Name:______Period:____
Chapter 4 Notes Packet on Quadratic Functions and Factoring
Notes #15: Graphing quadratic equations in standard form, vertex form, and intercept form.
A. Intro to Graphs of Quadratic Equations:
· A ______is a function that can be written in the form where a, b, and c are real numbers and a0. Ex:
· The graph of a quadratic function is a U-shaped curve called a ______. The maximum or minimum point is called the ______
Identify the vertex of each graph; identify whether it is a minimum or a maximum.
1.) 2.)
Vertex: ( , ) ______Vertex: ( , ) ______
3.) 4.)
Vertex: ( , ) ______Vertex: ( , ) ______
B. Key Features of a Parabola:
· Direction of Opening: When , the parabola opens ______:When , the parabola opens ______:
· Width: When , the parabola is ______than
When , the parabola is the ______width as
When , the parabola is ______than
· Vertex: The highest or lowest point of the parabola is called the vertex, which is on the axis of symmetry. To find the vertex, plug in and solve for y. This yields a point (____, ____)
· Axis of symmetry: This is a vertical line passing through the vertex. Its equation is:
· x-intercepts: are the 0, 1, or 2 points where the parabola crosses the x-axis. Plug in y = 0 and solve for x.
· y-intercept: is the point where the parabola crosses the y-axis. Plug in x = 0 and solve for y.
Without graphing the quadratic functions, complete the requested information:
5.)What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______/ 6.)
What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______
7.)
What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______/ 8.)
What is the direction of opening? ______
Is the vertex a max or min? ______
Wider or narrower than y = x2 ? ______
The parabola y = x2 is graphed to the right.
Note its vertex (___, ___) and its width.
You will be asked to compare other parabolas to this graph. /
C. Graphing in STANDARD FORM (): we need to find the vertex first.
Vertex- list a = ____, b = ____, c = ____
- find x =
- plug this x-value into the function (table)
- this point (___, ___) is the vertex of the parabola / Graphing
- put the vertex you found in the center of
your x-y chart.
- choose 2 x-values less than and 2 x-values more than your vertex.
- plug in these x values to get 4 more points.
- graph all 5 points
Find the vertex of each parabola. Graph the function and find the requested information
9.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____/ Vertex: ______
Max or min? ______
Direction of opening? ______
Axis of symmetry: ______
Compare to the graph of y = x2
______
10.) h(x) = 2x2 + 4x + 1
/ Vertex: ______
Max or min? ______
Direction of opening? ______
Axis of symmetry: ______
Compare to the graph of y = x2
______
11.) k(x) = 2 – x –x2
/ Vertex: ______
Max or min? ______
Direction of opening? ______
Axis of symmetry: ______
Compare to the graph of y = x2
______
12.) State whether the function y = -3x2 + 12x - 6 has a minimum value or a maximum
value. Then find the minimum or maximum value.
13.) Find the vertex of . State whether it is a minimum or maximum. Find that minimum or maximum value.
Another useful form of the quadratic function is the vertex form: ______.GRAPH OF VERTEX FORM y = a(x - h)2 + k
The graph of y = a(x - h)2 + k is the parabola y = ax2 translated ______h units and ______k units.
· The vertex is (___, ___).
· The axis of symmetry is x = ___.
· The graph opens up if a ___ 0 and down if a ___ 0.
Find the vertex of each parabola and graph.
13.)/ Vertex: ______
14.)
/ Vertex: ______
15.) Write a quadratic function in vertex form for the function whose graph has its vertex
at (-5, 4) and passes through the point (7, 1).
GRAPH OF INTERCEPT FORM y = a(x - p)(x - q):Characteristics of the graph y = a(x - p)(x - q):
· The x-intercepts are ___ and ___.
· The axis of symmetry is halfway between (___, 0) and ( ___ , 0)
and it has equation x =
· The graph opens up if a ___ 0 and opens down if a ___ 0.
16.) Graph y = -2(x - 1)(x - 5)
/ x-intercepts: ______, ______
Vertex: ______
Converting between forms:
From intercept form to standard form· Use FOIL to multiply the binomials together
· Distribute the coefficient to all 3 terms / Ex:
From vertex form to standard form
· Re-write the squared term as the product of two binomials
· Use FOIL to multiply the binomials together
· Distribute the coefficient to all 3 terms
· Add constant at the end / Ex:
HW #15: Pg. 202: 47-63 odd
pg. 240 #3-39 x 6’s
pg. 249 #4-40 x 4’s
Notes 16: Sections 4-3 and 4-4: Solving quadratics by Factoring
A. Factoring Quadratics
Examples of monomials:______Examples of binomials:______
Examples of trinomials:______
Strategies to use: (1) Look for a GCF to factor out of all terms
(2) Look for special factoring patterns as listed below
(3) Use the X-Box method
(4) Check your factoring by using multiplication/FOIL
Factor each expression completely. Check using multiplication.
1.) / 2.) / 3.)4.) / 5.) / 6.)
7.) / 8.) / 9.) 25t2 - 110t + 121
10.) / 11.) / 12.)
B. Solving quadratics using factoring
To solve a quadratic equation is to find the x values for which the function is equal to _____. The solutions are called the _____ or ______of the equation. To do this, we use the Zero Product Property:
Zero Product Property
List some pairs of numbers that multiply to zero:
(___)(___) = 0 (___)(___) = 0 (___)(___) = 0 (___)(___) = 0
What did you notice? ______
ZERO PRODUCT PROPERTYIf the ______of two expressions is zero, then ______or ______of the expressions equals zero.
Algebra / If A and B are expressions and AB = ____ , then A = _____ or B = __.
Example / If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is,
x = ______or x = ______.
Use this pattern to solve for the variable:
1. get the quadratic = 0 and factor completely
2. set each ( ) = 0 (this means to write two new equations)
3. solve for the variable (you sometimes get more than 1 solution)
Find the roots of each equation:
13.) / 14.) / 15.)Find the zeros of each equation:
16.) / 17.) v(v + 3) = 10 / 18.)
Find the zeros of the function by rewriting the function in intercept form:
19.) / 20.) / 21.)Graph the function. Label the vertex and axis of symmetry:
22.)/ Vertex: ______
Maximum or minimum value: ______
x-intercepts: ______
Axis of symmetry: ______
Compare width to the graph of y = x2
______
HW #16:
pg. 255 #4-52 x 4’s, pg. 263 #4-52 x 4’s, Pg. 265: #78-81
Study for Quiz on Sec. 4.1-4.4 (Quiz is tomorrow!)
Notes #17: Section 4-5 Solve Quadratic Equations by Finding Square Roots
A. Simplifying Square Roots:
· Make a factor tree; circle pairs of “buddies.”
· One of each pair comes out of the root, the non-paired numbers stay in the root.
· Multiply the terms on the outside together; multiply the terms on the inside together
Simplify:
1.) / 2.)B. Multiplying Square Roots:
· Simplify each radical completely by taking out “buddies”
· (outside • outside) or
· Simplify your answer, if possible
Simplify:
3.) / 4.)C. Simplifying Square Roots in Fractions:
· Split up the fraction:
· Simplify first by taking out “buddies” or reducing (you can only reduce two numbers that are both under a root or two numbers that are both not in a root)
· Square root top, square root bottom
· If one square root is left in the denominator, multiply the top and the bottom by the square root and simplify
OR If a binomial is left in the denominator, then multiply top and bottom by the conjugate of the denominator (exact same expression except with the opposite sign). Remember to FOIL on the denominator.
· Reduce if possible
Simplify:
5.) / 6.) / 7.)
D. Solving Quadratic Equations Using Square Roots
· Isolate the variable or expression being squared (get it ______)
· Square root both sides of the equation (include + and – on the right side!)
· This means you have ______equations to solve!!
· Solve for the variable (make sure there are no roots in the denominator)
8.) x2 = 25 9.) 3x2 = 81
10.) 4x2 – 1 = 0 11.)
12.) (2y + 3)2 = 49 13.)
HW #17: Pg. 202: 46-62 even, Pg. 258: 79-89 odd, pg. 269 #3-33 x 3’s
Notes#18: Sections 4-6 & 4-7 Complex Numbers and Completing the square
Section 4.6: Complex Numbers
A. Definitions
DefineComplex Numbers:
imaginary unit (i):
imaginary number :
B. Solving a quadratic equation with complex roots
· Isolate the expression being squared
· Square root both sides; write two equations Replace with i. Simplify
Solve
1.) x2 = -27 / 2.) 2x2 + 11 = -37 / 3.)C. Adding, subtracting, and multiplying complex numbers
· Distribute/FOIL. Combine like terms.
· Replace with (-1). Simplify.
Simplify
4.) (3 + 7i) - (8 - 2i) / 5.) (2 + 5i) + (7 - 2i) / 6.) (2 + i)(-5 + 2i) / 7.) (3 - i)(5 - 2i)B. Dividing complex numbers
· If i is part of a monomial on the denominator, multiply top and bottom by i. ex:
· If i is part of a binomial on the denominator, multiply top and bottom by the complex conjugate of the denominator (same expression but opposite sign). FOIL. ex:
· Replace with (-1). Simplify.
8.) / 9.) / 10.) / 11.)
Section 4.7: Completing the Square
B. Review: Solving Using Square Roots
· Factor and write one side of the equation as the square of a binomial
· Square root both sides of the equation (include + and – on the right side; 2 equations!
· Solve for the variable (make sure there are no roots in the denominator)
1) (k + 2)2 = 12 2.) x2 + 2x + 1 = 8 3.) n2 – 14n + 49 = 3
C. Completing the Square
· Take half the b (the x coefficient)
· Square this number (no decimals – leave as a fraction!)
· Add this number to the expression
· Factor – it should be a binomial, squared ( )2
4.) x2 + 6x + _____ 5.) m2 – 14m + ______
( )( )
( )2
Find the value of c such that each expression is a perfect square trinomial. Then write the expression as the square of a binomial.
6.) w2 + 7w + c 7.) k2 – 5k + c
Solving by Completing the Square:· Collect variables on the left, numbers on the right
· Divide ALL terms by a; leave as fractions (no decimals!)
· Complete the square on the left – add this number to BOTH sides
· Square root both sides (include a ______and ______equation!)
· Solve for the variable (simplify all roots – look for )
8.) x2 + 4x – 5 = 0 9.) m2 – 5m + 11 = 10
10.) 11.)
12.) 2x2 – 3x – 1 = 0 13.)
Cumulative Review: Solving Quadratics
Solve by factoring:
14.) 12k2 – 5k = 2 15.) 49m2 – 16 = 0
Solve by using square roots:
16.) 4w2 = 18 17.) 3y2 – 8 = 0
HW #18: pg. 279 #3-33 x 3’s pg. 288 #3-36 x 3’s pg. 1013: 6-32 even
Notes #19: Sec. 4-8 Use the Quadratic Formula and the Discriminant
A. Review of Simplifying Radicals and Fractions
· Simplify expression under the radical sign (); reduce
· Reduce only from ALL terms of the fraction.
(You can’t reduce a number outside of a radical with a number inside of a radical)
· Make sure that you have TWO answers
Simplify:
1.) / 2.)3.) / 4.)
5.) / 6.)
B. Solving Quadratics using the Quadratic Formula