🧮

Radical Functions Overview

Sep 16, 2025

Overview

This lecture focused on radical functions, including how to convert between radical and exponent forms, analyze their properties (domain, range, intercepts, end behavior), and perform basic graph transformations such as reflections and shifts. The lesson also included strategies for solving problems algebraically and graphically, and tips for exam questions.

Radical and Exponent Forms

  • Radical expressions can be rewritten using fractional exponents:
    • ( x^{1/2} ) is equivalent to ( \sqrt{x} ).
    • ( x^{1/n} ) is equivalent to the n-th root of ( x ).
  • The index is the root value in a radical (e.g., 2 for square root, 3 for cube root).
  • The radicand is the value under the radical sign.
  • When using graphing calculators like Desmos:
    • For square roots, type sqrt(x).
    • For cube roots, type cbrt(x).
    • For higher roots, use exponent notation (e.g., ( x^{1/4} ) for the fourth root).
  • Translating between radical and exponent forms is essential for graphing and simplifying expressions.

Properties of Radical Functions

  • Domain: The set of possible ( x ) values. For most radical functions, the radicand must be non-negative (for even roots).
  • Range: The set of possible output values (( y )). For square root functions, ( y \geq 0 ).
  • The parent function ( y = \sqrt{x} ):
    • Domain: [0, ∞)
    • Range: [0, ∞)
    • Intercept: Only at (0, 0)
    • The function is always increasing on its domain.
    • End behavior: As ( x \to \infty ), ( y \to \infty ).
  • For radical functions, negative values under the radical (for even roots) result in imaginary numbers, which are not considered in this context.

Working with Domain and Range

  • To find the domain of a radical function, set the radicand ( \geq 0 ) and solve for ( x ).
    • Example: For ( y = \sqrt{25 - x} ), set ( 25 - x \geq 0 ) so ( x \leq 25 ).
      • Domain: (−∞, 25]
      • Range: [0, ∞)
  • You can check domain values by substituting numbers into the radicand to ensure the result is non-negative.
  • The range is often determined by considering the possible outputs or by graphing the function.
  • For more complex functions, graphing is especially helpful for visualizing the range.

Transformations of Radical Functions

  • Reflections:
    • Negative sign outside the radical (( -\sqrt{x} )): reflects the graph over the x-axis.
    • Negative sign inside the radical (( \sqrt{-x} )): reflects the graph over the y-axis.
    • Negative signs both inside and outside: reflects over both axes.
  • Vertical Shifts:
    • Adding or subtracting a value outside the radical (( \sqrt{x} + k )): moves the graph up or down by ( k ).
  • Horizontal Shifts:
    • Adding or subtracting a value inside the radical (( \sqrt{x + h} )): moves the graph left or right by ( h ) (opposite the sign).
    • For example, ( \sqrt{x + 3} ) shifts left by 3; ( \sqrt{x - 3} ) shifts right by 3.
  • Parent Function: The original, simplest form of the function (e.g., ( y = \sqrt{x} )). Transformations create "child" functions by shifting, reflecting, or stretching the parent.
  • Vertical and horizontal shifts can be identified by whether the change is outside (vertical) or inside (horizontal) the radical.

Example: Solving for Increasing Interval

  • To determine where a radical function is increasing:
    • Factor the radicand to find critical points (where the function starts or stops being defined).
    • Graph the function to see where it rises.
    • Example: For ( y = \sqrt{x^2 - x - 12} ), factor the radicand to find roots at ( x = -3 ) and ( x = 4 ).
      • The function is increasing from ( x = 4 ) to ( \infty ).
  • Graphing is often the quickest way to identify intervals of increase or decrease.

Exam Strategy & Algebraic vs. Graphical Solutions

  • For domain questions, solve algebraically by setting the radicand ( \geq 0 ).
  • For range or intervals of increase/decrease, graphing is usually more effective.
  • To answer multiple-choice questions:
    • Use algebra to eliminate options based on domain restrictions.
    • Graph the remaining options to check range or behavior.
  • Always include endpoints in the domain if the radicand can equal zero (since ( \sqrt{0} = 0 ) is defined).

Simplifying Radical Functions

  • Factor numerators and denominators to simplify expressions.
  • Cancel common terms where possible.
  • The degree (highest exponent) of the function indicates the number of possible real roots or factors.
  • Always check for discontinuities or holes when simplifying.

Key Terms & Definitions

  • Radical: An expression involving a root, such as ( \sqrt{x} ).
  • Radicand: The value under the radical sign.
  • Index: The degree of the root (e.g., 2 for square root).
  • Parent Function: The simplest form of a function type, such as ( y = \sqrt{x} ).
  • Domain: All possible ( x )-values for a function.
  • Range: All possible ( y )-values for a function.
  • Reflection: Flipping a graph over an axis (x-axis or y-axis).
  • Vertical Shift: Moving a graph up or down.
  • Horizontal Shift: Moving a graph left or right.
  • Transformation: Any change to the parent function, including shifts, reflections, stretches, or compressions.

Action Items / Next Steps

  • Review lesson 104 in your course materials for more examples and explanations.
  • Complete assigned homework on radical functions and their transformations.
  • Practice graphing radical functions and identifying their domain and range.
  • Prepare for exam questions by practicing both algebraic and graphical approaches.
  • Take screenshots or notes of key tables and cheat sheets provided in the lesson for quick reference.

Full transcript