Polynomial Functions Topic 1: Graphs of Polynomial Functions
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- Slide 1
- Polynomial Functions Topic 1: Graphs of Polynomial
Functions
- Slide 2
- I can describe the characteristics of a polynomial function by
analyzing its graph. I can describe the characteristics of a
polynomial function by analyzing its equation. I can match
equations to their corresponding graphs.
- Slide 3
- Information A polynomial function is a function that consists
of one or more terms added together. Each term consists of a
coefficient, a variable, and a whole number exponent. Variables
cannot have negative or fractional exponents The variable cannot be
in the exponent, the demoninator, or under a radical sign
- Slide 4
- Information The equations of polynomial functions can be
written in standard form.
- Slide 5
- Information In a polynomial function, The leading coefficient
is the coefficient of the term with the highest exponent of x in a
polynomial function in standard form. The constant term is the term
in which the variable has an exponent of 0. The degree of a
function is the greatest exponent of the function. Polynomial
functions are named according to their degree: constant functions
have degree 0, linear functions have degree 1 quadratic functions
have degree 2 cubic functions have degree 3
- Slide 6
- Example 1 Identify which of the following functions is a
polynomial function. Explain. a) b) c) Identifying a polynomial
function NO, since there is a fractional exponent. NO, since there
is a variable inside a radical. NO, since there is a variable in
the denominator.
- Slide 7
- Example 1 (continued) Identify which of the following functions
is a polynomial function. Explain. d) e) f) Identifying a
polynomial function YES. NO, since the variable has a negative
exponent.
- Slide 8
- Example 2 Write the terms in each of the following polynomial
functions in descending order. Identify the degree and the name of
each function. a) b) c) Identifying the degree of a polynomial
function Degree 1 Linear Function Degree 2 Quadratic Function
Degree 0 Constant Function
- Slide 9
- Example 2 (continued) Write the terms in each of the following
polynomial functions in descending order. Identify the degree and
the name of each function. d) e) f) Identifying the degree of a
polynomial function Degree 3 Cubic Function Degree 2 Quadratic
Function Degree 3 Cubic Function
- Slide 10
- Information The graphs of polynomial functions have many
characteristics. The characteristics that will be explored in this
topic are as follows The x-intercept is the x-value of the point
where a function crosses the x-axis. The y-intercept is the y-value
of the point where a function crosses the y-axis. The domain is the
x-values for which the function is defined. The range is the
y-values for which the function is defined. The end behaviour of a
function is the description of the graphs behaviour at the far left
and far right.
- Slide 11
- Information A turning point of a function is any point where
the y- values of a graph of a function change from increasing to
decreasing or change from decreasing to increasing. An absolute
maximum is the greatest value in the range of a function. An
absolute minimum is the least value in the range of a function. A
local maximum is a maximum turning point that is not the absolute
maximum. A local minimum is a minimum turning point that is not the
absolute minimum.
- Slide 12
- Information
- Slide 13
- The upcoming slides contain exploratory activities that you
should work through independently. To do so, you need to remember
the following: x-intercepts are the places where the graph crosses
the x- axis (the horizontal axis). y-intercepts are the places in
which the graph crosses the y-axis (the vertical axis). These ones
are the x-intercepts (x = -1 and x = 3) These ones are the
y-intercepts (y = 3)
- Slide 14
- Information domain: the set of all possible x-values range: the
set of all possible y-values For example: For more practice with
domain and range go to http://goo.gl/EwBa x
- Slide 15
- Use technology to investigate the characteristics of the
following constant functions. Set your windows to X:[-5, 5, 1] and
Y: [-5, 5, 1]. Record your findings in the table in your workbook.
Identifying characteristics of a constant function Complete this
activity in your book before continuing! Explore
- Slide 16
- You should notice
- Slide 17
- Use technology to investigate the characteristics of the
following linear functions. Set your windows to X:[-5, 5, 1] and Y:
[-5, 5, 1]. Record your findings in the table in your workbook.
Identifying characteristics of a linear function Complete this
activity in your book before continuing! Explore
- Slide 18
- You should notice
- Slide 19
- Use technology to investigate the characteristics of the
following quadratic functions. Set your windows to X:[-5, 5, 1] and
Y: [-5, 5, 1]. Record your findings in the table below. Identifying
characteristics of a quadratic function Complete this activity in
your book before continuing! Explore
- Slide 20
- You should notice
- Slide 21
- Use technology to investigate the characteristics of the
following cubic functions. Set your windows to X:[-5, 5, 1] and Y:
[-5, 5, 1]. Record your findings in the table below. Identifying
characteristics of a cubic function Complete this activity in your
book before continuing! Explore
- Slide 22
- You should notice
- Slide 23
- Example 3 a) How is the maximum possible number of x-intercepts
related to the degree of any polynomial function? b) Do all
polynomials of degree 0, 1, 2, or 3 have only one y- intercept?
What is the y-intercept for any polynomial function? Summarizing
and analyzing the characteristics of graphs of polynomials The
maximum possible number of x-intercepts is equal to the degree of
the polynomial function. All polynomials of degree 0, 1, 2, or 3
have only one y-intercept, and it is equal to the constant term in
the equation.
- Slide 24
- Example 3 c) What are the domain and range for all polynomial
functions? d) Explain why some cubic polynomial functions have
turning points but not maximum or minimum values. Summarizing and
analyzing the characteristics of graphs of polynomials Since cubic
functions continue to go in both directions (up and down), their
turning points are not maximum or minimum values.
- Slide 25
- Example 3 e) How is the leading coefficient in any polynomial
function related to the graph of the function? Summarizing and
analyzing the characteristics of graphs of polynomials The leading
coefficient determines a graphs end behaviour. Equations with
positive lead coefficients increase to the right. Equations with
negative lead coefficients decrease to the right.
- Slide 26
- Example 4 Determine characteristics of each function using its
equation. Using an equation to determine characteristics of a graph
Since these are degree 1 polynomials, they each have only 1
x-intercept. 01 y = -3 y = 1 0 turning points The y-intercept is
equal to the constant term. 0 turning points Linear functions are
straight lines so there are no turning points. Q3 Q4 Q2 Q4 Linear
equations go from Q3 Q1 (positive lead coefficient) or Q2 Q4
(negative lead coefficient).
- Slide 27
- Example 4 Determine characteristics of each function using its
equation. Using an equation to determine characteristics of a graph
Since these are degree 2 polynomials, they each have up to 2
x-intercepts. 22 y = 8 y = -6 1 turning point The y-intercept is
equal to the constant term. 1 turning point Quadratic functions are
parabolas so there are 1 turning point. Q3 Q4 Q2 Q1 Linear
equations go from Q2 Q1 (positive lead coefficient) or Q3 Q4
(negative lead coefficient). EXTRA (not in workbook)
- Slide 28
- Example 4 Determine characteristics of each function using its
equation. Using an equation to determine characteristics of a graph
Since these are degree 3 polynomials, they can have up to 3
x-intercepts. 33 y = -10 y = 3 2 turning points The y-intercept is
equal to the constant term. 2 turning points Cubic functions are
curves that can have up to 2 turning points. Q3 Q1 Q2 Q4 Cubic
equations go from Q3 Q1 (positive lead coefficient) or Q2 Q4
(negative lead coefficient). EXTRA (not in workbook)
- Slide 29
- Example 5 Match each graph with the correct polynomial
function. Justify your reasoning. Matching polynomial functions to
their graphs
- Slide 30
- Example 5 Matching polynomial functions to their graphs This
equation is a cubic function with a negative lead coefficient (goes
from Q2 Q4 and has a y-intercept of -2. It matches with graph v.
This equation is a quadratic function with a positive lead
coefficient (goes from Q2 Q1 and has a y-intercept of -2. It
matches with graph iv. This equation is a cubic function with a
positive lead coefficient (goes from Q3 Q1 and has a y-intercept of
-2. It matches with graph i.
- Slide 31
- Example 5 Matching polynomial functions to their graphs This
equation is a linear function with a negative lead coefficient
(goes from Q2 Q4 and has a y- intercept of -3. It matches with
graph vi. This equation is a quadratic function with a negative
lead coefficient (goes from Q3 Q4 and has a y-intercept of -2. It
matches with graph iii. This equation is a linear function with a
positive lead coefficient (goes from Q3 Q1 and has a y-intercept of
-3. It matches with graph ii.
- Slide 32
- Example 6 For each set of characteristics below, sketch the
graph of a possible polynomial function. a)range: y-intercept: 4
Reasoning about the characteristics of the graphs of polynomial
functions Try it first! Your answer may be different, but it must
be a parabola that is opening upward. Its minimum must be at y =
-2, and it must have a y-intercept at 4.
- Slide 33
- Example 6 For each set of characteristics below, sketch the
graph of a possible polynomial function. b)range: turning points:
one in quadrant III and one in quadrant I Reasoning about the
characteristics of the graphs of polynomial functions Try it first!
Your answer may be different, but it must be a cubic graph that
goes from Q2 Q4. It has to have turning points in Q3 and Q1.
- Slide 34
- Need to Know A polynomial is a function that consists of one or
more terms added together. Each term consists of a coefficient, a
variable, and a whole number exponent. The leading coefficient is
the coefficient of the term with the highest exponent of x in a
polynomial function in standard form. The degree of a function is
the greatest exponent of the function. It helps determine the shape
of the graph of the function. The constant term is the term in
which the variable has an exponent of 0.
- Slide 35
- Need to Know We can identify some characteristics of a
polynomial function when the equation is written in standard form.
The maximum number of x-intercepts is equal to the degree of the
function. The maximum number of turning points is equal to one less
than the degree of the function. The end behaviour is determined by
the degree and leading coefficient. The y-intercept is the constant
term.
- Slide 36
- Need to Know A turning point of a function is any point where
the y- values of a graph of a function changes from increasing to
decreasing or change from decreasing to increasing. An absolute
maximum (minimum) is the greatest (least) value in the range. A
local maximum (minimum) is a maximum (minimum) that is not an
absolute maximum (minimum).
- Slide 37
- Need to Know
- Slide 38
- Slide 39
- Youre ready! Try the homework from this section. Linear and
cubic polynomial functions have similar end behaviour. if the lead
coefficient is positive, the graph extends from quadrant III to
quadrant I if the lead coefficient is negative, the graph extends
from quadrant II to quadrant IV Quadratic polynomial functions have
unique end behaviour. If the lead coefficient is positive, the
graph extends from quadrant II to quadrant I If the lead
coefficient is negative, the graph extends from quadrant III to
quadrant IV