Questions: Below is the graph of y=sqrt(x). Translate it to make it the graph of y=sqrt(x-3)-4.

Solution Steps

To translate the graph of \( y = \sqrt{x} \) to \( y = \sqrt{x-3} - 4 \), we need to perform two transformations:

1. Shift the graph 3 units to the right.
2. Shift the graph 4 units down.

The original graph given is \( y = \sqrt{x} \). This is a basic square root function, which starts at the origin \((0, 0)\) and increases gradually to the right.

The new function is \( y = \sqrt{x-3} - 4 \). This involves two transformations:

1. Horizontal Shift: The term \( x-3 \) inside the square root indicates a horizontal shift to the right by 3 units.
2. Vertical Shift: The term \(-4\) outside the square root indicates a vertical shift downward by 4 units.

To apply the horizontal shift, we move every point on the graph of \( y = \sqrt{x} \) to the right by 3 units. For example:

• The point \((0, 0)\) on \( y = \sqrt{x} \) moves to \((3, 0)\) on \( y = \sqrt{x-3}\).
• The point \((1, 1)\) on \( y = \sqrt{x} \) moves to \((4, 1)\) on \( y = \sqrt{x-3}\).

Next, we apply the vertical shift by moving every point on the graph of \( y = \sqrt{x-3} \) downward by 4 units. For example:

• The point \((3, 0)\) on \( y = \sqrt{x-3} \) moves to \((3, -4)\) on \( y = \sqrt{x-3} - 4\).
• The point \((4, 1)\) on \( y = \sqrt{x-3} \) moves to \((4, -3)\) on \( y = \sqrt{x-3} - 4\).

Final Answer