Examples of absolute value in the following topics:
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- 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.
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- 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:
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- 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!
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- 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 anxn will be bigger than nK times the absolute value of any other term, and bigger than K times the other terms combined!
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- 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, ∣x∣={−x,x,if x<0if x≥0, is the graph of the absolute value function.
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- 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.
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- 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.
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- As 4x3 tends to be much larger (in absolute value) than 43x2−23x−2 when x tends to positive or negative infinity, we see that y goes, like 4x3, 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).
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- 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 (x1,y1) and (x2,y2), 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.
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- 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).