H math 2/3
Power Models
In the "Power Models" unit, students learn to recognize types of situations which can be represented by power models, inverse power models, and quadratic models, and to compare patterns found in the tables and graphs of models of the form y = ax2, y = ax3, y = a/x2, y = a/x3, and y = ax2 + bx + c. In addition, students begin to develop an understanding of rate of change in the context of these models. Some specific skills are developed here: solving quadratics by graphical methods and simplifying exponential and radical expressions. In addition, the vocabulary and concepts of direct and inverse variation are introduced. Key Ideas from Course 2, Unit 4 Power model: a function with equation of the form y = axn or a/xn. Direct variation power model: a function with equation y = axn (n > 0), for example, the relationship between volume of a cube and edge length is modeled by a direct variation function. We say that the volume varies directly with the cube of the edge length, because volume is given by the equation v = s3. Thus, as s3 increases, v increases with it. This is a cubic, or power 3, model. Inverse variation power model: a function with equation y = a/xn (n > 0). For example, the relationship between time and speed, for a fixed distance traveled, is modeled by an inverse variation function. We say that time varies inversely with speed, because the time is given by the equation t = d/s, where d is the fixed distance. Thus, as s increases, t decreases. For example, if the fixed distance is 300 miles then traveling at 20 mph will lead to a trip time of 15 hours. Increasing speed to 50 mph will cause the time to decrease to 6 hours. Quadratic model: a function with equation in the form y = ax2 + bx + c. The relationship between height (in feet) of a kicked ball and its time in flight (in seconds) is modeled reasonably well by a quadratic function. For example, if h = -16t2 + 50t + 3, the ball's height in feet after t seconds depends on the initial height (3 feet in this example), the initial velocity of the ball (50 ft/sec in this example), and the effect of gravity (indicated by the -16 ft/sec in this example). Radical or fractional power model: a function with equation in the form y = xn where n is a fraction greater than 0. For example, the relationship between the length of a side of a square and the area of the square is given by S = or . Direct variation power model graph: passes through the origin. If the power is odd, then the end behavior is in opposite directions, and the curve has rotational symmetry around the origin (the point where the x- and y-axes intersect). If the power is even, then the end behavior is in the same direction, and the curve has reflection symmetry across the y-axis. Inverse variation power model graph: does not cross the axes; the axes are asymptotes. As with direct variation power models, the odd powers are symmetric about the origin. The even powers are symmetric across the y-axis (a > 0). Quadratic function graph: is a parabola. It will have a maximum or a minimum value, and be symmetric about a vertical axis. The rate of change is not constant. Radical or fractional power model graph: resembles the ones below.
Linear Systems
Linear systems are used to model all kinds of real-world situations. (I have a geologist friend who uses computers to solve linear systems with hundreds of thousands of equations in hundreds of thousands of variables - all to try to figure out where earthquake-prone fault lines are.) Here is a much, much simpler real-world application of linear systems. It sometimes takes some thought to translate the problem from English language to mathematical symbols, but once you're done with that, it's straightforward. Example: You drive 150 miles to a national park. It takes you three hours. Part of the time you're on a freeway, where the speed limit is 70 mph. The rest of the time you're on smaller country roads, where the speed limit is 30 mph. Supposing you drove exactly at the speed limit the whole way, how much time did you spend on each type of road? The two unknowns, x and y, will stand for the amount of time spent on the freeway and the amount of time spent on the country roads. Since the whole trip takes three hours, we have the first equation: x + y = 3 Now, since rate × time = distance travelled, 70x is the distance travelled on the freeway and 30y is the distance travelled on the country roads. Adding these, we have: 70x + 30y = 150 We will solve the system by the substitution method. First, solve the first equation for y: y = 3 - x Now substitute 3 - x for y in the second equation: 70x + 30(3 - x) = 150 Simplify and solve for x. 70x + 90 - 30x = 150 40x = 60 x = 1.5 Substituting back in x + y = 3, we have y = 1.5 So you spend one and a half hours driving on each type of road.
Cosine
ratio of the adjacent side to the hypotenuse of a right-angled triangle
Sine
ratio of the opposite side to the hypotenuse of a right-angled triangle
Sine
sin
Tangent
tan
Diameter
the length of a straight line passing through the center of a circle and connecting two points on the circumference
Circumference
the length of the closed curve of a circle. C=3.14(diameter)
Normal Curve
the symmetrical bell-shaped curve that describes the distribution of many physical and psychological attributes. Most scores fall near the average, and fewer and fewer scores lie near the extremes.
Multiple Variable Models
. Solve for y: 6y - x + z = 4 Solution: Begin isolating y by adding x and subtracting z from both sides. 6y - x + z = 4 Original equation. + x - z = 4 + x - z Add (x - z) to ---------------------- both sides. 6y = 4 + x - z Divide each term by 6 6y 4 x z -- = - + - - - 6 6 6 6 2 x z y = - + - - - 3 6 6 As you can see above, this process doesn't do much good because you still have variables in the answer. However, when you have more than one equation with the same variables, you can use the process described above to solve for all the variables and get a constant for an answer. When you have two or more equations that call for the same solution, you have a system of equations. When solving systems of equations, always remember that if a = b, you can substitute b for a or a for b. Example 1. Solve for 3x + 2y = 3 and x = 3y - 10 Solution: Replace x in the first equation with its equivalent, (3y - 10) from the second equation. 3x + 2y = 3 Top equation. 3(3y - 10) + 2y = 3 Replaced x with (3y - 10). 9y - 30 + 2y = 3 Multiplied out. 11y = 33 Simplified. y = 3 Divide each side by 11 to get answer. Now that y has a value, you can plug that value in either equation and find a value for x. Because the second equation has already been solved for x, it will be easier to plug 3 in for y in that equation. x = 3(3) - 10 x = 9 - 10 x = -1 The solution is the ordered pair (-1,3).
Geometric Proof
A proof relating to segments and angles, using properties, postulates, definitions, and theorems
Sector
A sector is a region bounded by two radius and an arc lying between the radius.
Segment
A segment is a region bounded by a chord and an arc lying between the chord's endpoints.
Characteristics of a Quadratic Function
1. Standard form is y = ax2 + bx + c, where a≠ 0. 2. The graph is a parabola, a u-shaped figure. 3. The parabola will open upward or downward. 4. A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point. Click , to view Parabola that opens upward and Parabola that opens downward. 5. The domain of a quadratic function is all real numbers. 6. To determine the range of a quadratic function, ask yourself two questions: Is the vertex a minimum or maximum? What is the y-value of the vertex? If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value. If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value. 7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form x = n, where n is a real number. Click More Images to view Parabola that opens upward. Its axis of symmetry is the vertical line x =0. 8. The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots, solutions, and solution sets. Each quadratic function will have two, one, or no x-intercepts.
Transformations
A Transformation is a change in the position, size, or shape of a geometric figure. The given figure is called the preimage and the resulting figure is called the image. A transformation maps a figure onto its image. Prime notation is sometimes used to identify image points., transition, rotation, reflection.The other important Transformation is Resizing (also called dilation, contraction, compression, enlargement or even expansion). The shape becomes bigger or smaller:
Chords
A chord of a circle is a line segment whose two endpoints lie on the circle. The diameter, passing through the circle's center, is the largest chord in a circle.
Quadratic Functions
A function that can be written in the form f(x) = ax2+bx+c, where a,b,and c are real numbers and a ≠ 0
Cosecant
CSC
Properties of Circles
Chords, radius, diameter, tangent, secant, sector, segment, arc;
Cosine
Cos
Cotangent
Cot
Right Triangle Trigonometry
Discover the patterns. Don't worry if you're confused about what everything means right now and don't freak out about memorizing everything. It's not too hard if you know the patterns: The abbreviations are always used when writing out the trig functions. You will never write out cotangent or secant. When you see the abbreviation, you should hear the name. Likewise, when you hear the name, you should hear the abbreviation. Notice that in every case except csc (cosecant), the abbreviation is the first three letters of the name. Csc is an exception because the first three letters are "cos" which is already used. So instead it's the first three consonants. You can remember the first three ratios by the following: Sohcahtoa. Just think of it as a name to remember. Make him an Aztec Chieftain if it helps you remember it, just make sure you remember how to spell it. It's basically the first letter of "sin opposite hypotenuse, cos adjacent hypotenuse, tan opposite adjacent" Notice that if you insert the word over between any two words that aren't trig ratios (i.e. adjacent and hypotenuse, not cos and adjacent), after the name of each trig function is its ratio. The last three are just reciprocals of the first three (not inverses). Remember that anything without a prefix "co" has an reciprocal with the prefix, and anything with a prefix "co" has a reciprocal function without the prefix. Therefore, the csc, sec, and cot trig ratios are the reciprocals of the sin, cos, and tan ratios respectively. For example, cot's ratio is adjacent over opposite. Know the parts of the triangle. You probably know what the hypotenuse is at this point, but you might be a bit confused about the opposite and adjacent sides. Look at the following diagram: These sides are correct when you are using angle C. If you wanted to use angle A, the words opposite and adjacent would be flipped in the diagram. Understand what the trig ratios are and when they are used. When right angled trig was first discovered, it was realized that when you have two right triangles that are similar (meaning the angle measures are the same), if you divide one side by another and do the same with the corresponding sides of the other triangle, you would get the same values. The trig functions were then developed so that you could find the ratio for any given angle. The side names were also given to make it easier to determine which angles to use. You can use the trig ratios to determine a side measure given one of the sides and an angle, or you can use them to determine an angle measure given two side lengths. Figure out what you want to solve. Mark the unknown value with an "x." This will help you set up the equation later. Also make sure you have enough information to solve the triangle. You need either an angle and a side or all three sides. Set up the ratio. Label the opposite side, adjacent side, and hypotenuse with respect to the marked angle (it doesn't matter if the mark is a number or an "x" from the previous step). Then write down which sides you either know or want to find. Without considering csc, sec, or cot, determine which ratio involves both of the sides you wrote down. You shouldn't use the reciprocal ratios because there's usually no calculator button for them. Even if you could, there will almost never be a situation where you will need to use them to solve a right triangle. Once you know which ratio to use, write it down, followed by the value or variable of the triangle. Then write an equals sign followed by the sides the ratio encompasses (still in terms of opposite, adjacent, and hypotenuse). Rewrite the equation, filling in the side lengths/variable in the ratio. Solve the equation. If the variable is outside the trig function (that means you were solving for a side), then just solve for the exact value of x then plug the expression into your calculator for a decimal approximation of the side length. If your variable is inside the argument of the trig function (that means you were solving for an angle), then you should simplify the expression on the right then plug the inverse of that trig function followed by the expression. For example, if your equation was sin(x)=2/4, then you would simplify the right hand side to get 1/2, then punch into your calculator "sin-1" (it's all one button, usually the second option for the trig function you want) followed by 1/2. Make sure when you do calculations, you are in the correct mode. If you want degrees, put your calculator in degree mode, if you want radians, put your calculator in radian mode, if you don't know what degrees or radians, put your calculator in degree mode. The value of x is the value of the side or angle you were trying to find.
The Unit Circle
IS A CIRCLE CENTERED AT (0,0) AND HAS A RADIUS OF 1 UNIT DEFINED BY THE EQUATION: X² + Y² = 1
Matrix Models
Matrices A Matrix is an array of numbers: A Matrix A Matrix (This one has 2 Rows and 3 Columns) We talk about one matrix, or several matrices. There are many things you can do with them ... Adding To add two matrices, just add the numbers in the matching positions: Matrix Addition These are the calculations: 3+4=7 8+0=8 4+1=5 6-9=-3 The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size. Example: a matrix with 3 rows and 5 columns can be added to another matrix of 3 rows and 5 columns. But it could not be added to a matrix with 3 rows and 4 columns (the columns don't match in size) Negative The negative of a matrix is also simple: Matrix Negative These are the calculations: -(2)=-2 -(-4)=+4 -(7)=-7 -(10)=-10 Subtracting To subtract two matrices, just subtract the numbers in the matching positions: Matrix Subtraction These are the calculations: 3-4=-1 8-0=8 4-1=3 6-(-9)=15 Note: subtracting is actually defined as the addition of a negative matrix: A + (-B) Multiply by a Constant You can multiply a matrix by some value: Matrix Multiply Constant These are the calculations: 2×4=8 2×0=0 2×1=2 2×-9=-18 We call the constant a scalar, so officially this is called "scalar multiplication". Multiplying by Another Matrix To multiply two matrices together is actually a bit more difficult ... so I have whole page just for that called Multiplying Matrices. Dividing And what about division? Well you don't actually divide matrices, you do it this way: A/B = A × (1/B) = A × B-1 where B-1 means the "inverse" of B. So you don't divide, instead you multiply by an inverse. And there are special ways to find the Inverse ... ... learn more about the Inverse of a Matrix. Transposing To "transpose" a matrix, just swap the rows and columns. We put a "T" in the top right-hand corner to mean transpose: Matrix Transpose Notation A matrix is usually shown by a capital letter (such as A, or B) Each entry (or "element") is shown by a lower case letter with a "subscript" of row,column: Matrix Notation column Rows and Columns So which is the row and which is the column? Rows go left-right Columns go up-down You can also remember that rows come before columns by the word "arc": ar,c Example: B = A Matrix Here are some sample entries: b1,1 = 6 (the entry at row 1, column 1 is 6) b1,3 = 24 (the entry at row 1, column 3 is 24) b2,3 = 8 (the entry at row 2, column 3 is 8)
Patterns of Association
Research further
Periodic Models of Change
Research further
Secant
Sec
Linear Programming
The process of finding the maximum or minimum values of a function for a region defined by inequalities
Families of Functions
There are three main categories that we can group functions. These three functions include linear, quadratic, and absolute value functions.
Arc
an arc of a circle is any connected part of the circle's circumference.
Patterns in Chance
Waiting-time distribution (also known as a geometric distribution): Occurs in situations in which someone is watching a sequence of independent trials and waiting for a certain event to occur. For example, the trials could be a person trying to shoot baskets and waiting for success. The shooter could be successful on the first try, or the shooter might have to wait for 10 shots for success to happen. The observer records the frequency with which the event occurred on the first trial, second trial, third trial, etc., in a frequency table. (See Course 1 Unit 1 for basic work with frequency tables.) Number of Trials Needed Frequency 1 2 3 etc. Independent trials: The probability of a success on each trial is unaffected by the outcome of a previous trial. For example, the basketball shooter has the same chance of success on each attempt, no matter how the last attempt turned out. Graph of a waiting-time distribution: Has a characteristic shape. For example, suppose the probability of success is 40% on the first trial, then 40% of the time you will be successful on the first trial. It will take 2 trials only if the first observation was a failure and the second was a success. The probability of fail then success is (0.60)(0.40) = 0.24. The graph will look as follows: Formula for the probabilities in a waiting-time distribution: P(x) = (1 - p)x - 1(p), where x is the number of attempts that had to be made to get a success and p is the probability of success on any one attempt. For the above example, where the probability of a basket is 40%, the probability of having to wait through 5 attempts would be P(5) = (0.6)4(0.4). Multiplication Rule: States that the probability of (A and B), where A and B are independent events, is P(A) times P(B). For example, since the probability of having to wait through 5 attempts in the basketball problem would be the same as the probability of "no and no, and no, and no, and yes," which is P(no)4P(yes) = (0.6)4(0.4). Average wait time, or expected value: The mean of the distribution or ∑ xP(x). Since the waiting-time distribution is infinite, this is an infinite series. Using the algebra of summing series, we find that the average wait time is 1/p.
Parabola
a U shaped line on a graph, the shape of the graph of a quadratic function
Standard Deviation
a measure of variability that describes an average distance of every score from the mean
Secant
a secant is an extended chord; a straight line cutting the circle at two points
Algebraic Proof
a set of logical algebraic statements used to prove an equation or expression
Radius
a straight line from the center to the perimeter of a circle (or from the center to the surface of a sphere) Two make up the diameter
Tangent
a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point, ratio of the opposite to the adjacent side of a right-angled triangle
