HS Math Solutions - Algebra: Absolute Value Functions

Algebra: Absolute Value Functions

Contents (4 topics • 34 videos)

Solving Absolute Value Equations (13 videos)

Solving and Graphing Absolute Value Inequalities (7 videos)

Graphing Absolute Value Functions (7 videos)

Writing Absolute Value Functions (7 videos)

Solving Absolute Value Equations

Introduction Videos

Solving Absolute Value Equations Algebraically

Solving Absolute Value Equations Graphically

Solving Double Absolute Value Equations

Solving Nested Absolute Value Equations

Theorems and Definitions

Solving Absolute Value Equations Absolute value measures the distance a number is from the origin. The absolute value equation, \( \big| x-k \big| = d \), can be solved by breaking it first into two distinct equations: \[ x-k=d \quad \quad \quad x-k=-d \] Note: In the original equation, \( \big| x-k \big| = d \), if \( d \lt 0 \), the equation has no solution.

Video Practice Problems Playlist on YouTube

Solve.
\[ \Big|n-4\Big|=7 \]

Solve.
\[ -7\Big|n-2\Big|+5=19 \]

Solve.
\[ \frac{2}{3} \Big|2c-3\Big|+5=11 \]

Solve.
\[ \frac{2\big|5y-2\big|}{-3} +5=-7 \]

Solve.
\[ 6+9\Big|7-4y\Big|=87 \]

Solve.
\[ \Big| 2x+7 \Big| =12- \Big| 5x-2 \Big| \]

Solve.
\[ \Big| 2x-1 \Big| + x = \Big| 3x+2 \Big| - 5 \]

Solve.
\[ \Big| |x+2| -8 \Big| = \frac{1}{2}x+6 \]

Write an absolute value equation whose solutions are
\[ x=-9, 3 \text{.} \]

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)

Graph Paper (.PDF)

Solving and Graphing Absolute Value Inequalities

Introduction Videos

Solving and Graphing Absolute Value Inequalities

Solving Double Absolute Value Inequalities

Video Practice Problems Playlist on YouTube

Solve and graph the inequality.
\[ -5 \Big| m \Big| \gt-30 \]

Solve and graph the inequality.
\[ \frac{\Big| 2t+3 \Big|}{-5} \leq -3 \]

Solve and graph the inequality.
\[ -5+ \Big| 4-3x \Big| \lt -25 \]

Solve and graph the inequality.
\[ \Big| 2c+1 \Big| \gt \Big| c+7 \Big| \]

Write an absolute value inequality whose solution set is \[ (-\infty, -3) \cup (9, +\infty) \text{.} \]

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)

Graphing Absolute Value Functions

Introduction Videos

Graphing Absolute Value Functions

Theorems and Definitions

Graphing Absolute Value Functions The general form of an absolute value function is \( f(x)=a \big| x-h \big| + k \). The absolute value function is a "v"-shaped graph. The graph of the parent function, \( f(x)=\big| x \big| \), is shown on the right.

\(a \gt 0\): the "v"-shape of the absolute value graph opens up
\(a \lt 0\): "v"-shape of the absolute value graph opens down (the graph has been reflected over the \(x\)-axis)
\( a \): the steepness of the legs of the graph (the slopes are \( \pm a \))
\( \big| a \big| \gt 1 \): the graph is stretched vertically
\( 0 \lt \big| a \big| \lt 1 \): the graph is compressed vertically
\(h\): horizontal shift in the graph of the function from the parent function, \(f(x)=\big| x \big|\)
\(k\): vertical shift in the graph of the function from the parent function, \(f(x)=\big| x \big|\)
\( (h, k) \) is the vertex of the graph

Video Practice Problems Playlist on YouTube

Graph the absolute value function.
\[ g(x)=2 \big| x+1 \big| \]

Graph the absolute value function.
\[ h(x)=- \frac{5}{2} \big| x \big| + 5 \]

Graph the absolute value function.
\[ f(x)= \frac{4}{3} \big| x-3 \big| - 1 \]

Graph the absolute value function.
\[ h(x)= \frac{ \big| x+2 \big| }{-2}-3 \]

Graph the absolute value function.
\[ f(x)=- \big| 4x-1 \big| +3 \]

Graph the absolute value function.
\[ f(x)= -2 \big| 2-3x \big| +4 \]

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)

Graph Paper (.PDF)

Writing Absolute Value Functions

Introduction Videos

Writing Absolute Value Functions from a Graph

Writing Absolute Value Functions Using its Vertex and a Second Point

Video Practice Problems Playlist on YouTube

Write an absolute value function whose vertex is point \( V \) and passes through point \( P \text{.} \)
\[ V(1, 4) \quad \quad P(-1, -2) \]

Write an absolute value function whose vertex is point \( V \) and passes through point \( P \text{.} \)
\[ V(-2, -5) \quad \quad P(4, 3) \]

Write an absolute value function whose intercepts are the given ordered pairs.
\[ (-5, 0) \quad (-1, 0) \quad (0, -2) \]

Write an absolute value function whose parent graph has been reflected over the \(x\)-axis and translated 4 units up.

Write an absolute value function whose parent graph has been compressed horizontally by a factor of \( \frac{3}{2} \), translated five units right, and translated two units down.

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)