Week 30: Harder factoring, moving graphs, and and-or

Field Book

109 to 112

Due

Test 27 due

Method

Paper first, every step

Trinomials get a number in front of the x squared this week, which breaks the quick method and calls for grouping instead. The graphs section is the fun one: once you know how a shape moves when you add, subtract, or negate, you can sketch a whole family of curves without plotting a single point.

What you have to be able to do

Factor a trinomial that has a leading coefficient
Shift and reflect a graph without plotting points
Tell a conjunction from a disjunction

The procedures

Trinomials with a leading coefficient

Multiply the first coefficient by the last constant. Call that the target.
Find two numbers that multiply to the target and add to the middle coefficient.
Split the middle term into those two pieces, then factor the four terms by grouping.

Shifts and reflections

A number added outside moves the graph up. Subtracted outside moves it down.
A number inside the parentheses moves it sideways, and the direction is the opposite of what it looks like: minus moves right.
A minus sign in front of the whole thing flips it upside down across the x axis.

And versus or

A conjunction is and. Both parts must be true, so you keep only the overlap.
A disjunction is or. Either part being true is enough, so you keep everything from both.
And makes the answer smaller. Or makes it bigger.

Worked

WORKED

Factor 2x² + 7x + 3.
Target: 2 × 3 = 6. Two numbers multiplying to 6 and adding to 7 are 6 and 1.
Split: 2x² + 6x + x + 3. Group: 2x(x + 3) + 1(x + 3).
ANSWER: (2x + 1)(x + 3)

WORKED

Describe y = x² + 3 and y = −x².
The 3 is outside, so the parabola moves up three.
The minus in front flips it.
ANSWER: shifted up 3, and flipped upside down

Terms

Term	Meaning
leading coefficient	the number multiplying the highest power term
conjunction	two statements joined by and, both must hold
disjunction	two statements joined by or, either may hold

Check yourself

1. Factor 3x² + 7x + 2

2. What does y = x² − 4 look like compared to y = x²?

3. Does and make the solution set larger or smaller?

reveal answers

(3x + 1)(x + 2) · shifted down 4 · smaller

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.