Week 9- Quadratic Functions

This week in math, we learned more about quadratic functions and how to find the x intercepts through factored form. We learned through general form we can find the y intercept and through standard form we can find the vertex and the axis of symmetry. Taking this information, we can take the general form of the equation and convert it into either standard form or factored form to find the information that we need.

General form: 

Standard form(completing the square): {y=a(x-p)^2}+q

Factored form: y = a (x-x1) (x-x2)

Ex) General form to factored and standard form

y= 2x^2 -14x + 16

  • y-intercept: 16
  • opens up, minimum value
  • congruent: y=2x^2
  • stretches

y= (2x-16) (x+1)

  • x-intercepts: (8, 0) (-1, 0)
  • axis of symmetry(x) = 3.5

y= 2x^2 -14x + 16

y= 2(x^2 -7x+49/4 -49/4)+ 16

y= 2(x^2 -7x+49/4)-98/4+ 64/2

y= 2(x -7/2)2 -34/4

y= 2(x -7/2)2 -17/4

  • vertex (3.5, 8.5)

Week 8- Graphing Quadratic Equations

This week we learned how to graph quadratic equations, which is an expansion of what we have learned so far.

The general form of the equation of a quadratic function

Vocabulary

  • Parabola: the curve of every quadratic function on the graph
  • Vertex: the highest or lowest point of the parabola
  • Minimum point: the lowest point(vertex) of the parabola as it opens upward
  • Maximum point: the highest point(vertex) of the parabola as it opens downward
  • Axis of symmetry: the line that divides the parabola equally in half
  • Domain: all the possible x-values
  • Range: all the possible y-values
  • Congruent: the pattern that moves up by 1, 3, 5..  as it moves across the graph (determines the shape of the parabola)
  • Compression: when the parabola is wider
  • Stretch: when the parabola is skinnier (pattern: 2, 6, 10..)

Standard form{y=a(x-p)^2}+q, indicates all the information needed to graph the parabola

a

  • opens up or down
  • stretches or compresses

p

  • horizontal translation
  • vertex(p,q)

q

  • vertical translation(not the y-intercept)
  • vertex(p,q)

 

Week 7- Discriminant of a Quadratic Equation

This week of math we learned about discrinants and In the quadratic equation, the discriminant tells us the number of real solutions. We can find the discriminant by plugging in the correct numbers of the quadratic equation into the radicand of the quadratic formula.

quadratic formula:

radicand of the formula: b^2 – 4ac

discriminant is a positive number(+#): 2 solutions

discriminant is 0: 1 solution

discriminant is a negative number(-#): 0 solutions/ unreal

steps:

1) verify that the equation equals to 0

2) identify a, b, and c

3) take the correct formula of the radicand and plug in the numbers

example:

x^2 + 8x + 7

a:1, b: 8, c: 7

b^2 – 4ac

8^2 – 4(1)(7)

64- 4(7)

64- 28

+38= 2 real solutions

Week 6- Solving Quadratic Equations

This week, we learned how to solve quadratic equations by factoring, completing the square, and using the quadratic formula.

Factoring

Factoring and using the zero product property is one of the easiest ways to solve a quadratic formula, and it works with rational roots.

  • A quadratic equation is any equation that can be written in the form ax^2 + bx + c= 0.
  • Zero product property: if ab=0, then a=0 and b=0, or both a, b=0

example:

3x^2 – 21x- 54= 0

  • Factor, remembering CDPEU

3(x^2  – 7x- 18)= 0

  • Use the zero product property

(x+2) (x-9)= 0

Either x+2= 0 or x-9= 0

x= -2, x= 9

Completing the Square

When the equation doesn’t factor, completing the square is a great way to solve the quadratic equation, and it works well with irrational roots.

  • Rearrange the equation if the coefficient of x^2 is negative
  • Make sure to get rid of the coefficient of x^2 by dividing each side by the coefficient (when the equation equals to 0)
  • Move the constant term to the right of the equation
  • complete the perfect square by dividing the 2nd term and multiplying and squaring it, and adding it to both sides of the equation
  • factor the perfect square, and simplify the right side
  • rake the square root of each side, and isolate the x

example:

12x= -2x^2 + 54

2x^2 + 12x= 54

x^2 + 6x= 27

x^2 + 6x + 9= 27+ 9

(x-3)^2 = \pm √36

x-3= \pm 6

x= 3 \pm 6

Quadratic Formula

The Quadratic formula is easy to use if it is arranged in the right form.

  • Quadratic formula: 
  • Identify the values of a,b, and c( the coefficients) ax^2 + bx+ c= 0
  • Plug a,b, and c in the formula and solve the equation

example:

Week 5- Factoring Polynomials

This week in pre-calculus 11, we reviewed and extended our understanding of factoring polynomials.

To remember the steps of factoring polynomials is simple. Just remember can divers pee easily underwater:)

Common GCF (remainder)

Difference of squares

Pattern

Easy (when the variable doesn’t have a coefficient)

Ugly (if the variable has a coefficient)

examples:

Week 4- Multiplying and Dividing Radical Expressions

This week in precalculus we learned how to solve redical equations. There are rules to remember when multiplying and dividing radical expressions.

Multiplication:

1) It is recommended to simplify first, because it is easier to simplify the smaller number

2) Remember to use the distributive property (FOIL aka the claw method)

Division:

1) There should not be a negative on the bottom of a fraction, so always move it to the top of the fraction

2) There should not be a square root sign in the denominator

monomial: multiply both the top and the bottom by the radical and the root of the denominator (with out coefficient)

binomial: multiply the conjugate of the denominator to both the top and the bottom of the fraction (remember that you need to foil when multiplying)

example:

Week 3- Absolute Value

On the third week of precalculus, we began a new unit on absolute value. Absolute value is the principal square root of a number, and it is the distance of the number from zero.

The absolute value of | -7 | and | 7 | are both 7.

Example: -| 5-9×3 | *remember to use BEDMAS

-| 5-9×3 |= -| 5-27 |= -| -22 |= -(22)= -22

1) use bedmas to find the number inside the absolute value symbol.

2) multiply 9 and 3 which equals to 27

3) take 27 away from 5, leaving -22

4) square -22 and then square root it, leaving 22

5) because there is a negative sign infront of the absolute value symbol, the answer will be -22

 

Week 2- Geometric Sequences and Series

On the second week of precalculus 11, we learned about geometric sequences and series. In a geometric sequence, each term is multiplied by a constant, known as the common ratio(r).

Geometric Sequences: 7,14,28,56,112..

t_{n}t_{1} (r^n-1)

t_{10} = 7 (2^9)

t_{10} = 7(512)

t_{10} = 3584

Geometric Series: 4+12+36+108+324..

Sn= a(r^n – 1) / r- 1

S_{9} = 4 (3^9 -1) / 3-1

S_{9} = 4 (19683-1) / 3-1

S_{9} = 4 (19682) / 2

S_{9} = 78728 / 2

S_{9} = 39364

Infinite Geometric Series: 16+ 4+ 1+ 0.25+…

S_{infinity} = a/1-r

S_{infinity} = 16/ 1- \frac{1}{4}

S_{infinity} = 16/ \frac{3}{4}

S_{infinity} = 16/ 1- \frac{1}{4}

S_{infinity} = 16/ \frac{3}{4}

S_{infinity} = \frac{64}{3}

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