The questionThe asks question to determine which of the asks exponential for equations reflects over the an exponential ** equation thaty satisfies the following transformations-axis:

1., Reflection over the y has a -axisvertical: This requires a negative exponent stretch. 2. by ** aVertical factor stretch of by a2 factor, and a horizontal shift right by 1 of. Let’s 2: This analyze each means the transformation:

function1 is. multiplied ** byReflection over2 the. 3. y -axisHorizontal shift: right This by requires a1 **negative: sign applied to the This involves base replacing of the x exponent \, i).e., with \( x - the 1 base \(4 \ in) the must exponent become.

( Breakdown of4^{- thex options}:

1).

. ( f(x) =2 2. (4Vertical) stretch^{- byx-1 a} factor \ of) 2: This means the equation is

• Reflection: multiplied by $2 Correct$ ($- (x$ causes reflection over the y-axis).
• Vertical stretch: Correct (multiplied by 2).
  the coefficient in front of the base -).

3. Horizontal Horizontal shift right by 1: To shift shift an equation horizontally right, replace $x$ with $(x - 1)$: in the exponent Incorrect.

(the exponent Analyze shifts the left given, choices not:

right1).

.2 . f \((x f)(x =) =2 -(24)(^{-x - 1}4:) Reflects^{ overx- y\1}-axis due) to
( 4 -^{- Reflectionx:}\ ),Incorrect but the (the reflection should affect the exponent, not the overall sign).
shift - Vertical stretch is: ** Correct.left
by -1 Horizontal**, shift not: Correct right (right. shift by 21).

3.. \( f(x) = -4(2)^{-x-1}

• Reflection: Correct.
• Vertical f stretch(x:) Incorrect = ( -mult2(4)ipl^{x - 1} \ied): No reflection over the ( byy 4,)- notaxis because2 the base is not (4).
  ^{- -x Horizontal shift}:). 3Incorrect. ( f ((xleft) shift, not right).

4. $f(x) = = -2(4) -4(2)^{-x - 1}$: Wrong base for the exponent, and reflection/shift^{x rules are not properly applied.
5. f+(x1)} =
   • Reflection: Incorrect (no - negative exponent for reflection).
   • Vertical stretch2(4:)^{ Correct.x +
     -1} \ Horizontal): Reflect shifts and shifts: horizontally ** Incorrectleft,** not right (.

Correctleft answer:

** shiftNone of the by given choices match all1 the).

required transformations properly### Correct. Answer**

Would you like: me toThe equation that satisfies explain all criteria is this in more detail ?$f(x) = 2(4)^{-x-1}$.

Would you like further clarification or steps broken down?

5 Related Questions:

1. What are the transformations involved in $f(x) = 2(4)^{x+1}$?
2. How does a reflection over the x-axis differ from one over the y-axis in exponential functions?
3. What is the impact of negative exponents on the graph of $f(x) = a(b)^x$?
4. How would the function change if the shift were left instead of right?
5. Can we rewrite $4^{-x-1}$ in terms of $2$? If so, how?

Tip:

When analyzing transformations, always carefully examine the base and the sign of the exponent, as they control reflections and horizontal shifts.