Algebra 1 UNIT 3 Quadratic Functions Test Study Guide

One Variable Quadratics

Name: ____________________________ Date: _______ Period: _______

Factor a = 1 Standard form: 𝑥2 + 𝑏𝑥 + 𝑐 Draw and x. Put b on the bottom and c on the top. Find 2 numbers that multiply to c and add to b. These numbers go on the right and left of the x. Finally, write out your factors in the form (x + #)(x + #). Set the factors equal to 0 to solve for x.

Ex: 𝒙𝟐 + 𝟑𝒙 − 𝟓𝟒 FACTORS: (𝑥 − 6)(𝑥 + 9) SOLUTIONS: 𝑥 = −9, 6 1. 𝑥2 + 18𝑥 + 45

2. 𝑥2 − 5𝑥 − 24

3. 𝑥2 − 17𝑥 + 60

Factor a ≠ 1 Standard form: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 Draw and x. Put b on the bottom and a*c on the top. Find 2 numbers that multiply to a*c and add to b. These numbers go on the right and left of the x. Write out your answer in the form (x + #)(x + #). Divide both numbers by a. Simplify the fractions. If there are any fractions left, move the denominator in front of x (“bottoms up”). Now solve for x.

Ex: 𝟔𝒙𝟐 + 𝟏𝟕𝒙 + 𝟏𝟐 (𝑥 + 8)(𝑥 + 9)

(𝑥 +8

6)(𝑥 +

9

6)

(𝑥 +4

3) (𝑥 +

3

2)

FACTORS: (3𝑥 + 4)(2𝑥 + 3)

SOLUTIONS: 𝑥 = −4

3, −

3

2

4. 5𝑥2 − 11𝑥 − 12

5. 2𝑥2 − 9𝑥 + 10

6. 4𝑥2 + 12𝑥 − 27

Removing GCF Look at a, b, & c. Can you divide all three numbers by the same number? If yes, do so. Write the new equation in parenthesis with the GCF out front.

Ex: 𝟏𝟓𝒙𝟐 − 𝟒𝟓𝒙 + 𝟑𝟎 Can divide 15

15(𝑥2 − 3𝑥 + 2) 7. 1230𝑥2 + 100𝑥 − 19400

8. 77ℎ2 + 169ℎ − 33

9. 28𝑛2 − 49𝑛 + 42

Difference of Square These are a special type of quadratic where b=0. If there is a GCF, take it out. Then make sure the remaining terms are perfect squares with subtraction. To factor, draw two sets of parenthesis. Take the square root of the first term and put it in front of both sets of parenthesis. Take the square root of the second term, and put it at the end of both sets of parenthesis. Make one + and one -.

Ex: 16𝑥4 − 196 Factor out GCF 4 4(4𝑥4 − 49) Answer 4(2𝑥2 − 7)(2𝑥2 + 7) 10. 9𝑐2 − 121 11. 100 − 64𝑘2

12. 𝑤6 − 400 13. 8𝑥2 − 200 14. 50𝑏2 − 162𝑦4

Algebra 1 UNIT 3 Quadratic Functions Test Study Guide

Solving with Factoring Standard form:𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 Before you factor, the equation MUST equal 0. Then factor as normal. You should end up with two sets of parenthesis set equal to 0. By the Zero Product Property, at least one of the parenthesis must equal 0. So set each set of parenthesis equal to 0 and solve for x. Your answer should be x=#, #.

Ex: 𝑥2 + 9𝑥 + 18 = 4 Subtract 4 𝑥2 + 9𝑥 + 14 = 0 Factor (𝑥 + 2)(𝑥 + 7) = 0 Split 𝑥 + 2 = 0 𝑥 + 7 = 0 Solve 𝑥 = −2, −7

15. 3𝑥2 + 10𝑥 + 3 = 0

16. 2𝑥2 − 6𝑥 = 176 17. 2𝑥2 + 5𝑥 − 30 = −6𝑥 − 35

Root Method The opposite of squaring something is to take the square root. Solve the equations like multi-step equations except you take the square root whenever you get to an entire side being squared.

Ex: 2(𝑚 + 4)2 = 8 Divide 2 (𝑚 + 4)2 = 4 Square root 𝑚 + 4 = ±2 Subtract 4 𝑚 = −4 ± 2 Split m= -6, -2 18. 𝑡2 − 3 = 13

19. 7𝑟2 + 15 = 92

20. 12(𝑝 − 5)2 = 108

Completing the Square The equation must have all the variables on the SAME side. 1. Move the constant terms to

the opposite side of the equation as the variables

2. If a≠1, divide every term by a

3. Calculate (𝑏

2)

2 and add that

number to both sides of the equation.

4. Factor the side with 𝑥2 and combine the terms on the other side.

5. Take the square root of both sides. Don’t forget the ± in front of the constant side.

6. Solve for the variable. If there are no square roots left, split the ± and solve for both numbers.

Ex: 2𝑓2 + 11𝑓 + 5 = 0 Subtract 5 2𝑓2 + 11𝑓 = −5

Divide 2 𝑓2 +11

2𝑓 = −

5

2

(𝑏

2)

2 (

11

2

2)

2

= (11

4)

2=

121

16

𝑓2 +11

2𝑓 +

121

16= −

5

2+

121

16

Factor (𝑓 +11

4)

2=

81

16

Square root 𝑓 +11

4= ±

9

4

Solve 𝑓 = −11

4±

9

4

Split 𝑓 = −1

2, −5

21. 𝑥2 − 4𝑥 − 21 = 0

22. 3𝑥2 + 34𝑥 + 40 = 0 23. 𝑥2 + 8 = 9𝑥 24. 4𝑡2 + 8𝑥 = −3

Quadratic Formula Must be in the form:

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 𝟎 Quadratic formula:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

Plug the numbers under the square root into the calculator first. Then take the square root. Then simplify (check triangle) or split.

Ex: 2𝑔2 − 6𝑔 − 20 = 0 a=2, b=-6, c=-20

𝑔 =6 ± √(−6)2 − 4(2)(−20)

2(2)

=6 ± √196

4=

6 ± 14

4

Split 6+14

4=

20

4= 5,

6−14

4= −

8

4=

−2 Answer 𝑔 = 5, −2

25. 2𝑥2 − 3𝑥 − 5 = 0 26. 2𝑚2 − 7𝑚 − 13 = −10

Algebra 1 UNIT 3 Quadratic Functions Test Study Guide

Two Variable Quadratics

Name: ____________________________ Date: _______ Period: _______

Graphing in STANDARD Form

Standard form: 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 a > 0 means a is positive & parabola opens up a < 0 means a is negative & parabola opens down “c” is the constant & is the y-int. If c = 0, you will not see a constant number in the equation Use AOS to calculate “x” of the

vertex: 𝒙 =−𝒃

𝟐𝒂; plug in for “x” to

solve for “y”: vertex ( x , y ) Get two more points on one side Reflect & Connect

1. Graph 𝒇(𝒙) = 𝒙𝟐 + 𝟐𝒙 − 𝟖

2. For the function 𝒇(𝒙) = −𝒙𝟐 − 𝟒𝒙

Identify the following for the graph:

vertex: (____,____) AOS: x = ______

x-int: ___________ y-int: ________

roots: (____.____) (____,____)

Graphing in VERTEX Form

Vertex form: 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 ( h , k ) are the coordinates of the vertex ( x , y ) ; but OPPOSITE SIGN for the number in the (parentheses). Plot the vertex. Get two more points on one side Reflect & Connect

3. Graph 𝒇(𝒙) = 𝟐(𝒙 − 𝟑)𝟐 + 𝟒

4. Identify the following for the graph:

vertex: (____,____) a = ______

Provide the equation in Vertex Form:

𝒇(𝒙) = _____________________________

Algebra 1 UNIT 3 Quadratic Functions Test Study Guide

Graphing in INTERCEPT Form

Vertex form: 𝒂(𝒙 − 𝒑)(𝒙 − 𝒒) p & q are the x-int. of the graph, but OPPOSITE SIGN for the number in the (parentheses). These are also called “roots”, “zeros”, “solutions”. Plot points. x of the vertex is half-way between these two root points; plug in for “x” to solve for “y”: vertex ( x , y ). Plot the vertex. Get one more point on one side Reflect & Connect

5. Graph 𝒇(𝒙) = −𝟑(𝒙 − 𝟐)(𝒙 − 𝟒)

6. Identify the following for the graph: x-int: ___________ a = ______ Provide the equation in Vertex Form:

𝒇(𝒙) = _____________________________

Converting

FROM STANDARD Form

TO VERTEX Use AOS

TO INTERCEPT Factor

**REFER TO GREEN FOLDABLE**

7. Convert 𝒇(𝒙) = 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟓 to Vertex Form

8. Convert 𝒇(𝒙) = 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟓 to Intercept Form

Converting FROM VERTEX Form

TO STANDARD Multiply

TO INTERCEPT Continue from

the Standard Form & Factor

**REFER TO GREEN FOLDABLE**

9. Convert 𝒇(𝒙) = (𝒙 + 𝟒)𝟐 − 𝟑𝟔 to Standard Form

10. Convert 𝒇(𝒙) = (𝒙 + 𝟒)𝟐 − 𝟑𝟔 to Intercept Form

Converting FROM INTERCEPT Form

TO STANDARD Multiply

TO VERTEX Continue from the

Standard Form & Use AOS

**REFER TO GREEN FOLDABLE**

11. Convert 𝒇(𝒙) = 𝟐(𝒙 − 𝟑)(𝒙 + 𝟏) to Standard Form

12. Convert 𝒇(𝒙) = 𝟐(𝒙 − 𝟑)(𝒙 + 𝟏) to Vertex Form

Algebra 1 UNIT 3 Quadratic Functions Test Study Guide

Transformations

“a”: the leading coefficient tells…

a > 1 STRETCH (narrow)

0