Question

describe the transformation of the parent function $h(x)=3^{x}$ to turn into $k(x)=2\cdot3^{-\frac{1}{2}x}-5$ (10 points)

reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.

reflection across the y-axis, horizontal stretch by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.

reflection across the x-axis, horizontal stretch by a factor of 2, vertical shrink by a factor of 2, vertical shift upwards by 5 units.

reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift upwards by 5 units.

Explanation:

Step1: Identify y-axis reflection

For $h(x)=3^x$, replacing $x$ with $-x$ gives $3^{-x}$, which is a reflection across the y-axis.

Step2: Identify horizontal stretch

The term $3^{-\frac{1}{2}x}=3^{-(\frac{1}{2}x)}$. For $f(bx)$, $0<|b|<1$ means horizontal stretch by $\frac{1}{|b|}$. Here $b=\frac{1}{2}$, so stretch factor is $2$.

Step3: Identify vertical stretch

Multiplying by 2: $2\cdot3^{-\frac{1}{2}x}$ is a vertical stretch by factor 2.

Step4: Identify vertical shift

Subtracting 5: $2\cdot3^{-\frac{1}{2}x}-5$ is a vertical shift down 5 units.

Answer:

Reflection across the y-axis, horizontal stretch by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.