Algebra II Test 2

2.1 Graphing the Parent Function: $y = (x)^2$

• Vertex: $(0,0)$

• Steps from Vertex:

  • Move right $1$ unit, up $1$ unit

  • Move right $2$ units, up $4$ units

  • multiply by number in front to solve

  • Ex: g(x)= -1/2(x+2)² → the vertex would be (-2,0)

  • Multiply 1 and 4 by -½ to get the points on the function

• Symmetry:

  • Always mirror points on the opposite side across the axis of symmetry.

Stretch vs. Compression

• Stretch if $a > 1$

  • Examples: $2(x+2)^2$, $4(x+3)^2$

• Compression if $0 < a < 1$

  • Examples: $\frac{1}{2}(x+2)^2$, $\frac{1}{3}(x+2)^2$

Determining features of A Quadratic Function

Domain: (infinity, infinity)

Range: [4, infinity)

Min: 3 or ¾. → Min would be in a graph like this, Max would be if the function was turned upside down

Axis of symmetry: x=3

Without a graph

f(x)= -(x+4)²-5

vertex- (-4,-5)

axis of symmetry: x=4

max: (-infinity, infinity)

domain: (-infinity, infinity)

range: (-infinity, -5]

Find the equation

The vertex is (-2,3) and the y- intercept is -1. Write the equation

Step 1: y= a(x-h)² + k → write the formula

Step 2: y=a(x+2)² + 3 → put in the vertex

Step 3: -1=a(0+2)² + 3 → because the y int. is (0,-1)

Step 4: -1= 4a+3 → 2²= 4, bring down the 3

Step 5: -4=4a → subtract the 3

Step 6: a= -1 → solve

Standard to Vertex Ex: with y=x²+6x+2

1. Constant on left → y-2=x²+6x+9

2. take ½ of the coefficient of x and ² it → 6/2= 3²= 9

3. Add to both sides 9+y-2=x²+6x+9

4. Rewrite trinomial as binomial squared → y+7= (x+3)² → 3 came from 6/2

5. Move constant to right y= (x+3)² -7

Ex: with y= 2x²-8x+3

y-3 = 2x²-8x+3

8+y-3 = 2(x²-4x+4)→ +4 came from -4/2=-2²=4 → +8 came from 2(4)

5+y = 2(x-2)²

y = 2(x-2)²-5

2.2

Solving for the Vertex when written in Standard Form

Formula: $-\frac{b}{2a}=h$

Ex: y = x²-6x+10

1. a= 1, b= -6, c= 10

2. $h=-\frac{-6}{2\left(1\right)}$

3. h = 3

4. 3x²-6(3)+10

5. = 9-18+10

6. =1

7. V = (3,1)

To find axis of symmetry look at what’s in the x

Ex: V= (3, 7) Axis of sym = 3

2.3

Factoring:

First Multiply the a and c: x²+7x+12

This would be 12

Find two numbers that multiply to 12 and add to 7 (the b)

This would be 3 and 4

(x+3) and (x+4) would the answers

Factoring with a number in front x²

Given: 2x²-5x-3

Follow all the same steps to get x= 6 and x=1

Then plug in for c

2x²-6x+1x-3. (add the negative sign to keep like og)

Then make 2 groups 2x(x-3) + 1(x-3)

The ones the the ( ) should be the same

Make another set using the outside numbers → (2x+1) (x-3)

set both sets equal to zero

final answers: x= 3 x= -1/2

*always check for GCF

If like this: 2x²= -9x+5 → move over 2x²+9x-5

2.4

Square roots

Perfect Square: x²=16 → x = + or - 4

Simplifying Radicals

Ex:

$\sqrt{18}=\sqrt{9\cdot2}$

9 can be squared so it will put on the outside of the radical, the 2 can’t so it will stay

$=3\sqrt2$

Ex:

$2\sqrt{28}$ take out what makes 28

$2\sqrt{4\cdot7}$ take out the 4 because it can be squared

$2\cdot2\sqrt7$ multiply with the existing 2

$4\sqrt7$ answer

Ex:

$\sqrt{25y^8}$ there are 4 pairs of y’s

answer- 5y⁴

Using i

$\sqrt{-1}=i$

$\left(\sqrt{-1}\right)=i^2$

$-1=i^2$

Ex: x²=-5

x² = 5 • -1

x² = 5$i^2$

x= + or - $i\sqrt5$

Adding and subtracting complex numbers

(4-7i) + (-11+9i) = -7 +2i