ becomes more negative (e.g., $-2.5, -7$), $g(x)$ becomes more negative.

Let's look at the options again with this understanding:

• Point A: $(-7, -10.206)$. $x = -7$ is very negative. $g(-7)$ should be very negative. $-10.206$ is negative, but my calculation showed it should be around $-43.70$. This suggests that $-10.206$ is too