Unit 1

Functions

Read

Part 1.2D corresponds to section 3.4, Equations and Graphs of Polynomial Functions, starting on page 136 of your Pre-Calculus 12 textbook.

At the beginning of this lesson, various types of polynomial functions were introduced. The shapes of the graphs were predicted by thinking about their end behaviour and the possible number of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts.

The zeros of a function 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 of the graph of that function. The zeros are the values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» for which the function would be equal to zero.

The factors of a polynomial function can be used to determine the zeros of the function and the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts of the function’s graph.

Use the graph to determine

  1. the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts of the function,
  2. the least possible degree of the function,
  3. the sign of the leading coefficient,
  4. the possible factors of the function of the least possible degree, and
  5. the intervals for which the function is positive and negative.

a.
The graph crosses 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»«mn»3«/mn»«/mstyle»«/math».
The graph touches 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». Because it just touches the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis, and does not cross it, there are an even number of zeros 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». The fewest possible number of zeros is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math».
b.
Because we do not see the entire graph, we cannot assume that there are no more «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts. However, from the section of the graph that we do see, we can determine the least possible degree of the function. We see at least three zeros represented, so the least possible degree of the function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math».
c.
The leading coefficient is positive based on the end behaviour of the graph. The left end is extending down into Quadrant III and the right end is extending up into quadrant I.
d.
Because the zeros are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math», the possible factors are «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»«mo»,«/mo»«mo»§#160;«/mo»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
e.
Split the graph into intervals or sections based on the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts. There are three intervals, as shown in the graph.


In the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»§#60;«/mo»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#60;«/mo»«mn»0«/mn»«/mstyle»«/math» .
In the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»1«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#60;«/mo»«mn»0«/mn»«/mstyle»«/math».
In the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»§#62;«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#62;«/mo»«mn»0«/mn»«/mstyle»«/math».