Math 6 Honors Periods 6 & 7 (Wednesday)

Areas of Triangles and Trapezoids 10-2

Any side of a triangle can be considered to be the base. The height is then the perpendicular distance from the opposite vertex to the base line.

Let us find the area of a triangle having base b and height h. The triangle and a congruent copy of it can be put together to form a parallelogram

Since the area of the parallelogram is bh—from yesterday’s lesson--the area of the triangle is half the area of the parallelogram so we have the following

Formula

Area of triangle = ½ · base · height

A = ½ bh

Find the area of each triangle

Look at the pictures in your textbook Page 325 and practice a few

Note that in the second example, the lengths of the sides of the right angle of the triangle were used as the base and the height. this can be done for any right triangle.

Trapezoids

The height of a trapezoid is the perpendicular distance between the parallel sides. These parallel sides are called the bases of the trapezoid. The method used to find the formula for the area of a triangle can be used to find the formula for the area of a trapezoid having bases b₁ and b₂ and height h. The trapezoid and a congruent copy of it can be put together to form a parallelogram. The area of the parallelogram is (b₁ + b₂)h and the area of the original trapezoid is half the area of the parallelogram.

Formula

Area of trapezoid = ½ · (sum of bases) · height

A = ½ (b₁ +b₂)h

Algebra Period 3 (Tuesday)

Quadratic Equations Review: Chapter 13


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 13-2 Solving Quadratics Using Square Roots -
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 PRINCIPAL SQUARE ROOT ANY MORE.

YOU NEED THE + OR - SYMBOL!!

EXAMPLE:
3x² = 18
divide both sides by 3 and get: x² = 6
square root each side and get x = + or - SQRT of 6

HARDER EXAMPLE:
(x - 5)² = 9
SQRT each side and get: x - 5 = + or - 3
+ 5 to both sides: x = 5 + 3 or x = 5 - 3
So, the 2 roots are x = 8 or x = 2

HARDEST EXAMPLE:
(x + 2)² = 7
SQRT each side and get: x + 2 = + or - SQRT of 7
-2 to both sides: x = -2+ SQRT 7 or x = -2 - SQRT 7

THE QUADRATIC FORMULA 13-4

-b plus or minus the square root of b squared minus 4ac all over 2a

Notice how the first part is the x value of the vertex -b/2a
The plus or minus square root of b squared minus 4ac represents
how far away the two x intercepts (or roots) are from the vertex!!!!

Very few real world quadratics can be solved by factoring or square rooting each side.
And completing the square always works, but it long and cumbersome!

All quadratics can be solved by using the QUADRATIC FORMULA.

(you will find out that some quadratics have NO REAL solutions, which means that there are no x intercepts - the parabola does not cross the x axis! Think about what kinds of parabolas would do this....ones that are smiles that have a vertex above the x or ones that are frowns that have a vertex below the x axis. You will find out in Algebra II that these parabolas have IMAGINARY roots)

So now you know 5 ways that you know to find the roots:

1. graph
2. factor if possible
3. square root each side
4. complete the square - that's what the quadratic formula is based on!
5. plug and chug in the Quadratic Formula -
This method always works if there's a REAL solution!

DON'T FORGET TO PUT THE QUADRATIC IN STANDARD FORM BEFORE PLUGGING THE VALUES INTO THE QUADRATIC FORMULA!

ax² + bx + c = 0

DISCRIMINANTS - a part of the Quadratic Formula that helps you to understand the graph of the parabola even before you graph it!
the discriminant is b² - 4ac

(the radicand in the Quadratic Formula, but without the SQRT)

Depending on the value of the radicand, you will know
HOW MANY REAL ROOTS IT HAS

1) Some quadratics have 2 real roots (x intercepts or solutions) - Graph crosses x axis twice
2) Some have 1 real root (x intercept or solution) - Vertex is sitting on the x axis
3) Some have NO real roots (no x intercepts or solutions) - vertex either is above the x axis and is a smiley face (a coefficient is positive) or
the vertex is below the x axis and is a frown face (a coefficient is negative)

In both of these cases, the parabola will NEVER CROSS (intercept) the x axis!

b² -4ac > 0 That is, if it's positive, then there are 2 roots
b² -4ac = 0 That is, if it's zero , then there is 1 root
b² -4ac <0>

Algebra Period 3 (Tuesday)

Review of how to graph a quadratic

Find the vertex ( x = -b/2a, then plug in to the equation to find the y coordinate)

Find the axis (line) of symmetry. Just use the x value of the vertex that is
x = -b/2a

Draw the line of symmetry using a dotted or dashed line

Find 2 more points in an x y table (go either left or right of the vertex and use 2 points close to the line of symmetry for the x, then plug into the equation to find the y values)

Find the 2 shadow (mirror) points by counting over from the axis at the exact same y value

Draw a U (not a V) through the points. Extend it past and put arrows at the end
Remember to label the x and y axes and put arrows on the axes.

Function vs. Relations: Functions are special relations where there is a unique x for each y
Therefore, there will never be 2 x’s that will repeat and if you use a vertical line on the graph, it will only hit one point on the graph.

Domain vs. Range: the domain for any quadratic function is ALL REAL NUMBERS
The range depends on where the vertex is and whether the quadratic is a smile or a frown.
Generally, it will be of the form:

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}

f(x) is just a different way of saying y. What is better about it? Without seeing the work before the solution, you can actually tell the x value as well as the y value in the solution.

For example, if the solution is f(-3) = 12 you know that the point is (-3, 12)
Compare that to the solution y = 12. For that solution, you would not know the x value unless you look back in the problem.

Math 6 Honors Periods 6 & 7 (Tuesday )

Areas of Rectangles and Parallelograms 10-1

In Chapter 4 (so long ago) we measured lengths of segments and found perimeters of various polygons. Now we will measure the part of the plane enclosed by a polygon. We call this measure the area of the polygon.

In the metric system the unit area often use is the square centimeter. cm²

Notice that the area of each rectangle is the product of the lengths of the two consecutive sides. These sides are called the length and the width of the rectangle The length names the longer side and the width names the shorter side. These sides are also named the base and height.

Formula

Area of rectangle = length · width

A= lw

The length and the width of the rectangle are called its dimensions

In the case of a parallelogram, we may consider either pair of parallel sides to the bases. (the word base is also used to denote the length of the base) the height is the perpendicular distance between the bases.

Formula

Area of parallelogram = base · height

A= bh

The unit areas used are square meters (m²), square millimeters (mm²) and square centimeters (cm²). For very large regions, such as the State of California, or entire countries (Canada), we could use square kilometers (km²)

Sometimes we work with an unspecified unit of length. Then the unit of area is simply called a square unit. It is vital, however, that you indicate the area is in square units.