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QuestionIs the transformation from y=3xy=3^{x} to y=5⋅3−(x−2)+9y=5 \cdot 3^{-(x-2)}+9 unique? List transformations to prove or disprove the claim.

Studdy Solution

STEP 1

Assumptions1. The parent function is y=3xy=3^{x} . The transformed function is y=5⋅3−(x−)+9y=5 \cdot3^{-(x-)}+9
3. The transformations listed by the classmate are - A vertical stretch by a factor of5 - A horizontal translation by units to the right - A reflection in yy-axis - A vertical translation by9 units up

STEP 2

Let's start by applying the transformations one by one to the parent function y=xy=^{x} and see if we get the transformed function.
The first transformation is a vertical stretch by a factor of5. The general form for a vertical stretch is y=a⋅f(x)y=a \cdot f(x), where aa is the stretch factor.y=5⋅xy=5 \cdot^{x}

STEP 3

The second transformation is a horizontal translation by2 units to the right. The general form for a horizontal translation is y=f(x−h)y=f(x-h), where hh is the number of units to the right.y=5⋅3x−2y=5 \cdot3^{x-2}

STEP 4

The third transformation is a reflection in the yy-axis. The general form for a reflection in the yy-axis is y=f(−x)y=f(-x).y=⋅3−x+2y= \cdot3^{-x+2}

STEP 5

The fourth transformation is a vertical translation by9 units up. The general form for a vertical translation is y=f(x)+ky=f(x)+k, where kk is the number of units up.y=5⋅3−x+2+9y=5 \cdot3^{-x+2}+9

STEP 6

The final transformed function after applying all the transformations is y=5⋅3−x+2+9y=5 \cdot3^{-x+2}+9.However, this is not the same as the given transformed function y=5⋅3−(x−2)+9y=5 \cdot3^{-(x-2)}+9. The difference is in the sign of xx in the exponent. In the given transformed function, the entire (x−2)(x-2) is negated, but in the function we obtained after applying the transformations, only xx is negated.Therefore, the sequence of transformations listed by the classmate is incorrect.

STEP 7

Now, let's find the correct sequence of transformations that transforms the parent function y=3xy=3^{x} into the given transformed function y=5⋅3−(x−2)+9y=5 \cdot3^{-(x-2)}+9.
The first transformation should be a vertical stretch by a factor of5. The general form for a vertical stretch is y=a⋅f(x)y=a \cdot f(x), where aa is the stretch factor.y=5⋅3xy=5 \cdot3^{x}

STEP 8

The second transformation should be a reflection in the yy-axis. The general form for a reflection in the yy-axis is y=f(−x)y=f(-x).y=5⋅3−xy=5 \cdot3^{-x}

STEP 9

The third transformation should be a horizontal translation by2 units to the right. The general form for a horizontal translation is y=f(x−h)y=f(x-h), where hh is the number of units to the right.y=5⋅3−x+2y=5 \cdot3^{-x+2}

STEP 10

The fourth transformation should be a vertical translation by9 units up. The general form for a vertical translation is y=f(x)+ky=f(x)+k, where kk is the number of units up.y=5⋅3−x+2+9y=5 \cdot3^{-x+2}+9

STEP 11

The final transformed function after applying all the transformations is y=5⋅3−x++9y=5 \cdot3^{-x+}+9, which is the same as the given transformed function.Therefore, the correct sequence of transformations is- A vertical stretch by a factor of5- A reflection in yy-axis- A horizontal translation by units to the right- A vertical translation by9 units up

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