Absolute Value Function
At a rod factory in Monterrey, the quality-control machine doesn't care whether a rod is too long or too short: it only cares how far it deviates from the target length. The absolute value function turns any signed difference into an always-positive distance, and that changes the shape of the graph in a very specific way. Which lengths pass inspection? We'll solve it at the end.
From absolute value to a piecewise-defined function
The reverse path: from pieces to absolute value
Sometimes you are given the piecewise function and asked to write it in compact form. You look for an expression $E(x)$ such that one piece is $E(x)$ and the other is $-E(x)$, and such that $E(x)\ge0$ exactly where the positive piece is used.
Express the function $f(x) = |3x - 6|$ as a piecewise function, clearly stating the intervals corresponding to each algebraic expression.
Express the function $f(x) = |3x - 6|$ as a piecewise function, clearly stating 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…
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 compactly using the modulus function.
In a laboratory, the temperature deviation of a chemical reaction from the ideal temperature of $22^\circ\text{C}$ is given by the function $D(t) = |2t - 12|$, where $t$…
In a laboratory, the temperature deviation 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.
The vertex: domain, range, and intercepts
The domain of these functions is $\mathbb{R}$. The range comes from the vertex: for $h(x)=|x|+5$ it is $[5, \infty)$.
Determine the domain and range of the function $h(x) = |x| + 5$. Justify your answer based on the properties of the modulus.
Determine the domain and range of the function $h(x) = |x| + 5$. Justify your answer based on the properties of the modulus.
Given the following graph of a modulus function, deduce its analytical expression in the form $y = |x - a|$.
Given the following graph of a modulus function, deduce its analytical expression in the form $y = |x - a|$.
Determine the exact coordinates of the axis intercepts of the function $h(x) = |x - 3| - 5$.
Determine the exact coordinates of the axis intercepts of the function $h(x) = |x - 3| - 5$.
When the coefficient is negative: maxima and models
If the absolute value is multiplied by a negative number, the "V" flips over: you get a "Λ" with the vertex as a maximum, and the branches open downward.
Determine the coordinates of the vertex and the axis intercepts of the function $f(x) = -|2x - 6| + 8$. Do the branches of this function open towards positive or…
Determine the coordinates of the vertex and the axis intercepts of the function $f(x) = -|2x - 6| + 8$. Do the branches of this function open towards positive or negative y-values?
In a billiards simulation, a ball bounces off the cushion located on the x-axis. The path of the ball follows the function $y = 4 - |x - 3|$, where $x$ and $y$ are…
In a billiards simulation, a ball bounces off the cushion located on the x-axis. The path of the ball follows the function $y = 4 - |x - 3|$, where $x$ and $y$ are measured in units of the simulator's grid.
Determine analytically (without drawing the graph, or using it only to check):
- a) The coordinates of the point where the ball reaches its maximum distance from the cushion.
- b) The coordinates of the points where the ball would hit the X-axis.
In signal processing, an inverted triangular wave can be modelled using the function $S(t) = A - |B t - C|$. Determine the parameters $A, B, C > 0$ knowing that the…
In signal processing, an inverted triangular wave can be modelled using 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 crosses the x-axis (value $0\text{ V}$) at times $t = 2\text{ s}$ and $t = 8\text{ s}$.
Average rate of change in a function with absolute value
The average rate of change of $f$ on the interval $[a, b]$ tells you how much $f$ changes, on average, for each unit that $x$ advances. It is the slope of the secant line joining $(a, f(a))$ and $(b, f(b))$.
With absolute value, the graph changes direction at the vertex. The average rate of change only uses the endpoints of the interval, so it does not "see" that turn.
Simplify the expression for the function $f(x) = 2x + |x - 4|$ by writing it as a piecewise function. Then, calculate the average rate of change on the interval $[2, 6]$.
Simplify the expression for the function $f(x) = 2x + |x - 4|$ by writing it as a piecewise function. Then, calculate the average rate of change on the interval $[2, 6]$.
Quadratic and rational functions with absolute value
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 function with absolute value $f(x) = |x^2 - 25|$ as a piecewise function.
Express the quadratic function with absolute value $f(x) = |x^2 - 25|$ as a piecewise function.
A company's profit margin, in thousands of pesos, relative to its annual target is modeled by the function $M(x) = |x^2 - 40x + 300|$, where $x$ is the number of months…
A company's profit margin, in thousands of pesos, relative 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 as a piecewise function to identify in which periods the company was exactly on target, below it (before applying the absolute value), or above it.
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.
Parameters and rational functions
When a graph "touches" the $x$-axis at just one point, the expression inside has a double root: the discriminant equals $0$. When the roots $r_1$ and $r_2$ of $x^2+bx+c$ are known, it factors as $(x-r_1)(x-r_2)$, and from that you get $b=-(r_1+r_2)$ and $c=r_1 r_2$.
Reading $b$ and $c$ from the graph of $|x^2+bx+c|$
The parabola $y=x^2+bx+c$ opens upward. When you apply absolute value, the part between the roots $r_1$ and $r_2$ (where it was negative) is reflected and forms an upside-down "arch". The vertex of the original parabola, which was a minimum at $x=\frac{r_1+r_2}{2}$, becomes a local maximum of $f$, with positive value $\frac{(r_2-r_1)^2}{4}$. Outside the roots, the graph follows the original parabola.
Why? The vertex of $y=(x-r_1)(x-r_2)$ is exactly halfway between the roots; there $y=-\frac{(r_2-r_1)^2}{4}$, and reflecting changes its sign.
For a rational function, such as $\left|\frac{2x-4}{x+1}\right|$, you study the sign of the quotient (numerator and denominator) and change the sign wherever it is negative. The domain excludes the values that make the denominator zero ($x\ne-1$).
Asymptotes
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Vertical: at $x=-1$ the denominator approaches $0$ and the numerator approaches $-6$. The quotient grows without bound in absolute value, so $|f(x)|\to\infty$ and the line $x=-1$ is a vertical asymptote.
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Horizontal: you look at what happens to $f(x)$ when $x$ becomes very large ($x\to\infty$) or very negative ($x\to-\infty$). This is an idea from limits that you will see in detail in LIM-INF; here the intuition is enough.
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 the modulus.
The graph shown corresponds to the function $f(x) = |x^2 + bx + c|$. From the observable key points (roots and local maximum), set up a system of equations and determine…
The graph shown corresponds to the function $f(x) = |x^2 + bx + c|$. From the observable key points (roots and local maximum), set up a system of equations and determine the values of the parameters $b$ and $c$.
Express the rational function with a modulus $f(x) = \left| \frac{2x - 4}{x + 1} \right|$ as a piecewise function. Next, determine its domain and the equations of its…
Express the rational function with a modulus $f(x) = \left| \frac{2x - 4}{x + 1} \right|$ as a piecewise function. Next, determine its domain and the equations of its horizontal and vertical asymptotes.
Equations and inequalities with 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 $f(x) = |2x - 8|$.
Find analytically the points of intersection between the modulus function $f(x) = |3x - 2|$ and the constant function $g(x) = 7$.
Find analytically the points of intersection between the modulus 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 rod lengths.
Find the domain of definition of the radical function $h(x) = \sqrt{8 - |x^2 - 8|}$. To do so, set up and solve the corresponding inequality involving modulus.
Find the domain of definition of the radical function $h(x) = \sqrt{8 - |x^2 - 8|}$. To do so, set up and solve the corresponding inequality involving modulus.
Two absolute values: cases by intervals
When there is more than one absolute value, you look for the points where each expression inside equals zero. Those points split the number line into intervals, and in each interval you decide the sign of each expression and remove the bars.
Absolute value inside another
Sum of two absolute values
Intersection of two functions with absolute value
The following graph shows a function with a nested modulus of the form $f(x) = ||x| - c|$. Observe the vertices and intercepts, deduce the value of $c$ and write the…
The following graph shows a function with a nested modulus of the form $f(x) = ||x| - c|$. Observe the vertices and intercepts, deduce the value of $c$ and write the expression for the piecewise function on the interval $x \in [-4, 4]$.
The altitude error of an inspection drone during a manoeuvre is modelled using the sum of two modulus expressions: $E(t) = |t^2 - 4t + 3| + |t - 2|$, where $t$ is time…
The altitude error of an inspection drone during a manoeuvre is modelled using the sum of two modulus expressions: $E(t) = |t^2 - 4t + 3| + |t - 2|$, where $t$ is time in seconds. Express the function $E(t)$ without using modulus signs 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: analyse the different cases depending on the sign of the…
Find analytically the points of intersection of the functions $f(x) = |x + 1|$ and $g(x) = |2x - 5|$. Hint: analyse the different cases depending on the sign of the expressions inside the modulus signs.
Back to the beginning: the rods that pass inspection measure between $248.5$ and $251.5$ mm.