Question

determine the parent function.
\\( y = x^3 \\)
\\( y = \sqrt{x} \\)
\\( y = \frac{1}{x} \\)

identify the equation of the function.
\\( y = \sqrt{x-1} + 3 \\)
\\( y = \sqrt{x+1} + 3 \\)
\\( y = \sqrt{x-1} - 3 \\)

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Identify the starting point of the graph

The parent square root function \( y = \sqrt{x} \) starts at the origin \( (0, 0) \).

Looking at the provided graph, the starting point (endpoint) of the curve has been shifted:

• \( 1 \) unit to the right
• \( 3 \) units up

The new starting point is at \( (1, 3) \).

Step 2: Apply the transformations to the equation

To shift a function \( y = f(x) \):

• Right by \( h \) units: replace \( x \) with \( (x - h) \)
• Up by \( k \) units: add \( k \) to the outside of the function

For \( h = 1 \) and \( k = 3 \):

$$ y = \sqrt{x - 1} + 3 $$

Answer:

\( y = \sqrt{x-1}+3 \)