The page Slope in Sound & Picture is useful in understanding slope.
The slope of the squaring function is 2x, twice the value of the x at a given point. Its reciprocal is 1/x². For x values greater than or equal to zero, its inverse is x.
Squaring is useful in determining areas of plane figures. It is a widely used function.
When gravity and initial velocity and starting point are considered, but not wind resistance and angle of projection, thrown objects make parabolic, quadratic, paths.
These graphs are often used in examples in calculus.
The figure at the right shows a family of parabolas, y = ax², where a takes values such as 4, 3, 2, 1, 1/2, 1/4, 1/10, -1/10, -1/4, -1/2, -1, -2, -3, and -4. Each parabola looks similar to the squaring function pictured above on this page. The larger the magnitude of the number, the skinner the parabola. The smaller the magnitude of the number, the fatter or flatter the parabola. If a is positive, the parabola is concave up (opens up). If a is negative the parabola is concave down (opens down). In each case the vertex, or point a which the U-shape bends, is (0,0).
Parabolas are quadratics. The quadratic equation and discriminants are useful in determining vertex and x-intercepts of the graph. The page The Quadratic Equation, Formula, & Discriminant should be very useful.
Parabolas are conic sections [discussed later]. Parabolas often describe situations in which area is computed or distance is computed. Displacement of objects shot into the air or dropped from a height are often described by quadratic functions.
The square root function is important because it is the inverse function for squaring. It tells what number must be squared in order to get the input x value.
The square root of a number, (n), may be explored using this link.
Its slope is 1/(2 x). When its domain is greater than or equal to zero, its inverse is the squaring function. When its domain is all real numbers, its range includes complex numbers such as i, -1. [This page does not deal with graphs of complex numbers.]
See The Laws of Exponents.
Though the symbol for square root is familiar to many, a fractional exponent is not.
Above, y = x or y = xn when n = .5.
This means x = x.5 or x1/2. Other exponents are possible.
Pictures from pages on square root and index may help with vocabulary.
The cube root of x to the fifth, 3(x5), may be written as x5/3.
In general, the bth root of x to the a power, b(xa), is written (and simplified as one might simplify fractions) as xb/a.
This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).
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© 2005, Agnes Azzolino