Unit 1

Functions

Zeros, Roots, and x-intercepts

The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts of a function’s graph are where the graph touches or crosses the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math».

The zeros of a function are the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-values for which the function is equal to zero «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math».

The roots of an equation are the solutions to the equation.

As such, a function’s real zeros correspond to the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts on the graph of that same function, and correspond to the solutions, or roots, of the related equation.

A function’s graph may have zero, one, or more than one «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept, and a function may have zero, one, or more real zeros. Likewise, an equation may have zero, one, or more real roots.

It is important to note that the graph of a function may not cross the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis at the function’s zero. It may, instead, appear to bounce off the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis at that point.

In the previous Example, the graph of the function simply touched the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math». When this occurs, we know the function has an even number of factors corresponding to that zero. This is referred to as the multiplicity of the zero.

The zero at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math» had a multiplicity of at least «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» (The multiplicity could have been «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»4«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»6«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»8«/mn»«/mrow»«/mstyle»«/math», etc., but we were asked to consider the function of least possible degree). As such, the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» occurred at least twice (it could have occurred four times, six times, etc.). At a zero of even multiplicity, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»4«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»6«/mn»«mo»,«/mo»«mo»§#8230;«/mo»«mo»,«/mo»«/mstyle»«/math», a function’s graph behaves differently than at a zero of multiplicity «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» or at a zero of odd multiplicity, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»7«/mn»«mo»,«/mo»«mo»§#8230;«/mo»«mo»,«/mo»«/mstyle»«/math». The table below summarizes some common behaviours.

You may wish to refer to your Pre-Calculus 12 textbook for more examples. Example 4 on page 145 solves a problem using a polynomial function and its graph as a model of the situation described.