Graphing Absolute Value Equations and Solving Equations with Two Absolute Values

Practice vocabulary flashcards covering graphing absolute value functions, writing piecewise functions, and solving absolute value equations algebraically and graphically.

Absolute Value Reflection Rule

The rule stating that when taking the absolute value of a function, any portion of the graph below the $x$-axis (where $y < 0$) is reflected above the $x$-axis, while points with positive or zero $y$-coordinates remain unchanged and $x$-coordinates stay the same.

Absolute Value of a Linear Function

The V-shaped graph resulting from taking the absolute value of a linear function, with its vertex positioned at the $x$-intercept of the original line.

Piecewise Function Form of an Absolute Value Function

A representation of an absolute value function without absolute value signs, formed by dividing the domain at the $x$-intercepts and placing a negative sign in front of the expression for any domain interval where the original graph was reflected upside down.

Steps to Find the Piecewise Form of a Parabola

1. Find the $x$-intercepts by setting $y = 0$ to separate the domain into Left, Middle, and Right intervals. 2. Write the quadratic expression without absolute value bars for each interval. 3. Identify which regions were reflected above the $x$-axis. 4. Place a negative sign in front of the equation (with brackets) for any reflected region.

Graph of the Absolute Value of a Quadratic Function

A graph formed by reflecting any part of a parabola lying below the $x$-axis above the $x$-axis, splitting the domain into three intervals separated by the parabola's $x$-intercepts.

Four-Case Method for Solving Two Absolute Values

An algebraic method used for equations containing two absolute value expressions, which considers four distinct cases corresponding to all positive and negative sign combinations for the two absolute value terms, solving each case separately.

Extraneous Solution

An algebraic result obtained from solving one of the case equations in an absolute value problem that does not satisfy the original equation when checked by substitution.

Graphical Method for Solving Equations with Two Absolute Values

A method where an equation with two absolute value expressions is rearranged by moving one absolute value term to the opposite side to form two separate functions $y_1$ and $y_2$; the $x$-coordinates of the intersection points of their graphs provide the solutions.

Piecewise Form of $y = \left| -\frac{3}{2}x + 3 \right|$

The piecewise function representation split at the $x$-intercept $x = 2$, written as $y = -\frac{3}{2}x + 3$ for $x < 2$ (left) and $y = -\left(-\frac{3}{2}x + 3\right)$ for $x \ge 2$ (right).

Graphical Solution of $|x + 3| + |x - 5| = 9$

The graphical solution process where the equation is rewritten as $|x + 3| = 9 - |x - 5|$, yielding two graph intersection points at $(-3.5, 0.5)$ and $(5.5, 8.5)$, corresponding to the solutions $x_1 = -3.5$ and $x_2 = 8.5$.