# Polynomial, Power, and Rational Functions by bfs11840

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```									Polynomial, Power, and
Rational Functions
Chapter 2
Functions and Modeling
2.1
Polynomial
Polynomials
A polynomial, in other words, cannot have x
raised to a fractional, zero, or negative
power. It also cannot have x in the power.
Is the Function a Polynomial?
F(x)=-9 + 2x
Yes since no powers of x are fractional and
x is not a power.
F(x)=13
Yes for the same reason as above
F(x)=3x
No, because x is not the base it is the
power.
Is the Function a Polynomial?
F(x)= x1/2
No, because x is raised to a fractional
power.
Linear Function
A linear function can be written in the form
Y=mx+b
Finding a Linear Function
1. Find the coordinates.
2. Find the slope or the average rate of change by
finding the change in y over change in x.
3. Plug the slope in for m in either y=mx+b or y-y1
= m(x-x1).
4. Plug in one of the points for x and y or x1 and
y1, and then solve for b or y.
5. Write out final equation with m and b plugged
in.
Linear Function, Example
Find the equation of a linear function given:
F(-3) = 5 and f(6) = -2
Graphing a Linear Function
If given two points, then plot each point and
connect.
Standard Form
i

Remember that x and y
decide which graph is the
function.
Match a Parabola to its Function
in Vertex Form
1. Find the vertex and look for the graph that
has that vertex.
2. Determine if a is positive or negative, so
the graph should point up or down.
3. Find the x and y intercepts and check
them with the graph.
Transformations of a Parabola
Remember:
-aF(-b(x-h)) + k
- Flips over the x-axis
A—vertical stretch if a>1, shrink if 0<a<1
- Flips over the y-axis
B—horizontal stretch if 0<b<1,shrink if b>1
H—moves left/right
K—moves up/down
Transformations of a Parabola
¼ x^2 – 1
The ¼ is a and so there is a vertical shrink by ¼.
The ¼ is positive, so the graph does not flip over
the x –axis.
There is no number being added to or subtracted
from x, so the graph does not move left or right.
The x is not negative so the graph does not flip
over the y-axis.
The -1 moves the graph down 1.
Vertex of a Parabola

The point of the
parabola that is the
max or the min.
Find the Vertex and Axis of
Symmetry
G(x) = -3(x+2)^2 – 1
Vertex is (h,k) or (-2,-1)
Axis of symmetry is x=h or x= -2
How to put a Quadratic Function
in Vertex Form
Use Completing the Square
Using Completing the Square to
Vertex Form
Put y or f(x) on the left side of the equation
and all other terms on the other side of the
equation. Simplify the right side if needed.

I will use this example to show what the
steps mean. We will rewrite Y=3x2+5x-4
in vertex form, which is Y = a(x-h)2 +k.
Using Completing the Square to
Vertex Form
Factor out the coefficient of the x^2 term, a,
unless it is one, from all the terms with x
on the right side of the equation.
In our example, since a is 3 we need to
factor a three out from the x^2 and the x
terms.
Y=3(x2+5/3x)-4
Y = a(x-h)2 +k
Using Completing the Square to
Vertex Form
Now, divide the coefficient of the x term by
two and then square the result.
(5/3)/2 = 5/6     (5/6)2=25/36
Using Completing the Square to
Vertex Form
Add the resulting number to the term in the
parenthesis. Note that you are adding ―a‖ times
this amount to the function, so you will then need
to subtract this same amount so that the function
is not changed.
In our example we are adding 25/36 to the function
3 times, so we will then subtract 3*25/36.
Y=3(x2+5/3x+25/36)-4-3*25/36
Using Completing the Square to
Vertex Form
Factor the quadratic part and simplify the
constant part and the function will be in
vertex form.
Y=3(x + 5/6)2-73/12
Y = a(x-h)2 +k
Write an Equation for the Parabola
Given the Vertex and a Point
Use y=a(x-h)^2 + k to help.
1. Plug the x value of the vertex in as h and
the y value in as k.
2. Take the other coordinate and plug in x
for x and y for y.
3. Solve this equation for a.
4. Write out the vertex form for the parabola
with the a, h, and k plugged in.
Modeling
• Enter and plot the data as a scatter plot.
• Find the regression model that best fits the
– Does the scatter look linear or curved?
• Place the regression equation into y1, and
observe the fit to double check.
• Use the regression model to make the
predictions called for in the problem.
Describing a Linear Correlation
• If the scatter points look like they are
clustered along a line, then they have a
linear correlation.
– Positive linear correlation—positive slope
– Negative linear correlation—negative slope
• r – correlation coefficient—measures the
strength and direction of the linear
correlation of the data set.
Linear Correlation Coefficient
Properties
• [-1,1]
• r>0 there is a positive linear correlation
• r<0 there is a positive linear correlation
• |r|=1, then there is a strong linear
correlation.
• r=0, there is a weak or no linear correlation
Describing the Strength and
Direction of a Linear Correlation
1. State strong or weak based on how linear
the points look. The more linear they
look, the stronger the linear correlation.
2. Then state positive or negative for
direction. Positive slope or negative
slope.

• Finding maximums and
minimums
–Calculator
• Use max/min function
–By hand
• Find the vertex and determine if it is a
max or a min by determining which
way the graph points.
Creating Vertical Free-Fall Motion
equations.
S(t) = -1/2 gt2 +vot+so
v(t) = - gt +vo
t is time, g=32ft/s2 or 9.8m/s2 is the acceleration
due to gravity.
Initial height—so
Initial velocity--vo
Fireworks are shot by remote control into the
air from a pit that is 10 ft below the earth’s
surface.
Find an equation that models the height of
an aerial bomb t seconds after it is shot
upwards with an initial velocity of 80ft/sec.
Example Continued
We need to find a distance equation.
S(t) = -1/2 gt2 +vot+so
so = 10
vo = 80
g = 32 ft/sec^2
S(t) = -1/2 (32)t2 +80t+10
S(t) = -16t2 +80t+10
Example Continued
What is the maximum height above ground
level that the aerial bomb will reach? How
many seconds will it take to reach that
height?
The maximum height will be at the peak of
the parabola or the vertex. So use the
max function on the calculator and the y
value will be the max height and the x will
be the time it takes to reach the height.
Example Continued
What is the maximum height above ground
level that the aerial bomb will reach? How
many seconds will it take to reach that
height?
By hand, find –b/2a and this will be the time
it takes to reach the height. Plug this
value into the function and the result will
be the max height.

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