Download Lesson 3 - Denton ISD

Pre-AP Algebra 2
Unit 3 – Lesson 3 – Converting Standard Form to Intercept Form
Objectives: The students will be able to
• Convert from standard form to intercept form by factoring quadratic functions
• Factor with any value as a in 𝑓 𝑥 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐
Materials: Do Now worksheet; pairwork; hw #3-3
Time
5 min
10 min
15 min
30 min
Activity
Review Homework
Show the answers to hw #3-2 on the overhead. Students correct their answers. Pass around the tally
sheets.
Homework Presentations
Review the top 2 or 3 problems.
Do Now
Students will practice multiplying binomials to help review the patterns they will need in factoring
Direct Instruction
Background:
• Factoring: to break a number or expression down into a product of factors (a multiplication
problem). For example: 12 = (3)(4), 2x + 6 = (2)(x + 3)
• Polynomial: a sum of monomial terms, such as 2x2 – 5x + 3
Concepts:
• When factoring a quadratic
1) If the GCF of the terms is greater than 1, factor it out.
2) If you have 2 terms, is it a Difference of Squares: a2 – b2. This is one of the most
important factoring patterns. Memorize it!
§ a2 – b2 = (a – b)(a + b).
3) If you have 3 terms, look at a, the coefficient of x2.
§ If a = 1 (i.e. no number in front of the x2), use the t-table method.
• Find the factors of c whose sum is b.
• Try it in your head first. If you can’t find it, make a t-table.
§ If a > 1, use the box/ grouping method
• find two numbers whose sum is b and product is ac.
• Break the middle term into two parts based on those factors.
§ Some trinomials can’t be factored: they are called “prime”, just like numbers
that can’t be factored.
§ Check result by multiplying factors.
Examples:
1.
2.
3.
4.
5.
6.
7.
8.
20 min
6x5 + 9x 3 (show how to write out prime factorizations as an aid)
2
Factor x − 25
2
Factor 64 − 9x
2
Factor: x − 4x − 12
2
Factor: x + 2x − 4 (prime)
2
Factor: 3x − 27x + 60 (factoring out the GCF makes this much easier to do)
2
Factor 6x −19x +15
2
Find the x-intercepts of f (x) = 14x + 5x −1
Factor:
Pairwork
Show students how to generate factorable trinomials by starting with a pair of binomials and multiplying
them. You can make it more difficult by distributing a number across the entire trinomial. Ask students
to generate 3 factorable trinomials, 1 of which has a GCF greater than 1, and one prime trinomial. Write
the problems on a sheet of binder paper and switch with their partner. Partners should then try to factor
the trinomials that they receive.
Pre-AP Algebra 2
Lesson #3-3: Do Now
Name: _______________________
Multiplying Binomials
Part 1: Find the greatest common factor and describe the process you used:
(1) 24, 36, 60
(2) 8x2y3, 12x3y, 20x2y2
(3)
Part 2: Expand the following
(1)
(x + 4)(x + 3)
(2)
(x – 4)(x – 3)
(3)
(x + 4)(x – 3)
(4)
(x – 4)(x + 3)
(5)
Discuss what makes the middle and last terms + or −.
Part 3: Expand the following
(1)
𝑥−3 !
(2)
2𝑥 + 3
(3)
𝑥−4
!
!
(4)
(𝑥 + 3)(𝑥 − 3)
(5)
2(𝑥 + 3)(𝑥 − 3)
(6)
2𝑥 + 3 2𝑥 − 3
(7)
(2𝑥 + 1)(3𝑥 − 5)
(8)
(2𝑥 − 1)(3𝑥 + 5)
Pre-AP Algebra 2
Lesson #3-3: Pairwork
Name: _______________________
Basic Factoring Practice
1) Convert to intercept form. Then, match each function to its graph.
𝑓 𝑥 = 𝑥! − 4
𝑓 𝑥 = 2𝑥 ! − 𝑥 − 15
𝑓 𝑥 = 𝑥 ! + 5𝑥 + 4
𝑓 𝑥 = 15𝑥 ! − 31𝑥 − 12
A
B
C
D
Pre-AP Algebra 2
Lesson #3-3: Pairwork
Name: _______________________
2) Convert to intercept form. Then, find the x-intercepts and vertex.
a. f (x) = x 2 − 15x + 56 .
b. 𝑓 𝑥 = 3𝑥 ! + 2𝑥 − 16
3) Here is a window into the mind of Ms. Nicewarner… oooohh.. scary… This is how I come up with
trinomial problems for you to factor:
I start with the answer I want (the two binomial factors), I multiply them together, and that’s it! To
make it more challenging, I may multiply all the terms in the trinomial by some number or expression
– that will create a trinomial with a GCF that can be factored out.
Now you try it. Create a mini-worksheet with three factorable trinomials. Make one of them have a
GCF greater than 1. Then, trade papers with a partner and factor each other’s trinomials. Do the work
on a different piece of paper and just write the trinomials in the spaces below.
Pre-AP Algebra 2
Lesson #3-3: Pair Work
Name: ________________________
My First Factoring Worksheet
Factor each of the following trinomials completely.
1)
2)
3)
Pre-AP Algebra 2
Lesson #3-3: Homework
Name: _______________________
HW #3-3: Quadratics in Intercept Form
Check for Understanding
Can you complete these problems correctly by yourself
1) Find the x-intercepts for each function by factoring. Then, match each function to its graph.
Remember, always check first to see if the GCF is greater than 1.
a. f (x) = x 2 + 2x − 48
b. f (x) = 10x 2 − 50x + 60
c. f (x) = 4x 2 − 4x −15
d. f (x) = 6x 2 − 34x −12
I
II
III
IV
Pre-AP Algebra 2
Lesson #3-3: Homework
Name: _______________________
2) For each problem, write a quadratic equation that has the given x-intercepts. Expand out your
equation (i.e. don’t leave it written in factored form). Also, make sure that all of the coefficients
are integers.
a. x = 5 and x = -5
b. x = ±½
c. x = 7 and x = -3
d. x = -¼ and x = ¾
Spiral
What do you remember from Algebra 1and our previous units? (these are skills we will need
in this unit) Work on a separate sheet a paper
1. Find the vertex and x-intercepts of 𝑓(𝑥) = − 𝑥 + 8 ! + 1. Then graph 𝑓(𝑥). Use your
graph to answer the remaining questions.
2. 𝑓 −7 =
3. 𝑓 −9 =
4. 𝑓 −5 =
5. What value(s) of x make the following true
a. 𝑓 𝑥 = 0
b. 𝑓 𝑥 > 0
d. 𝑓 𝑥 = −3
e. 𝑓 𝑥 ≥ −3
c. 𝑓 𝑥 ≤ 0
f. 𝑓 𝑥 < −3