Question

value: 1

which statement is true about the graph of the function \\(y = \sqrt3{x + 4} - 6\\)?

a. shift the graph 4 units left and 6 units up
b. shift the graph 4 units left and 6 units down
c. shift the graph 4 units right and 6 units up
d. shift the graph 6 units right and 4 units down

Explanation:

Identify parent function

The given function is \(y = \sqrt[3]{x + 4} - 6\).
The parent function is the Cube Root Function \(f(x) = \sqrt[3]{x}\).

Analyze horizontal shift

Using Function Transformations, we look at the term inside the radical: \(x + 4\).
Replacing \(x\) with \(x + h\) shifts the graph horizontally.
Since we have \(x + 4\), the graph is shifted \(4\) units to the left.

Analyze vertical shift

We look at the constant term outside the radical: \(- 6\).
Adding a constant \(k\) outside the function shifts the graph vertically.
Since we have \(- 6\), the graph is shifted \(6\) units down.

Combine transformations

Combining these transformations, the graph of \(y = \sqrt[3]{x + 4} - 6\) is obtained by shifting the parent graph \(4\) units left and \(6\) units down.
This matches option b.

Answer:

• a. Shift the graph 4 units left and 6 units up
• b. Shift the graph 4 units left and 6 units down (Correct answer)
• c. Shift the graph 4 units right and 6 units up
• d. Shift the graph 6 units right and 4 units down