Radical Functions Overview

Overview

This lesson introduces radical functions, focusing on how to graph, analyze, and transform them, particularly the basic form y = √x, and examines how changes affect domain and range.

Introduction to Radical Functions

• A radical function contains a variable inside a radical, commonly a square root.
• The simplest radical function is y = √x.
• In this unit, we analyze and graph radical functions, solve single-radical equations, and perform transformations.

Analyzing y = √x

• Values of x less than 0 are undefined for y = √x; only x ≥ 0 gives valid outputs.
• The domain is x ≥ 0 (x is any real number).
• The range is y ≥ 0 (y is any real number).
• The graph starts at (0,0) and curves gently upwards, resembling half a sideways parabola.

Transformations of Radical Functions

• General form: y - K = a√[B(x - H)]
• Vertical translation: y = √x + K shifts graph up/down by K units.
• Horizontal translation: y = √(x - H) shifts graph left/right by H units.
• Reflection in x-axis: y = -√x flips graph over x-axis.
• Reflection in y-axis: y = √(-x) flips graph over y-axis (domain becomes x ≤ 0).
• Vertical stretch/compression: y = a√x stretches (|a| > 1) or compresses (0 < |a| < 1) vertically.
• Horizontal stretch/compression: y = √(B x) stretches/compresses horizontally by 1/|B|.

Effects on Domain and Range

• Horizontal shifts affect the starting x-value of the domain.
• Vertical shifts affect the minimum y-value of the range.
• Reflections over axes change the sign direction of domain or range.
• Stretches do not affect which x-values are valid, only the rate of increase.

Examples of Transformations and Their Domains/Ranges

• y - 4 = √(x + 2): shifted 2 left, 4 up; domain x ≥ -2, range y ≥ 4.
• 2y = √(-x): vertical stretch by ½, reflection in y-axis; domain x ≤ 0, range y ≥ 0.
• y + 5 = -√x: reflection over x-axis, 5 down; domain x ≥ 0, range y ≤ -5.
• y = 3√(2x - 1): vertical stretch by 3, horizontal compression by ½, ½ right; domain x ≥ ½, range y ≥ 0.

Combined Effects of a and B in y = a√(Bx)

• a > 0, B > 0: domain x ≥ 0, range y ≥ 0.
• a < 0, B > 0: domain x ≥ 0, range y ≤ 0.
• a > 0, B < 0: domain x ≤ 0, range y ≥ 0.
• a < 0, B < 0: domain x ≤ 0, range y ≤ 0.

Key Terms & Definitions

• Radical Function — a function with the variable inside a radical (root) symbol.
• Domain — set of all x-values that give valid y-values for a function.
• Range — set of all possible y-values that can be output by a function.
• Vertical/Horizontal Translation — shifting the graph up/down or left/right.
• Vertical/Horizontal Stretch — stretching or compressing the graph in the y or x direction.
• Reflection — flipping the graph over the x-axis (y = -√x) or y-axis (y = √(-x)).

Action Items / Next Steps

• Complete the assignment analyzing radical function domains, ranges, and transformations.
• Review key transformations (shift, stretch, reflect) and how they affect domain/range.
• Practice graphing radical functions and verifying results with a calculator or graphing tool.