absolute value

Examples of absolute value in the following topics:

• Absolute Value

  • Absolute value can be thought of as the distance of a real number from zero.
  • For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0.
  • The term "absolute value" has been used in this sense since at least 1806 in French and 1857 in English.
  • Other names for absolute value include "numerical value," "modulus," and "magnitude."
  • The absolute values of 5 and -5 shown on a number line.

• Equations with Absolute Value

  • To solve an equation with an absolute value, first isolate the absolute value, and then solve for the positive and negative cases.
  • At face value, nothing could be simpler: absolute value simply means the distance a number is from zero.  
  • The absolute value of $-5$ is $5$, and the absolute value of $5$ is also $5$, since both $-5$ and $5$ are $5$ units away from $0$.
  • Recall that absolute value is a measure of distance, so it can never be a negative value.
  • The following steps describe how to solve an absolute value equation:

• Inequalities with Absolute Value

  • Inequalities with absolute values can be solved by thinking about absolute value as a number's distance from 0 on the number line.
  • More complicated absolute value problems should be approached in the same way as equations with absolute values: algebraically isolate the absolute value, and then algebraically solve for $x$.
  • It is difficult to immediately visualize the meaning of this absolute value, let alone the value of $x$ itself.
  • Now think: the absolute value of the expression is greater than –3.
  • Absolute values are always positive, so the absolute value of anything is greater than –3!

• The Leading-Term Test

  • All polynomial functions of first or higher order either increase or decrease indefinitely as $x$ values grow larger and smaller.
  • The properties of the leading term and leading coefficient indicate whether $f(x)$ increases or decreases continually as the $x$-values approach positive and negative infinity:
  • Intuitively, one can see why we need to look at the leading coefficient to see how a polynomial behaves at infinity: When $x$ is very big (in absolute value), then the highest degree term will be much bigger (in absolute value) than the other terms combined.
  • and the absolute value of $x$ is bigger than $MnK$, where $M$ is the absolute value of the largest coefficient divided by the leading coefficient, $n$ is the degree of the polynomial and $K$ is a big number, then the absolute value of $a_nx^n$ will be bigger than $nK$ times the absolute value of any other term, and bigger than $K$ times the other terms combined!

• Piecewise Functions

  • Example 1: Consider the piecewise definition of the absolute value function:
  • For all $x$-values less than zero, the first function $(-x)$ is used, which negates the sign of the input value, making the output values positive.
  • For all values of $x$ greater than or equal to zero, the second function $(x)$ is used, making the output values equal to the input values.  
  • After finding and plotting some ordered pairs for all parts ("pieces") of the function the result is the V-shaped curve of the absolute value function below.
  • The piecewise function, $\left | x \right |= \left\{\begin{matrix} -x, & if\ x<0\\ x, & if\ x\geq0 \end{matrix}\right.$, is  the graph of the absolute value function.  

• One-to-One Functions

  • To see this, note that the points of intersection have the same y-value, because they lie on the line, but different x values, which by definition means the function cannot be one-to-one.
  • Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.  
  • This is an absolute value function, which is graphed below.

• Complex Conjugates

  • The modulus symbol looks just like the absolute value symbol, which is okay because whenever $b=0$ so that $z=a+bi=a$ is a real number, we have that the conjugate is $a-bi=a$.
  • So the symbol is consistent with the use of the absolute value symbol.

• Basics of Graphing Polynomial Functions

  • As $\frac {x^3}{4}$ tends to be much larger (in absolute value) than $\frac {3x^2}{4} - \frac {3x}{2} - 2$ when $x$ tends to positive or negative infinity, we see that $y$ goes, like $\frac {x^3}{4}$, to negative infinity when $x$ goes to negative infinity, and to positive infinity when $x$ goes to positive infinity.
  • Another easy point to draw is the intersection with the $y$-axis, as this equals the function value in the point zero, which equals the constant term of the polynomial.
  • In general, the more function values we compute, the more points of the graph we know, and the more accurate our graph will be.
  • Conversely, we can easily read the constant term of the polynomial by looking at its intersection with the $y$-axis if its graph is given (and indeed, we can readily read any function value if the graph is given).
  • The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).

• Slope

  • The steepness, or incline, of a line is measured by the absolute value of the slope.
  • A slope with a greater absolute value indicates a steeper line.
  • Given two points $(x_1, y_1)$ and $(x_2, y_2)$, take a look at the graph below and note how the "rise" of slope is given by the difference in the $y$-values of the two points, and the "run" is given by the difference in the $x$-values.

• Relative Minima and Maxima

  • Relative minima and maxima are points of the smallest and greatest values in their neighborhoods respectively.
  • The absolute maximum is the y-coordinate which is $16$.
  • The absolute minimum is the y-coordinate which is $-10$.
  • The local minimum is at the $y$-value of−16 and it occurs when $x=2$.
  • For the function pictured above, the absolute maximum occurs twice at $y=16$ and the absolute minimum is at $(3,-10)$.