Final: Math ISU

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Final: Math ISU by Mind Map: Final: Math ISU

1. Trigonometric ratios of right triangles

1.1. SOCAHTOA

1.1.1. SOH stands for Sine equals Opposite over Hypotenuse. CAH stands for Cosine equals Adjacent over Hypotenuse. TOA stands for Tangent equals Opposite over Adjacent.

1.1.2. Example:

2. Congruent figures

2.1. Congruent figures are identical in shape and size

2.2. Congruent triangles have the following properties: exactly the same three sides and exactly the same three angles.

3. Geometric Properties

3.1. Properties of Triangles

3.1.1. - Every triangle has three vertices. - Altitude The altitude of a triangle is the perpendicular from the base to the opposite vertex. - The median of a triangle is a line from a vertex to the midpoint of the opposite side. The three medians intersect at a single point, called the centroid of the triangle. - the interior angles of a triangle equal to 180 degrees 7 - An exterior angle of a triangle is equal to the sum of the opposite interior angles

3.2. Properties of Quadrilaterals

3.2.1. Squares

3.2.1.1. All angles are equal (90°). All sides are of equal length. Opposite sides are parallel. The diagonals bisect each other at 90°. The diagonals are equal in length.

3.2.2. Rhombus

3.2.2.1. Diagonally opposite angles are equal. All sides are of equal lengths. Opposite sides are parallel. The diagonals bisect each other at 90°.

3.2.3. Rectangles

3.2.3.1. All angles are equal (90°). Opposite sides are of equal length. The diagonals are equal in length.

3.2.4. Parallelogram

3.2.4.1. Diagonally opposite angles are equal. Opposite sides are of equal length. Opposite sides are parallel. The diagonals bisect each other.

3.2.5. Trapazoid

3.2.5.1. One pair of opposite sides is parallel.

3.2.6. Kite

3.2.6.1. Two pairs of sides are of equal length. One pair of diagonally opposite angles is equal. Only one diagonal is bisected by the other. The diagonals cross at 90°.

4. Analytic Geometry

4.1. Geometric Properties

4.2. Linear Systems

4.3. Co-oddinate Geomertry

5. Linear System:Two or more equation that are taken into consideration at one time

5.1. Solving Linear Systems/ Solution(s) of Linear Systems

5.1.1. Subsitution

5.1.1.1. Meaning: The Method of Substitution is solving for a lineear system by subbing for one variable from one equation into the other

5.1.1.1.1. Ex: Find Solution X + Y = 5 (1) 3x - Y = 7 (2) X + Y = 5 - Solve one equation Y = 5 – X 3X - (5 - X) = 7 Subsitute that variable in the equation (2) to solve 3X - 5 + x = 7 4X = 12 X= 3 X + Y = 5 Subsitute it back into (1) 3 + Y = 5 Y = 5 - 3 Y = 2 Conclude -> x = 3 , y =2

5.1.2. Elimination

5.1.2.1. Meaning: Solving a Linear System by adding of subtraction to elimination on of the variable

5.1.2.1.1. Ex: Find Solution 3x + y = 19 Add colum vertically 4x - y = 2 _______ 7x = 21 Solve for X x = 21/4 x=3 3x + y = 19 Sub x=3 into either equation 9 + y = 19 y = 10 Conclude --> x=3, y=10

5.1.3. Problem Solving Linear Systems

5.1.3.1. To solve problems in linear systems you can solve all by graphing, by the method substitution, or elimination

5.1.3.1.1. Example 1: Mixture Problem Matt has a can of 5% acid and a bottle of 10% acid. How much of each should he use to make a 250ml of 8% acetic acid? Let f be the # of 5% acid in the 8% mixture. Let t be the # of 10% acid in the 8% mixture. f + t = 250 (1) .05f + .1t = 20 (2) .05f + .1t = 20 .05(250 – t) + .1t = 20 12.5 - .05t + .1t = 20 12.5 + .05t =7.5 t = 150 sub 150 into (1) f + t = 250 f + 150 = 250 f = 100 Therefore, Matt should mix 100mL of the 5% acid with the 150mL of the 10% acid to make 250mL of 10% acid.

6. Quadratic Relations

7. Trigonometry

8. Coordinate Geometry: A system of geometry where the position of points on a Cartesian plane ( grid with perpendicular axes) plane is described using an ordered pair of numbers.

8.1. Line Segment/Lines

8.1.1. A line segment is the part of the line that joins 2 points

8.1.1.1. Length of a line segment

8.1.1.1.1. The Length of a line segment can be calculated using the following formula: √(x2 - x1) ² + (y2 - y1) ²

8.1.1.2. Midpoint

8.1.1.2.1. The Midpoint is a point that divedes a line segment into two equal parts

8.1.2. Slope

8.1.2.1. The Slope is defined as a measurement of steepness of a line

8.1.2.1.1. The Slope can be calculated by the following formulas: m = rise/run or m = y2 - y1 / x2 - x1

8.1.3. Equation of Circle

8.1.3.1. A Circle is a 2-dimensional In which all points are equal in distance from the radius

8.1.3.2. The equation of the circle is x² + y² = r²

8.1.3.2.1. Example: Determine an equation and a radius for a circle with a center of (0,0) and passes through points 9(6, -8). x² + y² = r² 6² + (-8)² = r² √100 = √r² 10 = r An eqation for the circle is x² + y² = 100, with the raidus of 10

9. Similar Figures

9.1. A Similar Figure is identical in shape but different in size

9.2. Similar Triangles

9.2.1. Similar triangles have the following properties: Corresponding angles are equal and ratios of corresponding sides are equal

10. Sum of angles in a triangle

11. Pythgoream Theorem

11.1. Example:

11.2. In a right angled triangle the square of the long side is equal to the sum of the squares of the other two sides. It is stated in this formula: a2 + b2 = c2

12. Forms of Quadratic Relations

12.1. Standard

12.1.1. The Standard Form of a Quadratic Equation looks like this: ax^2 + bx + c = 0 a, b and c are known values. a can't be 0. "x" is the variable or unknown

12.2. Vertex

12.2.1. Vertex forms looks like this y=ax(x-h)^2 + k (h,k) is the vertex a is the stretch or compression factor direction of opening is either up if a>0 or down if a<0 values x may take a set of real numbers {XER} Values y may take if a<0 then y ≤ k or if a>0 then y ≥ k

12.3. Factored

12.3.1. The form of an algebraic expression in which no part of the expression can be made simpler by pulling out a common factor.

12.3.1.1. Example: (x+2)(x-1)

13. Solve Problems involving right triangles

14. ALL TRIGONOMETRY

15. Negative and Zero exponents

15.1. Negative

15.1.1. When a base is raised to a negative exponent it is equal to the reciprocal of the base raise to the positive of the exponent.

15.1.2. For example: 2^ - 3 = 1/2^3

15.2. Zero

15.2.1. When the base is raised to an exponents of zero the answer is 1.

15.2.2. For example: 65^0 = 1

16. Graphing Quadratic Relations

16.1. Vertex Form

16.2. Factored Form

17. Quadratic Expressions

17.1. Factoring

17.1.1. Factoring a number means to break it up into numbers that can be multiplied together to get the original number. It also means the reverse of expanding.

17.1.1.1. Common Factoring is finding the greatest common factor of all the terms and dividing each term by it, writing the result inside the brackets. ALWAYS COMMON FACTOR FIRST IF POSSIBLE!

17.1.1.1.1. Example: Example: factor 2y+6 Both 2y and 6 have a common factor of 2: 2y is 2 × y 6 is 2 × 3 2y+6 = 2(y+3)

17.1.1.2. Factor By Grouping is to be preferably used when they are no common factors in all 4 terms. Grouping is grouping the first two terms together and the last two, and factoring out from each group.

17.1.1.2.1. EXAMPLE: Factor x3 + 2x2 + 8x + 16 = (x3 + 2x2) + (8x + 16) = x2 (x + 2)+ 8(x + 2) = (x + 2)( x2 + 8)

17.1.1.3. Factoring Complex Trinomials when a does not equal 1 is done by working in reverse replacing the middle term whose integer coefficients have a products of a X c and equal b

17.1.1.3.1. Example: Factor 3x^2 + 8x + 4 3x^2 + 8x + 4 = 3x^2 + 2x + 6x + 4 = (4x^2 + 2x) + (6x + 4) =x(3x + 2) + 2(3x + 2) =(3x + 2)(x+2)

17.1.1.4. Factoring Simple Trinomials when a = 1, is done by finding two integers whose product = c and whose sum is bx

17.1.1.4.1. Example: Factor x^2 - 4x _ 21 b= -4 c= -21 = (x+3)(x-7)

17.1.1.5. Perfect Squares Trinomials can be factored as a^2 + 2ab + b^2 = (a + b)^2. You can verify that that it is a prefect square if the first and last terms are perfect squares.

17.1.1.5.1. Example: Factor x^2 + 6x + 9 x^2 + 6x + 9 = (x)^2 + 2(x)(3) + 3^2 = (x + 3)^2

17.1.1.6. Difference of Squares can be factored as a^2 - b^2 = (a+b)(a-b)

17.1.1.6.1. Example: Factor x^2 - 100 = x^2 - 100 = (x)^2 - 10^2 = (x + 10) (x - 10)

18. Quadratic Equations

18.1. Completing the Square is the process for expressing y=ax^2 + bx + c in the form y =a(x -h)^2 + K

18.1.1. Example: Complete the square y=x^2 + 8x + 5 y=(x^2 + 8x)+ 5 =(X^2 + 8x + 4^2 -4^2) + 5 =(x^@ + 8X + 4^2) - 4^2 + 5 =(X + 4)^2 - 11

18.2. Maximize Revenue Problem

18.2.1. Matt runs a ski rental business that charges $12/snowboard and a mean of 36 rentals/day. He discovered that for each $0.50 decrease in cost, his business rents out two additional snowboards per day. At what cost can Matt maximize his revenue. Let R be the total revenue $ Let x be the number of 0.50 dollar decrease Equation: (12 - 0.5x)(36 + 2x) Complete the Sqaure R= (12 - 0.5x)(36 + 2x) = -x^2 + 6x + 432 = -1(x^2 -6x) + 432 = -1(x^2 -6x +(-3x)^2) + 432 = -1(x-3)^2 + 9 + 432 = -(x - 3)^2 +441 Reaches max value of 442 when x = 3 12 - 0.5(3) = $10.50 Matt should decrease rent costs to $10.50 for max revenue

18.3. Quadratic Formula

18.3.1. The quadratic formula is a method that is used to find the roots of a quadratic equation. The formula is: x = [-b ± √(b2 - 4ac)]/2a

18.3.1.1. Example: Solve 14x^2+9x+1=0 A= 14 B= 9 C = 1 Quadratic formula: -b +- √b^2-4(a)(c) -------------------------------- 2(a) So... -9 +- √25 --------------- 28 -9+√25 -9-√25 ----------- ------------- 28 28 -9+5 -9-5 -------- ----------- 28 28 -4 -14 -------- ------------ 28 28 So, the final answers are: -1/7 and -1/2