Section 5.4: Graphing Radical Functions

This section covers graphing square root and cube root functions, including transformations such as shifts, stretches, compressions, and reflections.

Example 1: Graphing a Square Root Function

Graph \( f(x) = \sqrt{x-2} + 3 \).

Step 1: Identify parent function: \( y = \sqrt{x} \)

Step 2: Horizontal shift right by 2 → new start point at (2,0)

Step 3: Vertical shift up by 3 → new start point at (2,3)

Step 4: Sketch curve increasing to the right from (2,3)

Graph \( g(x) = -2 \sqrt{x+1} - 4 \).

Step 1: Parent function: \( y = \sqrt{x} \)

Step 2: Horizontal shift left by 1 → start at (-1,0)

Step 3: Vertical stretch by factor of 2 → slope steeper

Step 4: Reflection over x-axis → curve points downward

Step 5: Vertical shift down 4 → new start at (-1,-4)

Graph \( f(x) = \sqrt{x+3} - 2 \)
Graph \( g(x) = -\sqrt{x-4} + 1 \)
Graph \( h(x) = 3 \sqrt{x+2} \)
Graph \( k(x) = \sqrt{x} - 5 \)
Graph \( m(x) = -\sqrt{x-1} - 3 \)