MAT 117 Week 6 Lesson Plan
I. Objectives:
1. Quadratic Functions
a. Understand the definitions of polynomial functions, constant functions, linear
functions, and quadratic functions.
b. Understand the concepts of axis and vertex of a parabola, and use them to graph
quadratic functions.
c. Be able to put quadratic functions in standard form and use standard form to graph
quadratic functions.
d. Be able to construct and use a quadratic model.
2. Polynomial Functions of Higher Degree
a. Apply the Leading Coefficient Test.
b. Be able to graph polynomials. Use characteristics of graphs of polynomials:
continuity, no sharp corners.
c. Find real zeros of some polynomials and understand repeated zeros and multiplicity.
II. Motivation [5 min]
Section 2.5 discussed linear modeling. However, there are some situations for
which a linear model is inappropriate. For example, consider profit functions. If a profit
function were linear and upward-sloping, a firm could make an arbitrarily high amount of
profit just by producing more and more items! As another example, consider the
distance a falling object travels as a function of time. If the time doubles, the distance
more than doubles. Hence a linear model is clearly inappropriate in this situation.
Section 4.1 discusses quadratic functions. Quadratic functions are more flexible
than linear functions; for example, the rate by which a quadratic function increases is
not constant. They can model the distance of a falling object as a function of time.
They can also provide simple models of cost and profit functions.
Section 4.2 discusses polynomials of higher degree. These are even more
flexible than quadratic functions. They can have more than one turning point. However,
they are also more complicated than quadratic functions; for example, it is harder to find
roots of higher-degree polynomials algebraically. Higher-degree polynomials can
provide more sophisticated models of cost and profit functions.
1. Quadratic Functions
a. Understand the definitions of polynomial functions, constant functions, linear
functions, and quadratic functions. [15 min]
Warm-up example or activity:
Review linear modeling and discuss some of the limitations of linear models. Ask
students whether linear models would be appropriate in the following situations:
revenue functions, cost functions, distance traveled by a falling object vs. time,
population growth over time.
Formal Concept: Define polynomial function, constant function, linear function, quadratic
function, and parabola.
Examples:
Graph each of the following functions and discuss their graphs:
1. y  8
2. y  14 x  6
3. x 2  6 x  5
4. x 2  9 x  14
b. Understand the concepts of axis and vertex of a parabola, and use them to graph
quadratic functions. [15 min]
Discuss how the graph of a quadratic is symmetric about some vertical axis, and has
either max or min at the vertex. Use the standard form of a quadratic function to find
the axis of symmetry, identify the vertex, and determine if it is a min or a max.
Examples:
1. y  2( x  3)2  3
2. y  3( x  2)2  1
3. y  5( x  3)2  2
4. y  7( x  1)2  3
c. Be able to put quadratic functions in standard form and use standard form to graph
quadratic functions. [20 min]
Remind them of completing the square, and show them how it is used to find the
standard form of a quadratic function. Plot the vertex, determine if it is a min or a max,
then choose a y value to solve the equation for to find the specific shape of the graph.
Examples:
1. y  4 x 2  16 x  13
2. y  3x 2  4 x  7
3. y 
 x 2  6 x  12
2
4. y  6 x 2  7 x  15
d. Be able to construct and use a quadratic model. [30 min]
Discuss strategies for approaching word problems. Identify the information given,
determine if it is more appropriate to study the problem analytically or using the
graphing utility on your calculator.
Examples:
1. For Master Widget, Inc., profit is given by -.01x2 + 2x +100, where x is the number of
widgets sold. How many widgets should Master Widget, Inc., sell to maximize its profit.
2. A family wants to build an enclosure for their new dog. They have a type of fence
that costs 20 dollars per foot. What is the largest enclosure that the family can make for
their dog if they have 1500 dollars to spend?
2. Polynomial Functions of Higher Degree
a. Apply the Leading Coefficient Test. [15 min.]
Discuss the difference between the graphs of polynomials whose leading term has an
odd power and those with an even power. Show them how the sign of the leading term
affects the graph and outline the four possible cases.
Examples:
1. y  3x3  4 x 2  2 x  6
2. y  ( x  3)6  6
3. y  2 x 4  3x  4
4. y   x5  9 x  17
b. Be able to graph polynomials. Use characteristics of graphs of polynomials:
continuity, no sharp corners. [15 min.]
Discuss the general trends of high order polynomials, including the maximum number of
roots and the maximum number of zeros. Discuss continuity and the fact that
polynomials never have sharp corners.
The examples for part (a) can be reused.
c. Find real zeros of some polynomials and understand repeated zeros and multiplicity.
[25 min]
Discuss some general strategies for factoring polynomials, i.e. factoring by grouping,
difference of squares, difference of cubes, and sum of cubes. Discuss the idea of
repeated zeros and how the number of repeated roots affects the behavior at that point.
Examples:
1.
y  5x3  25x 2  x  5
2. y  ( x  4)5
3. y  3x5  6 x 4  6 x3  12 x 2
4. y = x^3 - 1
Follow-up Activity
Homework or quiz. For a quiz, it is a good idea to combine concepts in one question.
For example, set up a quadratic model and find the vertices and intercepts.