2.4.0 Parent functions Essays - Fields Of Mathematics, Mathematics

2.4.0 Parent functions Today we will look at the graphs, domains, and ranges of four parent functions. Parent functions are the base functions, upon which transformations are applied. The line Grade 9 math focussed on the line. In function notation, the basic line is defined by [pic]. |x |y | |-2 |-2 | |-1 |-1 | |0 |0 | |1 |1 | |2 |2 | This line continues forever to the left and right, up and down. [pic] The parabola Grade 10 focussed on the parabola. In function notation, the basic parabola is defined by [pic]. |x |y | |-2 |4 | |-1 |1 | |0 |0 | |1 |1 | |2 |4 | The parabola continues forever to the left and right, continues forever up, but has a minimum y value of zero. [pic] The radical function The radical function is related to the parabola. In function notation, the basic radical function is defined by [pic]. The radical function has serious restrictions on the domain and range. In the real number system, we cannot take the square root of a negative number, and the square root function yields only positive values. |x |y | |0 |0 | |1 |1 | |4 |2 | |9 |3 | |16 |4 | Starting at the origin, the radical function continues right forever and up forever. [pic] The reciprocal function: Rectangular hyperbola. In function notation, the basic reciprocal function is defined by [pic]. The reciprocal function has some interesting properties. Reciprocation does not cause a change in sign. Reciprocating a number close to zero yields a number far from zero, and reciprocating a number far from zero yields a number close to zero. Notice that we can not reciprocate zero, nor can a reciprocation yield zero. |x |y | |-4 |[pic] | |-1 |-1 | |[pic] |-4 | |0 |undefined| |[pic] |4 | |1 |1 | |4 |[pic] | Note the restriction, [pic]. This function continues left and right forever, up and down forever, but x can never be zero, and neither can y. [pic] The graph approaches the axes, but never crosses or touches. This behaviour is call asymptotic. A line that the graph approaches indefinitely is called an asymptote. Determining domain and range from equations This can be done from a sketch, if you know how to sketch. This can be done when using transformations, when you know how to transform. For now, search for problems in the equation (zeroes in the denominator, negatives under square roots, and maxima or minima). Determine the domain and range: Ex1. [pic] No problems here. [pic] Ex2. [pic] No problems for x. The smallest [pic] can be is 0, so the smallest y can be is 3. This can also be identified if you know this is a parabola that opens up with a vertex of [pic]. [pic] Ex3. [pic] [pic] cannot be negative, that is, [pic], so [pic]. The smallest [pic] can be is zero, so the smallest y can be is 2. [pic]