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Review of Elementary Functions Review of ry Functions

# Review of Elementary Functions Review of ry Functions -...

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© 2001 by CRC Press LLC Supplement: Review of Elementary Functions In this supplement, we review the basic features and characteristics of the simple elementary functions. S.1 Affine Functions By an affine function, we mean an expression of the form y ( x ) = ax + b (S.1) In the special case where b = 0, we say that y is a linear function of x. We can interpret the parameters in the above function as representing the slope-intercept form of a straight line. Here, a is the slope, which is a measure of the steepness of a line; and b is the y -intercept (i.e., the line intersects the y -axis at the point (0, b )). The following cases illustrate the different possibilities: 1. a = 0: this specifies a horizontal line at a height b above the x -axis and that has zero slope. 2. a > 0: the height of a point on the line (i.e., the y -value) increases as the value of x increases. 3. a < 0: the height of the line decreases as the value of x increases. 4. b > 0: the line y -intercept is positive. 5. b < 0: the line y -intercept is negative. 6. x = k: this function represents a vertical line passing through the point ( k , 0). It should be noted that: If two lines have the same slope, they are parallel. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. (It is easy to deduce this

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© 2001 by CRC Press LLC property if you remember the relationship that you learned in trigonometry relating the sine and cosine of two angles that differ by π /2.) See Section S.4 for more details. S.2 Quadratic Functions Parabola A quadratic parabolic function is an expression of the form: y ( x ) = ax 2 + bx + c where a 0 (S.2) Any x for which ax 2 + bx + c = 0 is called a root or a zero of the quadratic func- tion. The graphs of quadratic functions are called parabolas. If we plot these parabolas, we note the following characteristics: 1. For a > 0, the parabola opens up (convex curve) as shown in Figure S.2 . 2. For a < 0, the parabola opens down (concave curve) as shown in Figure S.2 . FIGURE S.1 Graph of the line y = ax + b (a = 2, b = 5).
© 2001 by CRC Press LLC 3. The parabola does not always intersect the x -axis; but where it does, this point’s abscissa is a real root of the quadratic equation. A parabola can cross the x -axis in either 0 or 2 points, or the x -axis can be tangent to it at one point. If the vertex of the parabola is above the x -axis and the parabola opens up, there is no intersection, and hence, no real roots. If, on the other hand, the parabola opens down, the curve will intersect at two val- ues of x equidistant from the vertex position. If the vertex is below the x -axis, we reverse the convexity conditions for the existence of two real roots. We recall that the roots of a quadratic equation are given by: (S.3) When b 2 – 4 ac < 0, the parabola does not intersect the x -axis. There are no real roots; the roots are said to be complex conjugates. When b 2 – 4 ac = 0, the x -axis is tangent to the parabola and we have one double root.

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Review of Elementary Functions Review of ry Functions -...

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