Chapter 4 Factoring and Parabolas 📝

Any time you graph a quadratic equation you end up with a

parabola

Maximum Height

the highest y value that the parabola reaches

Vertex

the name of the highest point that a parabola reaches

Axis of Symmetry

A line that divides a plane figure or a graph into two congruent reflected halves

the x intercepts where the angry birds hit the ground

roots

Concave up Parabola

flipped upside down angry bird trajectory

Concave down Parabola

normal angry bird trajectory

Parabolas are

the shape that graphs of quadratic equations take

Which of the following is TRUE about the vertex of a parabola?
It is the line that goes straight down through the middle of the parabola.
It is the point that intersects with the x-axis.
It is the point where the parabola reaches the maximum or minimum depending on whether it opens down or up
It is the point that intersects with the y-axis.

It is the point where the parabola reaches the maximum or minimum depending on whether it opens down or up

Which of the following equations will be the graph of a parabola?
y = x^2 - 2x + 3
x = y^2 +2x - 3
y = 2x + 3
y = 2x + 4

y = x^2 - 2x + 3

y = x^2 - 2x + 3

8

Which one of the following divides the parabola into two equal halves?
Point of symmetry
Axis of equality
Axis of symmetry
Point of equality

Axis of symmetry

Which phrase describes a parabola that looks like an upside down U?
Concave down
Maximum point
Concave up
Axis of symmetry

Concave down

How do you multiply two things in parentheses that are next to each other

multiply the first thing in the first parentheses by the first and second thing in the second parentheses, then multiply the second thing in the first parentheses by the first and second thing in the second parentheses

There are _______ forms that actually have unique uses in parabolas

3

What are the three forms used to graph parabolas

standard form, intercept form, vertex form

Standard form is the

easiest and most basic to write, most quadratic forms will be in standard form from the beginning

Standard form

using a can tell you whether the parabola is concave up/positive or down/negative Using Standard form you can find the y intercept easily by substituting a and b for 0 and then c is the y intercept, if it is easier for the formula to be manipulated into standard form, one way you can calculate the axis of symmetry is = -b/2a

Intercept Form

the a value also tells you concave up or down, and you can substitute 0 for x and find the y intercept, you can also find the y intercept by substituting p and q for x and find your y intercept.

Vertex Form

you have to complete the square, the a value tells concave up or down, y intercept is easily found by substituting 0 for x, it will instantly tell you the vertex h will be the x coordinate and k will be the y coordinate.

Standard written FORMULA

y = ax^2 + bx + c

Intercept written FORMULA

y = a(x - p) (x - q)

Vertex written FORMULA

y = a(x - h)^2 + k

Given the following equation, y = 3(x + 5)(x + 9), x = 5 & x = 9 are _____.
not the x-intercepts
the square values
the x-intercepts
the y-intercepts

not the x-intercepts

Which of the following graphs represents the equation, y = -2x2 + 2x + 6?

the vertical line
the upside down u
the diagonal line
the horizontal line

The following equation is written in which parabolic form?
y = 2x2 - 3x + 12
Intercept
Parenthetical
Vertex
Standard

Standard

The following equation is written in which parabolic form?
y = -(x - 3)2 - 1
Intercept
Parenthetical
Vertex
Standard

Vertex

What is the equation of the axis of symmetry of the parabola given by the equation y = 6(x + 1)(x - 5)?
y = -2
x = -2
y = 2
x = 2

x = 2

monomial

algebraic expression with one term

binomial

algebraic expression with two terms

anytime you multiply two binomials

it is really just four mini multiplication problems

Foil, method for solving binomials Stands for

firsts
outsides
insides
lasts

Multiply using the FOIL method. Make sure to collect terms and simplify.
(3x - 2)(2x + 1)
6x2 -x - 2
3x2 + 5x +2
6x2 - 2x + 3x + 1
3x2 + 3x - 4x - 2

6x2 -x - 2

Using the FOIL method, what four terms do you have after multiplying the binomial (x - 2)(4x + 1)?
4x2, x, 8x, 2
x2, 4x, x, -2
5x2, 2x, -1, x
4x2, x, -8x, -2

4x2, x, -8x, -2

Multiply (2x + 8) (2x - 6)
4x - x + 2
4x2 + 16x -12
4x2 + 4x - 48
4x2 - 4x + 48

4x2 + 4x - 48

Use the area method to multiply the binomial (3x + 2)(2x - 4). Do not forget to combine like terms for your final answer!
5x2+x-2
5x2-8x-8
6x2-8x-8
6x2-8x+8

6x2-8x-8

Multiply (x + 4)(x - 3).
x2 + x - 12
2x + x + 12
x + 2x + 12
x - 12x - 12

x2 + x - 12

Conjugates

Two binomials are conjugates if they have the same two terms, but opposite signs on the second one

If you have a conjugate you can

skip the middle part of the problem and multiply the first by the first and the second by the second

If there is a squared sign on the outside of the parentheses you have to

NOT MULTIPLY EVERYTHING BY TWO, technically squared is multiplying something by itself so you duplicate the first section and it becomes a normal binomial

Solve this problem using the appropriate method:
(x + 3) (x2 - 2x + 1)
2x3 + 2x2 - 5x -3
x3 + x2 - 5x + 3
2x3 + x2 + 5x + 3
x3 + x2 + 5x + 3

x3 + x2 - 5x + 3

Solve this multiplication problem:
(2x - 1) (x + 9)
2x3 + 18x -9
2x2 + 19x + 9
2x2 + 17x - 9
3x2 + 17x + 9

2x2 + 17x - 9

Use the FOIL method to solve the following:
(x + 2) (x - 1)
x2 + x - 2
2x2 + x + 2
x2 - x - 2
x2 + 2x + 2

x2 + x - 2

Solve the following multiplication problem: (x^2 + 9) (x^2 - 9)
x^2 + 81x - 81
x^4 - 81
x^4 + 81
2x + 18x^2 + 18

x^4 - 81

Which expression is the conjugate of x+11?
-x-11
x+121
x-121
-x+11
x-11

x-11

Factor

the reverse of a multiplication problem, taking the product and turning it into the problem

How do you factor a quadratic equation

find two numbers that add up to the number in the middle and multiply into the number on the end

If you have a quadratic expression that looks weird you can

divide different things outside of the parentheses solve it like normal and leave the weird things on the outside

In addition to 1 and 21, what are the factors of 21?
3 and 7
1 and 20
3 and 6
10 and 11

3 and 7

The following factors correspond to which one of the following equations: (x + 1) (x - 6)?
x^2 + 5x + 6
x^2 + 7x + 7
x^2 - 5x - 6
x^2 - 7x - 5

x^2 - 5x - 6

What are the factors of x2 + 7x - 30?
(x + 10)(x - 3)
(x - 10)(x + 3)
(x + 6)(x - 5)
(x - 6)(x+ 5)

(x + 10)(x - 3)

Factor x^2 + 10x + 12.
(x + 5) (x + 7)
(x + 5) (x + 5)
None of the answers are correct; this equation cannot be factored further.
(x + 10) (x - 12)

None of the answers are correct; this equation cannot be factored further.

Factor the following expression: x^2 +9x + 8
(x - 1) (x + 8)
(x + 1) (x + 8)
(x + 1) (x - 8)
(x - 1) (x - 8)

(x + 1) (x + 8)

What would the first step be in completing the square of the following equation?
Add 7 to both sides of the equation.
Factor 2 out of each term in the trinomial.
Add 55 to both sides of the equation.
Subtract 55 from both sides of the equation.
Factor the trinomial into a perfect square binomial.

Factor 2 out of each term in the trinomial.

Which equation below contains a trinomial that can be readily rewritten as a perfect square binomial?

https://study.com/academy/practice/print-quiz-worksheet/quiz-worksheet-practice-problems-for-completing-the-square.html?format=print&showCorrectAnswers=true

ind the value for c that must be added to both sides, and rewrite the equation in the vertex form.

https://study.com/academy/practice/print-quiz-worksheet/quiz-worksheet-practice-problems-for-completing-the-square.html?format=print&showCorrectAnswers=true

Expressing the equation below in the vertex form will give a value of y = _____. y = xsquared - 11x - 6

132.25
-11
11
-138.25
5.5

5.5

Rewrite the following equation in vertex form.

https://study.com/academy/practice/print-quiz-worksheet/quiz-worksheet-practice-problems-for-completing-the-square.html?format=print&showCorrectAnswers=true

Find the solution to the equation below.
0 = x^2 + 7x +12

x = -3 or x = -4
x = -12 or x = -1
x = 3 or x = 4
x = -3 or x = 4

^

What does the zero product property tell us?
If we get a product of zero, neither one of the factors could have been zero.
There are only two ways to get a product of zero.
If we get a product of zero, at least one of the factors must have been zero.
If we get a product of zero, both the factors must have been zero.

If we get a product of zero, at least one of the factors must have been zero.

Find the solution to the equation below.
0 = -3x^2 + 13x + 10

x = 2/3 or x = -5
x = -2 or x = 3
x = -2 or x = 5
x = -2/3 or x = 5

x = -2/3 or x = 5

If the equation below told you a ball was 'h' feet in the air after 't' seconds, what would be the first step to finding out when it hit the ground?
h = -t^2 + 21t + 3
Substitute t = 0 into the equation.
Substitute h = 0 into the equation.
Divide out the negative in front of the t2.
Undo the +3 from both sides of the equation.

Substitute h = 0 into the equation.

Which one of the following equations is in intercept form?
y = x^2 + 5x + 3
y = -x^2 + 2x + 4
y = -2x^2 + 2x + 4
y = 2 (x - 2) (x + 2)

y = 2 (x - 2) (x + 2)