the function $f(x) = -\frac{2}{7}\\left(\\frac{5}{3}\\right)^x$ is reflected over the $y$-axis to create…

the function $f(x) = -\frac{2}{7}\\left(\\frac{5}{3}\\right)^x$ is reflected over the $y$-axis to create $g(x)$. which points represent ordered pairs on $g(x)$? choose three correct answers. $(-2.5, -4.474)$ $(4, -0.017)$ $(9, -0.003)$ $(0, -1.773)$ $(-7, -10.206)$ $(0.5, -0.221)$ this question requires at least 3 answers.

the function $f(x) = -\frac{2}{7}\\left(\\frac{5}{3}\\right)^x$ is reflected over the $y$-axis to create $g(x)$. which points represent ordered pairs on $g(x)$? choose three correct answers. $(-2.5, -4.474)$ $(4, -0.017)$ $(9, -0.003)$ $(0, -1.773)$ $(-7, -10.206)$ $(0.5, -0.221)$ this question requires at least 3 answers.

Answer

Explanation:

Step1: Find the formula for ( g(x) )

To reflect a function ( f(x) ) over the ( y )-axis, we use the transformation ( g(x)=f(-x) ). Given ( f(x)=-\frac{2}{7}\left(\frac{5}{3}\right)^x ), then ( g(x)=f(-x)=-\frac{2}{7}\left(\frac{5}{3}\right)^{-x}=-\frac{2}{7}\left(\frac{3}{5}\right)^x ) (since ( a^{-n}=\frac{1}{a^n} ), so ( \left(\frac{5}{3}\right)^{-x}=\left(\frac{3}{5}\right)^x )).

Step2: Test each point in ( g(x) )

  • For point ( (-2.5, -4.474) ): Substitute ( x = -2.5 ) into ( g(x) ). Wait, no—wait, ( g(x) ) is ( -\frac{2}{7}\left(\frac{3}{5}\right)^x ). Wait, no, wait: when we reflect over ( y )-axis, ( g(x)=f(-x) ), so if the original ( f(x) ) has ( x ), then ( g(x) ) has ( -x ) in the exponent. Wait, let's re-express:

Original ( f(x)=-\frac{2}{7}\left(\frac{5}{3}\right)^x ). Reflect over ( y )-axis: replace ( x ) with ( -x ), so ( g(x)=-\frac{2}{7}\left(\frac{5}{3}\right)^{-x}=-\frac{2}{7}\left(\frac{3}{5}\right)^x ).

Now test each point ((x,y)) by plugging ( x ) into ( g(x) ) and seeing if ( y ) matches.

  1. Point ( (-2.5, -4.474) ): Wait, ( x=-2.5 ). Wait, no—wait, ( g(x) ) is a function of ( x ), so for ( x=-2.5 ), ( g(-2.5)=-\frac{2}{7}\left(\frac{3}{5}\right)^{-2.5} )? Wait, no, wait: no, wait, I made a mistake. Wait, the reflection over ( y )-axis: if ( f(x) ) is the original, then ( g(x) = f(-x) ). So ( f(-x)=-\frac{2}{7}\left(\frac{5}{3}\right)^{-x}=-\frac{2}{7}\left(\frac{3}{5}\right)^x ). So ( g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x ). So when ( x ) is the input to ( g(x) ), we use ( x ) in ( \left(\frac{3}{5}\right)^x ). Wait, no—wait, let's take each point ((x,y)) and check if ( y = -\frac{2}{7}\left(\frac{3}{5}\right)^x ).

Wait, let's correct the transformation:

Reflection over ( y )-axis: ( (x,y) ) on ( f(x) ) becomes ( (-x,y) ) on ( g(x) )? No, no: the function transformation: if ( f(x) ) has a point ( (a,b) ), then ( g(x)=f(-x) ) has a point ( (-a,b) )? Wait, no. Wait, the graph of ( y = f(-x) ) is the reflection of ( y = f(x) ) over the ( y )-axis. So for a point ( (x, f(x)) ) on ( f(x) ), the corresponding point on ( g(x)=f(-x) ) is ( (-x, f(x)) ). Wait, I think I messed up the transformation earlier. Let's re-express:

Let me re-express the transformation correctly. To reflect a function ( y = f(x) ) over the ( y )-axis, the new function is ( y = f(-x) ). So if ( f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x ), then ( g(x) = f(-x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x} = -\frac{2}{7}\left(\frac{3}{5}\right)^x ). So ( g(x) = -\frac{2}{7}\left(\frac{3}{5}\right)^x ). Now, to check if a point ((x, y)) is on ( g(x) ), we substitute ( x ) into ( g(x) ) and see if ( y ) equals the result.

Let's test each point:

  1. Point ( (-2.5, -4.474) ): Wait, ( x = -2.5 ). Wait, no—wait, ( g(x) ) is ( -\frac{2}{7}\left(\frac{3}{5}\right)^x ). So plug ( 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} ). Let's compute ( \left(\frac{5}{3}\right)^{2.5} ). ( 2.5 = \frac{5}{2} ), so ( \left(\frac{5}{3}\right)^{5/2} = \sqrt{\left(\frac{5}{3}\right)^5} = \sqrt{\frac{3125}{243}} \approx \sqrt{12.86} \approx 3.586 ). Then ( -\frac{2}{7} \times 3.586 \approx -1.025 ). Wait, that's not -4.474. Wait, maybe I messed up the transformation. Wait, maybe the reflection is ( (x, y) ) on ( f(x) ) becomes ( (-x, y) ) on ( g(x) ). So let's find points on ( f(x) ) and then reflect them over ( y )-axis.

Let's take ( f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x ). Let's compute ( f(x) ) for some ( x ):

  • For ( x = 2.5 ): ( f(2.5) = -\frac{2}{7}\left(\frac{5}{3}\right)^{2.5} ). ( \left(\frac{5}{3}\right)^{2.5} = \left(\frac{5}{3}\right)^{5/2} = \sqrt{\left(\frac{5}{3}\right)^5} = \sqrt{\frac{3125}{243}} \approx 3.586 ). Then ( -\frac{2}{7} \times 3.586 \approx -1.025 ). No, that's not matching. Wait, maybe my initial transformation was wrong. Wait, no—wait, the problem says "reflected over the ( y )-axis to create ( g(x) )". The rule for reflecting a function over the ( y )-axis is ( g(x) = f(-x) ). So let's re-express ( f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x ), so ( g(x) = f(-x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x} = -\frac{2}{7}\left(\frac{3}{5}\right)^x ). Now let's test each point:
  1. Point ( (4, -0.017) ): ( x = 4 ). ( g(4) = -\frac{2}{7}\left(\frac{3}{5}\right)^4 = -\frac{2}{7} \times \frac{81}{625} = -\frac{162}{4375} \approx -0.0369 ). Not -0.017. Hmm.

Wait, maybe I made a mistake in the transformation. Wait, maybe the original function is ( f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x ), and reflecting over ( y )-axis gives ( g(x) = f(-x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x} = -\frac{2}{7}\left(\frac{3}{5}\right)^x ). Let's try ( x = -7 ): Wait, no, the point is ( (-7, -10.206) ). Wait, ( x = -7 ) in ( g(x) ): ( g(-7) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-7} = -\frac{2}{7}\left(\frac{5}{3}\right)^7 ). ( \left(\frac{5}{3}\right)^7 = \frac{78125}{2187} \approx 35.72 ). Then ( -\frac{2}{7} \times 35.72 \approx -10.206 ). Oh! That matches. So ( (-7, -10.206) ) is on ( g(x) ).

Next, point ( (-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} ). ( \left(\frac{5}{3}\right)^{2.5} = \left(\frac{5}{3}\right)^{5/2} = \sqrt{\left(\frac{5}{3}\right)^5} = \sqrt{\frac{3125}{243}} \approx 3.586 )? Wait, no, ( \left(\frac{5}{3}\right)^5 = \frac{3125}{243} \approx 12.86 ), square root is ( \approx 3.586 ). Then ( -\frac{2}{7} \times 3.586 \approx -1.025 ). No, that's not -4.474. Wait, maybe ( x = 2.5 ) in ( f(x) ): ( f(2.5) = -\frac{2}{7}\left(\frac{5}{3}\right)^{2.5} \approx -1.025 ), so reflecting over ( y )-axis would be ( (-2.5, -1.025) ), which is not the point given. Wait, maybe I messed up. Wait, let's check the point ( (0, -1.773) ): ( x = 0 ). ( g(0) = -\frac{2}{7}\left(\frac{3}{5}\right)^0 = -\frac{2}{7}(1) = -\frac{2}{7} \approx -0.2857 ). Not -1.773. Wait, ( f(0) = -\frac{2}{7}\left(\frac{5}{3}\right)^0 = -\frac{2}{7} \approx -0.2857 ), so reflecting over ( y )-axis, ( g(0) = f(0) = -\frac{2}{7} \approx -0.2857 ), so ( (0, -1.773) ) is not on ( g(x) ).

Wait, let's try point ( (9, -0.003) ): ( x = 9 ). ( g(9) = -\frac{2}{7}\left(\frac{3}{5}\right)^9 ). ( \left(\frac{3}{5}\right)^9 = \frac{19683}{1953125} \approx 0.01007 ). Then ( -\frac{2}{7} \times 0.01007 \approx -0.00288 \approx -0.003 ). That matches! So ( (9, -0.003) ) is on ( g(x) ).

Point ( (4, -0.017) ): ( x = 4 ). ( g(4) = -\frac{2}{7}\left(\frac{3}{5}\right)^4 = -\frac{2}{7} \times \frac{81}{625} = -\frac{162}{4375} \approx -0.0369 ). Not -0.017.

Point ( (0.5, -0.221) ): ( x = 0.5 ). ( g(0.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{0.5} = -\frac{2}{7} \times \sqrt{\frac{3}{5}} \approx -\frac{2}{7} \times 0.7746 \approx -0.221 ). Oh! That matches. So ( (0.5, -0.221) ) is on ( g(x) ).

Wait, let's recap:

  • ( (-7, -10.206) ): ( g(-7) = -\frac{2}{7}\left(\frac{3}{5}\right)^{-7} = -\frac{2}{7}\left(\frac{5}{3}\right)^7 \approx -10.206 ) ✔️
  • ( (9, -0.003) ): ( g(9) = -\frac{2}{7}\left(\frac{3}{5}\right)^9 \approx -0.003 ) ✔️
  • ( (0.5, -0.221) ): ( g(0.5) = -\frac{2}{7}\left(\frac{3}{5}\right)^{0.5} \approx -0.221 ) ✔️

Wait, let's check ( (-2.5, -4.474) ) again. ( 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} ). ( \left(\frac{5}{3}\right)^{2.5} = \left(\frac{5}{3}\right)^{5/2} = \sqrt{\left(\frac{5}{3}\right)^5} = \sqrt{\frac{3125}{243}} \approx 3.586 ). Then ( -\frac{2}{7} \times 3.586 \approx -1.025 ), not -4.474. So that's out.

Point ( (0, -1.773) ): ( g(0) = -\frac{2}{7} \approx -0.2857 ), not -1.773. Out.

Point ( (4, -0.017) ): ( g(4) \approx -0.0369 ), not -0.017. Out.

So the correct points are ( (-7, -10.206) ), ( (9, -0.003) ), and ( (0.5, -0.221) ). Wait, but let's check ( (-2.5, -4.474) ) again. Wait, maybe I made a mistake in the transformation. Wait, maybe the original function is ( f(x) = -\frac{2}{7}\left(\frac{5}{3}\right)^x ), and reflecting over ( y )-axis is ( g(x) = f(-x) = -\frac{2}{7}\left(\frac{5}{3}\right)^{-x} = -\frac{2}{7}\left(\frac{3}{5}\right)^x ). Wait, when ( 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} ). Let's compute ( \left(\frac{5}{3}\right)^{2.5} ):

( \left(\frac{5}{3}\right)^2 = \frac{25}{9} \approx 2.777 ), ( \left(\frac{5}{3}\right)^{0.5} = \sqrt{\frac{5}{3}} \approx 1.291 ). So ( \left(\frac{5}{3}\right)^{2.5} = \frac{25}{9} \times 1.291 \approx 2.777 \times 1.291 \approx 3.586 ). Then ( -\frac{2}{7} \times 3.586 \approx -1.025 ), which is not -4.474. So that's not correct.

Wait, maybe the point ( (-2.5, -4.474) ) is on ( f(x) ) when ( x =