Question

consider the function $f(x) = -\frac{1}{3}log_{2}(4x) + 2$. which of the following correctly describes the transformations applied to the parent function $g(x) = \log_{2}(x)$ to obtain $f(x)$? (10 points)

reflection across the x-axis, vertical stretch by a factor of $\frac{1}{3}$, horizontal stretch by a factor of $\frac{1}{4}$, vertical downwards up 2 units.

reflection across the x-axis, vertical shrink by a factor of 3, horizontal shrink by a factor of 4, vertical shift downwards 2 units.

reflection across the y-axis, vertical shrink by a factor of 3, horizontal shrink by a factor of 4, vertical shift downwards 2 units.

reflection across the x-axis, vertical shrink by a factor of $\frac{1}{3}$, horizontal shrink by a factor of $\frac{1}{4}$, vertical shift up 2 units.

Explanation:

1. Start with the parent function \(g(x) = \log_2(x)\).
2. Replace \(x\) with \(4x\): \(g_1(x) = \log_2(4x)\) is a horizontal shrink by a factor of \(\frac{1}{4}\).
3. Multiply by \(\frac{1}{3}\): \(g_2(x) = \frac{1}{3}\log_2(4x)\) is a vertical shrink by a factor of \(\frac{1}{3}\).
4. Multiply by \(-1\): \(g_3(x) = -\frac{1}{3}\log_2(4x)\) is a reflection across the x-axis.
5. Add 2: \(f(x) = -\frac{1}{3}\log_2(4x) + 2\) is a vertical shift up 2 units.
6. Match this sequence to the given options.

Answer:

D. Reflection across the x-axis, vertical shrink by a factor of \(\frac{1}{3}\), horizontal shrink by a factor of \(\frac{1}{4}\), vertical shift up 2 units.