Question

compared with the graph of the parent function, which equation shows only a vertical compression by a factor of \\(\frac{1}{3}\\) and a shift downward of 4 units?

• \\(y = \frac{1}{3}\sqrt3{x} - 4\\)
• \\(y = -\frac{1}{3}\sqrt3{x - 4}\\)
• \\(y = -\frac{1}{3}\sqrt3{x} - 4\\)
• \\(y = \frac{1}{3}\sqrt3{x + 4}\\)

compared with the graph of the parent function, which equation shows a vertical stretch by a factor of 6, a shift of 7 units right, and a reflection over the x-axis?

• \\(y = 6\sqrt{x + 7}\\)
• \\(y = -\frac{1}{6}\sqrt{x - 7}\\)
• \\(y = -6\sqrt{x + 7}\\)
• \\(y = -6\sqrt{x - 7}\\)

Explanation:

Analyze Question 1 transformations

Using the Vertical Dilation and Vertical Translation knowledge points

• Parent function: \(f(x) = \sqrt[3]{x}\)
• Vertical compression by a factor of \(\frac{1}{3}\): multiply the function by \(\frac{1}{3}\), giving \(\frac{1}{3}\sqrt[3]{x}\).
• Shift downward of 4 units: subtract 4 from the function, giving \(y = \frac{1}{3}\sqrt[3]{x} - 4\).

Analyze Question 2 transformations

Using the Vertical Dilation, Horizontal Translation, and Function Reflections knowledge points

• Parent function: \(f(x) = \sqrt{x}\)
• Vertical stretch by a factor of 6: multiply the function by 6, giving \(6\sqrt{x}\).
• Shift of 7 units right: replace \(x\) with \(x - 7\), giving \(6\sqrt{x - 7}\).
• Reflection over the x-axis: multiply the entire function by \(-1\), giving \(y = -6\sqrt{x - 7}\).

Answer:

Question 1

• (A) \(y = \frac{1}{3}\sqrt[3]{x} - 4\) (Correct answer)
• (B) \(y = -\frac{1}{3}\sqrt[3]{x - 4}\)
• (C) \(y = -\frac{1}{3}\sqrt[3]{x} - 4\)
• (D) \(y = \frac{1}{3}\sqrt[3]{x + 4}\)

Question 2

• (A) \(y = 6\sqrt{x + 7}\)
• (B) \(y = -\frac{1}{6}\sqrt{x - 7}\)
• (C) \(y = 6\sqrt{x} + 7\)
• (D) \(y = -6\sqrt{x - 7}\) (Correct answer)