Algebra 2 Honors - Unit 2 (Quadratic Functions)

2.1 - Quadratic Functions

Quadratic Function: a function of the form f\left(x\right)=ax^2+bx+c where a does not equal 0

• graphs of quadratic functions are called parabolas

Vertex form of a quadratic function: f\left(x\right)=a\left(x-h\right)^2+k

How to Graph a Quadratic Function with its Transformations with y=-\frac12\left(x+3\right)^2+4

1. Find the vertex and line of symmetry

   1. Here, the y is +4, and the x is -3.

   2. Since x = -3, that’s the line of symmetry.

2. Find the direction the parabola is facing.

   1. If the a is positive, the parabola opens up

   2. If the a is negative, the parabola opens down

   3. Here the a is -1/2, so it opens down

3. Graph 4 more points to finish drawing the parabola

   1. REFERENCE QUESTION 1 OF PAGE 1 FOR BETTER INSTRUCTIONS ON HOW TO FIND THOSE POINTS

4. Find the minimum value and maximum value

   1. Here, the maximum value is x=-3, while the minimum value is x=-\infty

2.1b - Polynomial Expressions and 2.3b

Formulas:

1. Difference of squares: \left(a+b\right)\left(a-b\right)=a^2-b^2

2. Squaring Binomials: \left(a+b\right)^2=a^2+2ab+b^2

3. \left(a-b\right)^2=a^2-2ab+b^2

4. \left(a+b\right)^3=a^3+3a^2b+3ab^2+b^3

Note:

• When doing combinations or in general simplifying, the end result must have the terms multiplying against each other, no adding or subtracting

• Ex: For x^2-w^2-16+8w , the end result isn’t x^2-\left(w-4\right)^2 since there’s subtraction. So the real answer is \left(x+w-4\right)\left(x-w-4\right)

PRACTICE PROBLEMS ON NOTES WORKSHEET FOR BOTH (2.3b HAS MORE FACTORING STUFF)

2.2 - Vertex Form Quadratic Function

How to Complete the Square with a quadratic equation with 2x^2+12x+7=0 :

1. Move the constant to the other side

   1. 2x^2+12x=-7

2. Divide the entire equation by a if a is not equal to

   1. Because a is not equal to 1, we get x^2+6x=-\frac72

3. Complete the square with \left(\frac{b}{2a}\right)^2 and add that to both sides

   1. x^2+6x+9=-\frac72+9

4. Factor the left and simplify the right

   1. \left(x+3\right)^2=\frac{11}{2}

5. Square root the problem, then isolate x.

   1. x+3=\pm\sqrt{\frac{11}{2}}

   2. x=-3\pm\sqrt{\frac{11}{2}}

How to convert from Standard Form to Vertex Form with f\left(x\right)=5x^2-20x+13

1. Group and factor out a = 5

   1. f\left(x\right)=5\left(x^2-4x\right)+13

2. Complete the square inside with \left(\frac{b}{2a}\right)^2

   1. it’s 4, so add and subtract it inside the parenthesis

   2. f\left(x\right)=5\left(x^2-4x+4-4\right)+13

3. Distribute and simplify

   1. First remove the -4 from the parenthesis, and don’t forget to multiply it by the a (a=5 here)

      1. f\left(x\right)=5\left(x^2-4x+4\right)-20+13

   2. Then simplify the parenthesis and add the terms together to get the final result

      1. f\left(x\right)=5\left(x-2\right)^2-7

Ask about how to do questions like this, where we use the line of symmetry to help us:

Joe Blow owns a commuter airline transport business. He transports about 800 passengers a day between Chicago and Fort Wayne. A round-trip ticket is $300. Joe has figured out that for every $5.00 increase in the ticket price, 10 passengers would be lost to the competition. What ticket price should Joe charge to maximize his income and what would his maximum income be?

2.3c

1. Trinomials:

   1. x^2+bx+c=\left(x+A\right)\left(x+B\right) where

      1. A\cdot B=c

      2. A+B=b

   2. ax^2+bx+c=\left(Ax+B\right)\left(Cx+D\right) where

      1. A\cdot C=a

      2. A\cdot D+B\cdot C=b

      3. B\cdot D=c

2. Two Cubes:

   1. Sum of Two Cubes: a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)

   2. Difference of Two Cubes: a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)

PRACTICE QUESTIONS LIKE THESE ON NOTES WORKSHEET

2.3d - Solving Quadratics Factoring

Notes:

1. When Factoring, factor by squares than cubes if you have a choice between the two

1. Zero Product Property: if a product of a real-number factors is 0, then at least one of the factors must be 0

   1. Ex: if ab = 0, then a = 0, b = 0, or both equal 0

2. Equation of Motion: s\left(t\right)=-16t^2+v_0t+s_0 where

   1. v_0 = initial velocity

   2. s_0 = initial height

   3. s\left(t\right) = height at time t

   4. -16t^2 = acceleration of gravity (ft/sec²)

   5. KNOW THIS EQUATION, IT POPS UP IN A COUPLE OF PROBLEMS

IN GENERAL PRACTICE THE PROBLEMS ON NOTES WORKSHEET

2.3e - Factored Form of a Quadratic Function

NOTES:
- Every solution for a quadratic function is an x-intercept on the graph

Factored Form of the Quadratic Function: y=a\left(x-p\right)\left(x-q\right) where

• the x-intercepts are p and q

• the axis of symmetry is halfway between (p,0) and (q,0)

PRACTICE PROBLEMS ON NOTES WORKSHEET

Pearson 2.4 - Imaginary and Complex Numbers

Imaginary number: invented so negative numbers would have square roots

• they consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1

Ex: What is i^{63} ?

• i^{63} = \left(i^4\right)^{15}\cdot i^3 =-i

When operating on imaginary numbers:

1. Always take the i out of the radical first.

2. Treat i as a variable

3. Never write i with a power greater than 1

Complex numbers: consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. So the real part is a and the imaginary part is bi.

• All real numbers are complex numbers

• We assume that i behaves like a real number, that is it obeys all the rules of real numbers

1. Addition and Subtraction:

   1. We just combine like terms

2. Multiplication:

   1. Multiply like usual, but rewrite all i’s at the end with a power greater than one to have a power of 1

      1. Ex: rewrite i^2 to just -i

   2. \left(A+Bi\right)\left(A-Bi\right)=A^2+B^2

3. Equality of Complex Numbers:

   1. a+bi=c+di if and only if a = c and b = d

4. Division:

   1. Write as a complex number in standard form with no imaginary numbers in the denominator

2.4b - Complex Numbers

For complex numbers, if a + bi and a - bi are solutions, then \left\lbrack x-\left(a+bi\right)\right\rbrack\left\lbrack x-\left(a-bi\right)\right\rbrack=0

Find the square root of \sqrt{5-12i}

1. Remember that \sqrt{5-12i}=a+bi so 5-12i=\left(a+bi\right)^2

2. Reference the Notes for the rest of the steps.

3. PRACTICE QUESTIONS LIKE THIS

2.5 - Completing the Square

ASK CHAT GPT HOW TO COMPLETE THE SQUARE

2.6a - Quadratic Formula/Nature of Solutions

Quadratic Formula: If ax^2+bx+c=0 , thenx=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Example: 3x^2+xy+4y^2-9=0

• Here, A = 3, B = y, and C = 4y^2-9

• Then just plug it into the formula to solve

Nature of Solutions:

Discriminant:

• B^2-4AC<0 = 2 complex conjugate solutions

• B^2-4AC=0 = 1 real solution

• B^2-4AC>0 = 2 real distinct solutions

For the equation ax^2+bx+c=0 the sum of the solutions is -\frac{B}{A} and the product of the solutions is \frac{C}{A}

Formula: x² - (sum)x + product = 0

PRACTICE PROBLEMS ON NOTES WORKSHEET

2.6b - Solutions of Quadratics Applications

PRACTICE PROBLEMS, SO USE NOTES WORKSHEET FOR PRACTICE

2.7b - Finding a Quadratic Function

PRACTICE PROBLEMS, SO USE NOTES WORKSHEET FOR PRACTICE