Review of how to graph a quadratic
Find the vertex ( x = -b/2a, then plug in to the equation to find the y coordinate)
x = -b/2a
Draw the line of symmetry using a dotted or dashed line
Function vs. Relations: Functions are special relations where there is a unique x for each y
Domain vs. Range: the domain for any quadratic function is ALL REAL NUMBERS
y such that y is either greater than or equal to (≥ ) or less than or equal to (≤) the y value of the vertex. {y l y≥ n} or {y l y ≤ n}
Chapter 9-3: Polygons
Polygon - closed figure with at least 3 sides- no curves - no overlapping lines
Named by the number of sides (called laterals) or their number of angles
Polygon literally means MANY (poly) ANGLES (gon)
Triangles - literally means 3 angles (and sides) - Each has 2 names
By angles: Acute, Obtuse, Right
By sides: Scalene, Isosceles, Equilateral
Triangles have 180 degrees
Quadrilaterals - literally means 4 laterals (sides) and angles - 360 degrees (2 triangles!)
Trapezoids - Only 1 set of parallel lines
Kite - no parallel lines - 2 adjacent sides are congruent
Parallelogram - Oppositie sides parallel and congruent
Types of parallelograms:
Rhombus - all sides congruent
Rectangle - 4 right angles
Square - Rectangle with all sides congruent (so it's also a rhombus)
A square is a regular rectangle
Regular polygons - all sides and angles congruent
2 famous ones - equilateral triangle and squares
ALL REGULAR POLYGONS ARE SIMILAR!
So to find the PERIMETER of a regular figure, you just need to know one side and multiply by the total number of sides.
Chapter 9-5: Congruent polygons
If two polygons are congruent, they all their corresponding parts are congruent
We use tick marks to denote congruent sides and angles.
There are 3 ways to show that triangles are congruent:
SSS = side, side, side = all corresponding sides are congruent
SAS = side, angle, side = 2 sides with the angle that is between them
ASA = angle, side, angle = 2 angles with the side between them
THERE IS NO SUCH THIS AS ASS OR AAA!!!!!!!!
If all angles are congruent, then they may be congruent or SIMILAR
Quadratic Equations: 13-1
But first… Finding the x intercepts of a parabola:
x intercepts = roots = solutions = zeros of a quadratic
You can find these one of two ways:
1) Read them from the graph
2) Set y or f(x) = 0 and then solve
Where the graph crosses the x axis is/are the x intercepts.
(Remember, y = 0 here!)
The x intercepts are the two solutions or roots of the quadratic.
When we factored in Chapter 6 and set each piece equal to zero,
We were finding the x value when y was zero.
That means we were finding these two roots!
You can find these roots (solutions, x intercepts, zeros) by several different methods.
You already know the following 2 methods:
1) Graphing and see where the graph crosses the x axis
2) Factoring, then using the zero products property to solve for x for each factor
ANOTHER METHOD:
Chapter
You did this in Ch 11 for Pythagorean Theorem!
If there is no middle x term, it's easiest to just square root both sides to solve!
BUT THE DIFFERENCE FROM PYTHAGOREAN--NOT LOOKING FOR JUST THE
Chapter 12-4 Quadratic functions
A QUADRATIC FUNCTION is not y = mx + b
(which is a LINEAR function),
but instead is
y = ax2 + bx + c
OR
f(x) = ax2 + bx + c
where a, b, and c are all real numbers and
a cannot be equal to zero because
it must have a variable that is squared ( degree of 2)
Quadratics have a squared term, so they have TWO possible solutions also called roots You already saw this in Chapter 6 when you factored the trinomial and used zero products prop.
If the domain is all real numbers, then you will have a PARABOLA which looks like
a smile when the a coefficient is positive or
looks like a frown when the a coefficient is negative.
Graphing quadratics:
You can graph quadratics exactly the way you graphed lines
by plugging in your choice of an x value and using the equation to find your y value.
Because it's a U shape, you should graph 5 points as follows:
Point 1) the vertex - the minimum value of the smile or the maximum value of the frown
The x value of the VERTEX = -b/2a
We get the values for a and b from the actual equation
f(x) = ax2 + bx + c
Plug that into the equation and then find the y value of the vertex
Next, draw the AXIS OF SYMMETRY : x = -b/2a
a line through the vertex parallel to the y axis
Point 2) Pick an x value to the right or left of the axis and find its y by plugging into the equation.
Point 3) Graph its mirror image on the other side of the axis of symmetry by counting from axis of symmetry
Points 4 and 5) Repeat point 2 and 3 directions with another point even farther from the vertex
JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END
Parabolas that are functions have domains that are ALL REAL NUMBERS
Their ranges depend on where the vertex is and also if the a coefficient is positive or negative
EXAMPLE: f(x) = -3x2 (or y = -3x2)
the a coefficient is negative so it is a frown face
the x value of the vertex (maximum) is -b/2a or 0/2(-3) = 0
the y value of the vertex is 0
So the vertex is (0, 0)
The domain is all real numbers.
The range is y is less than or equal to zero
To graph this function:
1) graph vertex (0, 0)
Draw dotted line x = 0 (actually this is the y axis!)
2) Pick x value to the right of axis of symmetry, say x = 1
Plug it in the equation: y = -3(1) = -3
Plot (1, -3)
3) count steps from axis of symmetry and place another point to the LEFT of axis in same place
(-1, -3)
4) Pick another x value to the right, say x = 2
Plug it in the equation to find y: y = -3(22) = -12
(2, -12)
5) count steps from axis of symmetry and place another point to the LEFT of axis in same place
(-2, -12)
JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END
CHAPTER 9
GEOMETRY
I will go over in class the symbols of each of the following
Point:(symbol is a dot or just a letter) Location in space - no size
Ray: (arrow pointing to the right) - one endpoint and one direction- Named by its endpoint first
Line: (generally, line with arrows on both ends above 2 points on the line) - Series of points that goes on infinitely in both directions - named either direction
Line segment: (a line with no arrows on either end) - a piece of a line with 2 endpoints in either direction
Lines can be parallel (2 vertical lines) or intersecting in the same plane
Parallel lines are lines in the same plane that never meet
If they intersect at exactly 90 degrees, then they are perpendicular
If they don't intersect but are in two different planes, they are skew
angle: (angle opening to the right) two rays that meet at the same endpoint
named by either just the vertex, or 3 points on the angle in either direction with vertex in middle
adjacent angles share one ray
vertical angles are opposite each other and congruent (equal)
Complementary sum to 90 degrees and
Supplementary sum to 180 degrees
acute is greater than 0 degrees and less than 90 degrees
90 degrees is right angle
obtuse is greater than 90 degrees but less than 180 degrees
180 degrees is a straight angle (line)
to write the measure of an angle you write m<
Chapter 12-1 and 12-2
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.
FUNCTIONS: Relations where there is just one y value for each x value IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
In the real world, I have a good example...pizza prices.
You can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.
VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function, but a vertical line function, x = 4 is not.
Any line, y = mx + b, is a function.
INPUTS: x values
OUTPUTS: y values
f(x) means the value of the function at the given x value
You can think of f(x) as the y value
Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)
Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain
EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number
EXAMPLE:
f(x) = x2 has the domain again of all real numbers, BUT the range is greater than or = to zero
because when a number is squared it will never be negative! So f(x) will always be 0 or positive
EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative
EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator. The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them
EXAMPLE:
f(x) = x - 10
thisthisx + 3
Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?
