Absolute Value Functions
Your thermostat is set to $70^\circ\text{F}$. It doesn't care whether the room is 3 degrees too warm or 3 degrees too cold. It only cares how far the room is from 70. That "distance from the target" is an absolute value, and it gives graphs a very recognizable shape. By the end of this page you'll be able to say exactly which temperatures keep the thermostat happy.
From Absolute Value to a Piecewise Function
The corner of the "V" is where the inside expression equals $0$. To rewrite $|E(x)|$ piecewise:
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Solve $E(x) = 0$. That is where the definition changes.
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Decide the sign of $E(x)$ on each side (test a point).
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Where $E(x) \ge 0$ write $E(x)$. Where $E(x) < 0$ write $-E(x)$.
Going backward, and evaluating
To write a piecewise function with absolute value, look for the $x$-value where the branches switch. Then see which branch is the opposite of the other. The branch that is valid on the "$\ge$" side is the inside expression.
To evaluate at a number, substitute, simplify the inside completely, and only then take the absolute value. For $h(x) = |2x - 7|$: $h(1) = |-5| = 5$ and $h(5) = |3| = 3$.
Average rate of change
Average rate of change on $[a, b]$ is still $\dfrac{f(b) - f(a)}{b - a}$. Just evaluate at the endpoints.
Express the function $f(x) = |3x - 6|$ as a piecewise function, clearly indicating the intervals corresponding to each algebraic expression.
Express the function $f(x) = |3x - 6|$ as a piecewise function, clearly indicating the intervals corresponding to each algebraic expression.
Given the following piecewise function: $f(x) = \begin{cases} x + 4 & \text{if } x \ge -4 \\ -x - 4 & \text{if } x < -4 \end{cases}$ Write its analytical expression in a…
Given the following piecewise function:
$f(x) = \begin{cases} x + 4 & \text{if } x \ge -4 \\ -x - 4 & \text{if } x < -4 \end{cases}$
Write its analytical expression in a compact form using absolute value.
Simplify the expression of the function $f(x) = 2x + |x - 4|$ by writing it as a piecewise function. Next, calculate the average rate of change in the interval $[2, 6]$.
Simplify the expression of the function $f(x) = 2x + |x - 4|$ by writing it as a piecewise function. Next, calculate the average rate of change in the interval $[2, 6]$.
The Vertex, Domain, Range and Intercepts
Slide $h$ and $k$ and watch the vertex move. Notice that $|x - 3|$ shifts the vertex to the right ($h = 3$), and the $-5$ at the end shifts it down.
If the function is just $f(x) = |ax + b|$, the vertex sits on the $x$-axis. Solving $|ax + b| = 0$ gives one solution, $x = -\tfrac{b}{a}$. The graph touches the axis at the vertex without crossing it.
Determine the domain and the range of the function $h(x) = |x| + 5$. Justify your answer based on the properties of absolute value.
Determine the domain and the range of the function $h(x) = |x| + 5$. Justify your answer based on the properties of absolute value.
Solve the equation $|2x - 8| = 0$ and explain what this result represents geometrically with respect to the graph of the function $f(x) = |2x - 8|$.
Solve the equation $|2x - 8| = 0$ and explain what this result represents geometrically with respect to the graph of the function $f(x) = |2x - 8|$.
Determine the exact coordinates of the points of intersection with the Cartesian axes of the function $h(x) = |x - 3| - 5$.
Determine the exact coordinates of the points of intersection with the Cartesian axes of the function $h(x) = |x - 3| - 5$.
Graphing and Writing Absolute Value Functions
If the inside has a coefficient on $x$, factor it out first: $|2x - 6| = |2(x - 3)| = 2|x - 3|$.
Given the following graph of an absolute value function, deduce its analytical expression in the form $y = |x - a|$.
Given the following graph of an absolute value function, deduce its analytical expression in the form $y = |x - a|$.
In a billiards simulation, a ball bounces against the cushion located on the x-axis. The trajectory of the ball follows the function $y = 4 - |x - 3|$. Observe the…
In a billiards simulation, a ball bounces against the cushion located on the x-axis. The trajectory of the ball follows the function $y = 4 - |x - 3|$. Observe the representation and analytically determine the coordinates of the point where the ball reaches its maximum separation from the cushion and the points where it would impact the x-axis.
Determine the coordinates of the vertex and the points of intersection with the axes of the function $f(x) = -|2x - 6| + 8$. Do the branches of this function open upward…
Determine the coordinates of the vertex and the points of intersection with the axes of the function $f(x) = -|2x - 6| + 8$. Do the branches of this function open upward or downward?
Absolute Value of a Quadratic
To write $|ax^2 + bx + c|$ piecewise, find the zeros of the quadratic, then check the sign of the quadratic on each interval by testing a point.
Evaluating works as before. For $g(x) = |-x^2 + 6x|$: $g(-2) = |-4 - 12| = 16$, $g(0) = 0$, $g(3) = |-9 + 18| = 9$ and $g(7) = |-49 + 42| = 7$.
Calculate the numerical value of the function $g(x) = |-x^2 + 4x|$ for the points $x = -1$, $x = 0$, $x = 2$ and $x = 5$.
Calculate the numerical value of the function $g(x) = |-x^2 + 4x|$ for the points $x = -1$, $x = 0$, $x = 2$ and $x = 5$.
Express the quadratic absolute value function $f(x) = |x^2 - 25|$ as a piecewise function.
Express the quadratic absolute value function $f(x) = |x^2 - 25|$ as a piecewise function.
Express the function $f(x) = |x^2 - 5x + 6|$ as a piecewise function. State in which interval or intervals the original parabola $y = x^2 - 5x + 6$ has been reflected…
Express the function $f(x) = |x^2 - 5x + 6|$ as a piecewise function. State in which interval or intervals the original parabola $y = x^2 - 5x + 6$ has been reflected across the x-axis.
Rational Expressions and Parameters
The same idea works for any expression inside the bars. Find where the inside is zero (or undefined), build a sign chart, and write one formula for each interval.
Finding a parameter
A frequent question is: "for what value of $k$ does $|x^2 + bx + k|$ touch the $x$-axis at exactly one point?" The graph touches the axis exactly where the quadratic is $0$. One touching point means a double root, so the discriminant $b^2 - 4ac$ must be $0$.
If the discriminant is negative the graph never touches the axis; if it is positive the graph touches it at two points.
A company's profit margin, in thousands of dollars, with respect to its annual target is modeled by the function $M(x) = |x^2 - 40x + 300|$, where $x$ is the number of…
A company's profit margin, in thousands of dollars, with respect to its annual target is modeled by the function $M(x) = |x^2 - 40x + 300|$, where $x$ is the number of months elapsed. Express the function piecewise to identify in which periods the company was exactly on target, below it (before taking the absolute value) or above it.
Given the function $f(x) = |x^2 - 2x + k|$, determine the value of the parameter $k$ knowing that the graph of the function touches the x-axis at a single point. For…
Given the function $f(x) = |x^2 - 2x + k|$, determine the value of the parameter $k$ knowing that the graph of the function touches the x-axis at a single point. For that value of $k$, express $f(x)$ without using absolute value.
Express the rational function with absolute value $f(x) = \left| \frac{2x - 4}{x + 1} \right|$ as a piecewise function. Then, determine its domain and the equations of…
Express the rational function with absolute value $f(x) = \left| \frac{2x - 4}{x + 1} \right|$ as a piecewise function. Then, determine its domain and the equations of its horizontal and vertical asymptotes.
Absolute Value Equations and Inequalities
Always isolate the absolute value first. Graphically, $|E(x)| = c$ asks where the graph of $y = |E(x)|$ meets the horizontal line $y = c$.
Back to the thermostat: the room is fine when its distance from $70^\circ\text{F}$ is at most $2$ degrees. That is $|T - 70| \le 2$.
In a laboratory, the deviation of the temperature of a chemical reaction from the ideal temperature of $22^\circ\text{C}$ is given by the function $D(t) = |2t - 12|$,…
In a laboratory, the deviation of the temperature of a chemical reaction from the ideal temperature of $22^\circ\text{C}$ is given by the function $D(t) = |2t - 12|$, where $t$ is the time in minutes. Express $D(t)$ as a piecewise function and determine at what instant the deviation is zero.
Find analytically the points of intersection between the absolute value function $f(x) = |3x - 2|$ and the constant function $g(x) = 7$.
Find analytically the points of intersection between the absolute value function $f(x) = |3x - 2|$ and the constant function $g(x) = 7$.
A machine cuts metal rods with a target length of $250\text{ mm}$. The cutting error function is defined as $E(x) = |x - 250|$. If quality control requires that…
A machine cuts metal rods with a target length of $250\text{ mm}$. The cutting error function is defined as $E(x) = |x - 250|$. If quality control requires that $E(x) \le 1.5$, set up the corresponding inequality and determine the interval of valid lengths for the rods.
Several Absolute Values and Absolute Values Inside Radicals
More than one absolute value
When an expression has several absolute values, find every place where an inside expression is zero. Those values cut the number line into intervals. On each interval, every inside has a fixed sign, so every bar can be removed.
If a problem gives you a specific interval, just check the sign of each inside on that interval. This works for quadratic insides too, such as $|t^2 - 4t + 3|$: factor to find its zeros and see where the interval sits relative to them.
Two absolute values in an equation
$|A| = |B|$ means $A = B$ or $A = -B$. Solve both and check.
Absolute value inside a radical
A square root needs a nonnegative radicand. So the domain of $\sqrt{E(x)}$ is the solution of $E(x) \ge 0$, which here is an absolute value inequality.
The altitude error of an inspection drone during a maneuver is modeled by the sum of two absolute values: $E(t) = |t^2 - 4t + 3| + |t - 2|$, where $t$ is the time in…
The altitude error of an inspection drone during a maneuver is modeled by the sum of two absolute values: $E(t) = |t^2 - 4t + 3| + |t - 2|$, where $t$ is the time in seconds. Express the function $E(t)$ without using absolute values in the time interval $t \in [1, 3]$.
Find analytically the points of intersection of the functions $f(x) = |x + 1|$ and $g(x) = |2x - 5|$. Hint: analyze the different cases according to the sign of the…
Find analytically the points of intersection of the functions $f(x) = |x + 1|$ and $g(x) = |2x - 5|$. Hint: analyze the different cases according to the sign of the expressions inside the absolute value.
Find the domain of definition of the radical function $h(x) = \sqrt{8 - |x^2 - 8|}$. To do this, set up and solve the corresponding absolute value inequality.
Find the domain of definition of the radical function $h(x) = \sqrt{8 - |x^2 - 8|}$. To do this, set up and solve the corresponding absolute value inequality.
Finding the Formula from Key Points
When a graph or a story gives you special points (vertex, zeros, a maximum), turn each one into an equation and solve the system.
Nested absolute values: $f(x) = ||x| - c|$
The inner $|x|$ makes the function even (symmetric about the $y$-axis). The outer bars reflect whatever dips below the $x$-axis. The graph looks like a "W": the zeros are at $x = \pm c$ and the middle peak is $(0, c)$. So $c$ can be read from the $x$-intercepts or from the $y$-intercept.
Fitting $A - |Bt - C|$
The vertex is at $t = \tfrac{C}{B}$ and its height is $A$. Each zero gives an equation.
Fitting $|x^2 + bx + c|$
The zeros of the graph are the zeros of the quadratic, and the local maximum is the reflected vertex. Plugging each known point into the formula gives a linear system in $b$ and $c$.
The following graph shows a nested absolute value function of the form $f(x) = ||x| - c|$. Observe the vertices and points of intersection, deduce the value of $c$ and…
The following graph shows a nested absolute value function of the form $f(x) = ||x| - c|$. Observe the vertices and points of intersection, deduce the value of $c$ and write the expression of the piecewise function for the interval $x \in [-4, 4]$.
In signal processing, an inverted triangular wave can be modeled with the function $S(t) = A - |B t - C|$. Determine the parameters $A, B, C > 0$ knowing that the signal…
In signal processing, an inverted triangular wave can be modeled with the function $S(t) = A - |B t - C|$. Determine the parameters $A, B, C > 0$ knowing that the signal reaches its maximum value of $12\text{ V}$ at time $t = 5\text{ s}$, and that it crosses the x-axis (value $0\text{ V}$) at times $t = 2\text{ s}$ and $t = 8\text{ s}$.
The graph shown corresponds to the function $f(x) = |x^2 + bx + c|$. From the observable singular points (roots and local maximum), set up a system of equations and…
The graph shown corresponds to the function $f(x) = |x^2 + bx + c|$. From the observable singular points (roots and local maximum), set up a system of equations and determine the values of the parameters $b$ and $c$.