Question

24: the graph of the function is a translation of the graph of (f(x) = \sqrt{x}).
graph the function (h(x) = \sqrt{x - 3} + 2)

Explanation:

Identify parent function and transformations

To graph the function \(h(x) = \sqrt{x - 3} + 2\), we analyze its transformations relative to the parent function \(f(x) = \sqrt{x}\).
The term \(x - 3\) inside the square root represents a horizontal shift to the right by 3 units.
Using the Vertical Translations knowledge point:
The term \(+ 2\) outside the square root represents a vertical shift upwards by 2 units.

Determine key points on the graph

We apply these shifts to the standard reference points of the parent function \(f(x) = \sqrt{x}\):

• The starting point \((0, 0)\) shifts to \((0 + 3, 0 + 2) = (3, 2)\).
• The point \((1, 1)\) shifts to \((1 + 3, 1 + 2) = (4, 3)\).
• The point \((4, 2)\) shifts to \((4 + 3, 2 + 2) = (7, 4)\).

Plot the transformed function

We plot the starting point at \((3, 2)\) and draw a smooth curve passing through \((4, 3)\) and \((7, 4)\), extending to the right.

Answer:

To graph the function \(h(x) = \sqrt{x - 3} + 2\), translate the parent graph of \(f(x) = \sqrt{x}\) by shifting it 3 units to the right and 2 units up.

The key points to plot are:

• Starting point: \((3, 2)\)
• Second point: \((4, 3)\)
• Third point: \((7, 4)\)

Draw a smooth curve starting at \((3, 2)\) and passing through \((4, 3)\) and \((7, 4)\) extending to the right.