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Chapter 2
Abby, Lauren, Eric, Raquel, Peyton, Wyatt
2.1 Power and Radical Functions
Vocabulary:
•Power Function: any function of the form f(x)=ax^n, where a and n are nonzero
constant real numbers.
•Monomial Function: any function that can be written as f(x)=a or f(x)=ax^n
where a and n are nonzero constant real numbers.
•Radical Function: a function that can be written as f(x)=4th sqroot of x^p, where
n and p are positive integers greater than 1 and have no common factors.
Monomial Functions
2.1 Analyze Monomial Function
f(x)=5x^2
• Domain: (-∞, ∞)
• Range: [0,∞)
• Intercept: 0
• Lim (x → -∞) f(x)=∞
• Lim (x → ∞) f(x)=∞
• Continuous for all real numbers
• Decreasing (-∞, 0)
• Increasing (0, ∞)
2.1 Functions with Negative
Exponents
f(x)=2x^-4
• Domain: (-∞,0) U (0,∞)
• Range: (0,∞)
• No Intercepts
• Lim (x → -∞) f(x)=0
• Lim (x → ∞) f(x)=0
• Infinite Discontinuity at x=0
• Increasing: (-∞, 0)
• Decreasing: (0, ∞)
2.1 Rational Exponents
2.1 Power Regression
1. Create a scatter plot from the data
2. Write a polynomial function to model the
data set. State the correlation coefficient
3.Use the equation to predict the rate for…….
2.1 Graph Radical Functions
2.1 Solve Radical Equations
2.2 Polynomial Function
Polynomial Function: sum of terms that consist of a coefficient that is a real number to the
non-negative powers
Leading Coefficient: The coefficient of the variable with the greatest exponent
Quartic Function: 4th degree polynomial function
Turning Point: indicates where the graph of a function changes from increasing to decreasing
and vice versa. The Highest possible number of turning points in the function is n-1
Quadratic Form: a function that can be described by an equation of the form f(x) =ax^2 + bx + c,
where a ≠ 0.
Repeated Zero: the highest possible value number of real zeros is equal to the highest degree of
the function
Multiplicity: how many times a particular number is zero for a given polynomial
2.2
Leading Term Test:
Uses the power and coefficient
of a term to determine end
behavior
2.2
Polynomial Function of degree n: the degree of the term in the polynomial that has the
highest value for n where a sub n is not equal to 0.
2.2 continued...
Calculators Suck, Here’s how to use them…
TI-nspire
Finding your relative or local mins and maxs
Step 1: Press [MENU] → Points & Lines → Point On to open
the Point On tool
Step 2: Use the Touchpad to move the cursor and click on the
graph
Step 3: Press [ENTER] or [CLICK] to create the point
Step 4: Press [ESC] to exit the Point On tool
TI-84
Finding your relative mins and max
Step 1: plug equation into Y=
Step 2: go to the graph
Step 3: Click “2nd Trace”
To find minimum→ click 3
To find maximum→ click 4
Step 4: left bound and right bound
and guess to find your relative
minimum or maximum → the y-value
is the value for the relative min or
max
2.3 Dividing Polynomials
vocab
Synthetic Division: A shortcut for dividing a polynomial by a linear factor, in
order to simplify it.
Depressed Polynomial: The quotient when a polynomial is divided by one of
it’s factors.
Synthetic Substitution: Using Synthetic division to evaluate a function
Let’s (Re)Learn synthetic Division
how would we rewrite
in synthetic?
1. first we write the coefficients of each term in the numerator remembering
to leave zeros as placeholders like so
2. for the denominator we negate the first term
and change the sign or the second term
3. bring down the first term multiply it by the divisor and
add that to the second term
4. continue this multiply and add process until all
terms are complete
5. the last number is your reminder
now begun with xo and move from right to left
until each term has its x
2.4 Zeros of Polynomial Functions
Vocab:
Rational Zero Theorem: If f is a polynomial function of the form
f(x)= an xn + an-1xn-1 +...+ a2x2 + a1x + a0, with degree n≥ 1, integer coefficients,
and a0≠ 0, then every rational zero of f has the form p/q.
Lower Bound: a real number “a” that is less than or equal to the least real
zero of a polynomial function.
Upper Bound: a real number “b” that is greater than or equal to the greatest
real zero of a polynomial function.
Descartes’ Rule of Signs: a rule that gives information about the number of
positive and negative real zeros of a polynomial function by looking at a
polynomial’s variation in sign.
2.4 continued...
Fundamental Theorem of Algebra: a polynomial function of degree n, where
n>0, has at least one zero (real or imaginary) in the complex number system.
Linear Factorization Theorem: if f(x) is a polynomial function of degree n>0,
then f has exactly n linear factors and f(x)=an(x–c1)(x–c2)...(x–cn), where an are
the complex zeros (including repeated zeros) of f.
Conjugate Root Theorem: when a polynomial equation in one variable has a
zero of the form a+bi, where b=0, then its complex conjugate, a–bi, is also a
root.
Complex Conjugates: two complex numbers of the form a+bi, where a and b
are real numbers and i is the imaginary unit.
Irreducible over the Reals: a quadratic expression that has real coefficients
but no real zeros associated with it.
2.4 continued...
Formulas
p/q- “p” is an integer of the constant a0 and “q” is an integer factor of the
leading coefficient.
f(x)= 3x3–7x2–22 + 8… 3 is the leading coefficient(q) and 8 is the constant(p).
Factors of 8/Factors of 3
Descartes’ rule of signs
+ to –
The original function has two
g(x) = -3x3+ 2x2– x –1
variations is sign, g(–x) has one. By
Descartes’ rule g(x) has either 2 or
– to +
0 positive zeros and one negative
3
2
g(–x)= –3(–x) + 2(–x) – (–x) – 1
real zero.
= 3x3 + 2x2 + x – 1
+ to –
2.5 Rational Functions
Terms:
Rational Function - The quotient of two polynomial functions a(x) and b(x),
where b is non zero.
Vertical Asymptote - The line that the function approaches 0 as they tend
to infinity. Occur at the real zeros of b(x).
Horizontal Asymptote - Horizontal lines that a function approaches as x
tends to plus or minus infinity.
Oblique Asymptote - This type of asymptote occurs when the numerator of
a rational function is exactly one degree greater than the denominator.
Holes - When a value of x sets both the denominator and the numerator of
a rational function equal to 0, there is a hole in the graph. A single point at
which the function has no value.
2.5 Graphing Rational Functions
1.) Find Vertical and Horizontal Asymptotes
f(x)=(x+5)/(x-2)
● The function is undefined at b(x) = (x-2). The real zero of b(x) is 2. Therefore, the domain of f is
all real numbers except x=-5.
● x=2 is a point of infinite discontinuity. It is the Vertical Asymptote.
● When checking for the Horizontal Asymptote look at the degrees of the function
f(x)=axn+...bxm+...
n>m y= None
n=m y=a/b
n<m y= 0
●
●
●
To find the y-int set the x’s to 0. So here the y-int (0,-5/2)
In order to graph we have to know our domain. There is a discontinuity at x=2, making
the domain: (-∞,2)(2,∞)
Now, add the asymptotes to the graph and plot the x and y intercepts. Then draw your curve
based off the domain and discontinuities
2.5 Solving Rational Functions
4/x^2-6x+8=(3x/x-2)+(2x/-4)
The Lowest Common Denominator is (x-2)(x-4), which is x^2-6x+8.
(x-2)(x-4)4/x^2-6x+8=(x-2)(x-4)(3x/x-2)+(2x/-4)
4=3x(x-4)+2(x-2)
4=3x^2-10x-4
0=3x^2-10x-8
0=(3x+2)(x-4)
x= -⅔ or x=4
Because the original equation is not defined when x=4, you can eliminate this extraneous
solution. So, the only solution is -2/3.
2.6
Thanks for Listening!