Name_________________________________________
Guided Notes 2.1
Pre-Calculus Honors
Objective 2.1: Polynomial Zeros
Do Now:
1. Find f(-5) if f(x) = a(x + 5)
2.
Does the value of ‘a’ matter? Why?
Power Functions and Polynomials:
power functions:
polynomials:
f(x) =
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0
f(x) = an(x – zn)(x – zn-1)…(x – z2)(x – z1)
where a and b are any number
ex:
where n is a non-negative integer (not a
fraction, decimal, or _______________________
number) and the a and z values are any
numbers
ex:
What do they look like? Make at least three observations about each group. You might want to sketch the general shape
of each group as well.
Non-Polynomial Power Functions:
Polynomial Power Functions:
Non-Power Polynomial Functions:
Polynomial Intercepts:
x-intercept: the point where the function intersects the ______-axis
y-intercept: the point where the function intersects the ______-axis
X-intercepts:
1. What is the y-value of the x-intercept? Is this always the case? Why or why not?
2.
What happens when you substitute the x-value of the x-intercept into the function? Try this with both a polynomial in
standard form and a polynomial in factored form.
3.
The x-value of the x-intercept is sometimes called a zero of a function f(x) because when you substitute it in, it makes
the function, f(x), equal _______________. The zero is also sometimes called the solution or root of the equation f(x) = 0.
4.
What do you notice about the x-values of the x-intercepts and how they relate to the polynomial?
5.
Is it easier to find the zeros from the factored form or standard form of the polynomial?
6.
Find the zeros of the function. Why are these the zeros? f(x) = 4(x + 2)(x – 7)
7.
Name_________________________________________
Find the roots of the equation f(x) = 0. Why are these the roots? f(x) = x2(x – z)(2x + 4)
8.
Find the zeros of the function. Why are these the zeros? g(x) = -3(x – k)(x + z)?
9.
How far apart are the x-intercepts of the function f(x) = 3(x – 3)(2x + 2)?
10. Find the x-intercept(s) of the function: f(x) = x2 - 11x + 30
11. Find the zeros of the function: f(x) = x3 – 2x2 – 63x
Sum it up:
Y-intercepts:
12. What is the x-value of the y-intercept? Is this always the case? Why or why not?
13. What happens when you substitute 0 in for x in the function? Try this with both a polynomial in standard form and a
polynomial in factored form.
14. Find the y-intercept: f(x) = -2(x – 2)(x + a)
15. Find the y-intercept: f(x) = 3x4 – 2x2 + b
16. Is it easier to find the y-intercepts from the factored form or standard form of the polynomial? Why?
17. What do you notice about the y-values of the y-intercepts and how they relate to the polynomial?
18. Challenge: Let’s say x = (y – 3)(y – 2). Find the x-intercept (think about what y has to be). Find the y-intercepts (think
about what x has to be!)
Sum it up:
For the x-intercept, the y-value is _______
For the y-intercept, the x-value is _______
Homework:
1. Find the zeros without graphing: f(x) = (x – 3)(x +2)
2. Find the x-intercepts and y-intercept of the function: f(x) = 3(x – 2)(3x + a)
3. Do these functions have different zeros: f(x) = (x – d)(x + w) and f(x) = 2(x – d)(x + w)
4. How far apart are the x-intercepts of the function (without graphing!) f(x) = x2 – 2x - 15