Patterns, Algebra and Functions

Janice made $40 during the first 5 hours she spent babysitting. She will continue to earn money at this rate until she finishes babysitting for 3 more hours. Find how much money Janice earned babysitting and how much she earns per hour

$8/hour. $64 total

Convert 1.4 meters to centimeters Convert 218 centimeters to meters

*write conversion factors and use proportions to convert to given units* 140 cm 2.18 m

Convert 4/5 to a decimal and to a percentage

0.8 and 80%

According to a survey, about 82% of engineers were highly satisfied with their job. If 145 engineers were surveyed, how many reported that they were highly satisfied?

118.9 --> 119

What is 14.5% of 96?

13.92

A child was given 100 mg of chocolate every two hours. How much chocolate will the child receive in five hours?

250 mg using proportions/ratios

Write each number in words: 29 478 (9,435) (98,542) (302,876)

29: twenty-nine 478: four hundred seventy-eight (9,435): nine thousand four hundred thirty-five (98,542): ninety-eight thousand five hundred forty-two (302,876): three hundred two thousand eight hundred seventy-six

Convert 3 (2/5) to a decimal and to a percentage

3.4 and 340%

Simplify the following: (1/2) + ( (3 3/4) - 2) +4

3/8

What is 150% of 20?

30

Solve 45% / 12% = 15% / x

4% (show work) *MAKE SURE TO CONVERT TO DECIMALS FROM PERCENTAGES*

Define polynomials and describe how to simplify polynomial expressions

A polynomial is a group of monomials added or subtracted together. Simplifying polynomials requires combining like terms. The like terms in a polynomial expression are those that have the same variable raised to the same power. It often helps to connect the like terms with arrows or lines in order to separate them from the monomials. Once you have determined the like terms, you can rearrange the polynomial by placing them. together. Remember to include the sign that is in front of each term. Once the like terms are placed together, you can apply each operation and simplify. When adding and subtracting polynomials, only add and subtract the coefficient, or the number part; the variable and exponent stay the same.

Describe functions and their properties

A quadratic function is a function in the form y = ax^2 + bx + c, where a does not equal 0. While a linear function forms a line, a quadratic function forms a parabola: a u-shape opening either upwards or downwards. If upwards, it's positive, and downwards is negative. The shape of the parabola depends ion the values of a, b, and c. All parabolas have a vertex which is either the maximum or minimum y-value (depending on if it is positive or negative). A quadratic function can have zero, one, or two solutions therefore 0, 1, or 2 x-intercepts. Recall that x-intercepts are referred to as the zeros or roots of a function. A quadratic function will have only one y-intercept. Understanding the basic components of a quadratic function can give you and idea of its graph

Discuss the characteristics of a quadratic function

A quadratic function is a polynomial function that follows the equation pattern f(x) = y = a*x^2 + b*x +c Where a, b, and c are real numbers and a is non-zero The domain and range are all reals, but the range is only those in the subset of the domain that satisfy the equation. The roots can be found through factoring or the quadratic equation.

Solve the following systems of equations: y = 2x + 7 y = -x +1

Can use either substitution (equation 1 = equation 2), subtracting/elimination (line up all the matching terms), or graphing (look for point of intersection) *SHOW WORK* (-2, 3)

Describe the shape and position of parabolas and how they change with the coefficients of a quadratic function

Changing a in the equation changes the direction the parabola faces (i.e. a = 1 or a = -1), and the width of its opening (the closer a is to 0, the wider the parabola will be). The values of b and c both affect the position of the parabola on the graph. The effect from changing b depends on the sign of a. If a is negative, decreasing b moves it to the left and increasing b moves it to the right. If a is positive, changes on b have an opposite effect. The value of c represents the y-intercept therefore changing c moves the parabola up and down the y-axis. The larger c is, the higher the parabola is on the graph

Demonstrate solving one variable linear equations

Example: a*x + b = 0, where a ≠ 0 Move b to the other side of the equal sign through and inverse operation. Then divide both sides by a

Explain exponential functions and logarithmic functions and the relationship between the two

Exponential functions are equations that have the format y = b^x, where base b>0 and b is non zero. Logarithmic functions are equations that have the format y = log_b (x). Base b may be anything except 1. 10 and e are the most common bases. See card for more info

How do you solve for x in the proportion 0.50 / 2 = 1.50 / x

First cross multiply to get 0.5*x = 2 * 1.5 Then you have 0.5*x = 4.5 Now divide by 0.5 on both sides such that x is left with a coefficient of 1. x = 6 Or if you notice any common factors/multiples, you can go off that

Explain the process for matrix addition and subtraction

For two matrices to be conformable for addition or subtraction, they must be of the same dimension; otherwise the operation is not defined. If matrix M is 3 x 2 and N is a 2 x 3 matrix, the operations M + N and M - N are meaningless. I f matrices M and N are the same size, the operation is as simple as adding or subtracting all of the corresponding elements. The result is a matrix of the same dimension as the two original matrices involved in the operation.

Discuss the requirements for and the results of matrix multiplication

For two matrices to be conformable for multiplication, they need not be of the same dimension, but specific dimensions must correspond. Taking the example of two matrices M and N to be multiplied M x N, matrix M must have the same number as columns are matrix N has rows. The resulting matrix will have the same amount of rows as M and the same number of columns as N. Square matrices of the same size are always conformable for multiplication and their product is always a matrix of the same size

Define greatest common factor (GCF) and least common multiple (LCM)

GCF: The largest number that evenly divides the numbers at hand LCM: The smallest product that the numbers in question have in common. The LCM is always greater than or equal to m*n where m and n are numbers in question

Explain how to find the transformation of an augmented matrix

I don't understand, look at the card or google

Discuss linear functions

In linear functions, the value of the function changes in direct proportion to x. The rate of change (aka slope) is constant throughout. Standard form is ax+by=c, where a, b, and c are real numbers. As an equation, it is commonly written y=mx+b

Find a quadratic equation whose real roots are x = 2 and x = -1

Multiply (x + 2) and (x - 1) to get it... x^2 - x - 2

Define the following common arithmetic terms specific to numbers: integers, prime, composite, even, and odd.

Numbers: basic blocks of mathematics. Specific features of numbers are identified by the following terms: Integers: The set of positive and negative numbers, including zero, not including fractions, decimals, or mixed numbers Prime Number: A whole number greater than 1 whose only two factors are one and itself. A number that is evenly divisible only by 1 and itself Composite Number: A whole number greater than 1 that has more than two different factors. Any whole number that is not a prime number. Even Number: Any integer that can be divided 2 with no remainder Odd Number: Any integer that cannot be divided evenly by 2 (i.e. 2n+1)

Graph the inequality 10 > -2x +4

Open circle. Need to solve the equation first to see how big/small x can be before -2x + 4 = 10 x > -3

Describe solving a system of linear equations graphically

Plot both equations on the same graph. The solution of the equation is the point where both lines cross each other. If the lines do not cross (i.e. they are parallel) then there is no solution

Describe rounding and how can it be useful

Rounding is reducing the digits in a number while still trying to keep the value similar. The result will be less accurate, but will be in a simpler form, and will be easier to use. Whole numbers can be rounded to the nearest ten, hundred or thousand

Explain how to use matrices to find the dilation of a planar figure

See card

Explain how to use matrices to find the reflection of a planar figure over each axis and over the line x = y

See card

Explain how to use matrices to find the rotation of a planar figure about the origin

See card

Explain how to find the sum, difference, product, or quotient of two functions

Sum: (f + g)(x) = f(x) + g(x) Difference: (f - g)(x) = f(x) - g(x) Product: (f * g)(x) = f(x) * g(x) Quotient: (f / g)(x) = f(x) / g(x)

Name the four geometric transformations and discuss how they relate to matrices

The four geometric transformations are translations, reflections, rotations, and dilations. When geometric transformations are expressed as matrices, the process of performing that transformation is simplified. For calculations of the geometric transformations of a planar figure, make a 2 x 2 matrix, where n is the number of vertices in the planar figure. Each column represents the rectangular coordinates of one vertex of the figure, with the top row containing the values of the x-coordinates and the bottom row containing the values of the y-coordinates. For example, given a planar triangular figure with coordinates (x_1, y_1), (x_2, y_2), and (x_3, y_3) the corresponding matrix is {x_1 x_2 x_3 ... y_1 y_2 y_3]. You can then perform the necessary transformations on this matrix to determine the coordinates of the resulting figure

Define main diagonal as it applies to a matrix

The main diagonal of a matrix is the set of elements on the diagonal from the top left to the bottom right of the matrix. Because of the way it is defined, only square matrices will have a main diagonal.

Explain graphing in the cartesian coordinate plane

The vertical line is the y-axis and the horizontal line is the x-axis. They function as number lines, where the intersection of the two axes is the zero point in which x=y=0. Cartesian coordinate points are written as (x,y) where the x and y values tell you how many steps to take in each direction.

Explain horizontal shift and vertical shift as they apply to the graphs of functions

These shifts happen when values are added to or subtracted from ALL of the x or y values. When a constant k is added/subtracted to/from the y-values, the graph shifts vertically. If k is added/subtracted to/from the x-values then the graph shifts horizontally.

Discuss matrix multiplication as a series of vector multiplications

To multiply larger matrices, treat each row from the first matrix and each column from the second matrix as individual vectors and follow the pattern for multiplying vectors. The scalar value found from multiplying the first row vector by the first column vector is placed in the first row, first column of the new matrix. The scalar value found from multiplying the second row vector by the first column vector is placed in the second row, first column of the new matrix. Continue this pattern until each row of the first matrix has been multiplied by each column of the second vector.

Explain the difference between variable that vary directly and those that vary inversely

Variables that vary directly are those that either both increase at the same rate or both decrease at the same rate. For example, in the functions f(x) = kx or f(x) = k*n^n, where k and n are positive, the value of x decrease Variables that vary inversely are those where one increase while the other decreases. For example, in the functions f(x) = k/x or f(x) = k/x^n where k is a positive constant, the value of y increases as the value of x decreases and the value of y decreases as the value of x increases. In both cases, k is the constant of variation

Multiply (2x^4)^2 * (xy)^4 * 4y^3 using the laws of exponents

Video Code: 532558 Laws of exponents states that when a power is raised to a power, you multiply the exponents. 16x^12 * y^7

Explain the process of dividing with decimals

Video Code: 560690 In a division problem, the divisor must always be a whole number. So, if the divisor is a decimal, make it a whole number by moving the decimal point accordingly. Then, divide as usual. With the quotient, move the decimal point the number of times and in the same direction as you did when you made the divisor a whole number. Another way is to move the decimal point of the dividend the same number of times and direction as you did the divisor. In this case, you would not have to move the decimal point after dividing. If the divisor is a whole number and the dividend is a decimal you do not have to adjust anything. Divide as usual

What is 30% of 120?

Video Code: 932623 36

Describe addition, subtraction, multiplication, and division with positive and negative numbers

When adding same signed numbers, combine their absolute values while maintaining their sign. When adding opposite signed numbers: let the larger addend be M and the smaller be N. Then |M|-|N|=P. Lastly, multiply P by either 1 or -1 such that the sign of P matches M's. When subtracting, multiply the number after the subtraction sign by -1, changing its sign, and change the original subtraction sign to an addition one. Now follow the procedures for addition. When multiplying, begin by multiplying the absolute values of the given numbers. Then, if the amount of original negative numbers is zero or positive, then the final product will positive. If the amount is odd, then the product is negative. The same logic applies for division.

Explain how to multiply two binomials

You can either use the F.O.I.L. method or the box method. F.O.I.L. stand for first, outer, inner, last. The box method is like Punit squares

Discuss rational functions

a function that can be constructed as a ratio of two polynomial expressions. ie f(x) = [g(x)] / [h(x)]. The domain is all reals except for the x-value in which h(x)=0 There will be vertical asymptotes where h(x) = 0. If the degree of the polynomial in the numerator is greater than the denominator's, then the x-axis will be a horizontal asymptote. If the degrees are equal, then the horizontal asymptote will not be y=0. If the numerator's degree is exactly 1 more than the denominator's, then the asymptote will be oblique The equation of the asymptote line is y = (p_n / q_(n-1)*x) + (p_(n-1) / q_(n-1))

Discuss square root functions

a function that contains a radical and is in the format f(x) = sqrt(a*x +b). The domain domain is all real numbers that yield a positive or zero radicand. The range is all real positive numbers including zero.

Discuss absolute value functions

in the format of f(x) = |a*x+b| The domain is all reals and the range is all positive reals. AKA piecewise function

Solve for x in the equation 40 / 8 = x / 24

x = 120 (show work...cross multiply. Isolate x, etc.)

A patient was given 100 mg of a certain medicine. The patient's dosage was a later decreased to 88 mg. What was the percentage decrease?

12%

Discuss four basic operations between matrices and scalars

Add the addend to every element in the matrix. Every element keeps the same position. Same logic applies to subtraction, multiplication and division M + 4 = [m_11 + 4 m_12 + 4 ...] M - 4 = [m_11 - 4 m_12 - 4 ...] M * 4 = [m_11 * 4 m_12 * 4 ...] M ÷ 4 = [m_11 ÷ 4 m_12 ÷ 4 ...]

Determine whether (-2,4) is a solution to the inequality y ≥ -2x + 3

Plug in x = -2 and y = 4. Carry out the operations to see if the inequality is true. No, false: 4 ≥ 7

Estimate the solution to 345,932 + 96,369

Start by rounding so there is only one non-zero digit. In this case, round to the nearest hundred thousand. So, 345,932 becomes 300,000 and 96,369 becomes 100,000. Then add the two rounded numbers: 300,000+100,000=400,000 The exact answer is 442,301. so the estimate is close to the estimate.

Demonstrate how to subtract 189 from 525 using regrouping

This some new math stuff using columns. Just look at the flashcard tbh

Order the following rational numbers from greatest to least: 0.3, 27%, sqrt(100), (72/9), (1/9), 4.5

Video Code: 419578 sqrt(100), (72/9), 4.5, 0.3, 27%, (1/9)

Use a model to represent the decimal 0.24 Write 0.24 as a fraction

Video Code: 449454 See card for explanation 24/100, 12/50, 6/25

Discuss how to determine if an equation is a function

Video Code: 822500 Substituting different values in for x (input values = domain). The output is the range. For a function to exist, every x-value must have its own unique y-value. Looking at a table of inputs and outputs, if the same input has more than one output, then it is not a function

Explain scientific notation

Video Code: 976454 It is a way of writing large numbers in a shorter way. The form (a * 10^n) is used in scientific notation, where (1<= a <10), and n is the number of places the decimal must move to get from the original number to a. n is positive if a is less than the original number n is negative if a is greater than the original number.

Describe rational, irrational, and real numbers

Video code: 461071 Rational Numbers: all integers, decimals, and fractions. Any terminating or repeating decimal number is a rational number Irrational Number: Cannot be written as fractions or decimals because the number of decimal places is infinite and there is no recurring pattern of digits within the number. For example, pi begins with 3.14159 and continues without terminating or repeating, so pi is an irrational number Real Numbers: the set of all rational and irrational numbers

Discuss roots and explain how they relate to exponents

Video code: 795655 and 114162 A root is another way of writing a fractional exponent. Instead of using a superscript, roots use the radical symbol to indicate the operation. A radical will have a number underneath the bar, and may have a number in the upper left indicating which root the number inside should be taken to. If no number is present, assume you are taking the square root (root 2). This means to are looking for how many times you need to multiple the root by itself to get the number inside the radical. A perfect square is a number that has an integer for it square root (ie 1, 4, 9, 16, 25, 36, 49, 64, 81, 100)

Simplify the following: 0.22 + 0.5 - (5.5 + 3.3 ÷ 3)

-5.88

Simplify the following: 1.45 + 1.5 + (6 - 9 ÷ 2) + 45

49.45

Discuss constant functions and, identity functions

Constant functions are given by the equation y=b or f(x)=b, where b is a real number. There is no independent variable present in the equation, so the function has a constant value for all x. The graph is a horizontal line with a slope of 0. Identity functions are identified by the equation y=x, where every value of y is equal to its corresponding x. The only zero point is the point (0,0). The graph is a diagonal line with slope = 1.

Discuss vector multiplication

The simplest type of matrix multiplication is a 1 x 2 matrix ( a row vector) times a 2 x 1 matric (a column vector). These will multiply such that m_11 * n_11 + m_12 * n_21 The two matrices are conformable for multiplication because matrix M has the same number of columns as N does rows. Because the other dimensions are both 1, the result is a scalar. Expanding our matrices to 1 x 3 and 3 x 1, the process is the same: m_11 * n_11 + m_12 * n_21 + m_13 * n_31 Once again, the result is a scalar. This type of basic matrix multiplication is the building block for multiplication of larger matrices.

Demonstrate how to subtract 477 from 620 using regrouping

This some new math stuff using columns. Just look at the flashcard tbh

Write 24.36% as a fraction and then as a decimal. Explain how you made these conversions

Video Code: 287297 2436 / 10000 = 609 / 2500 0.2436

A patient was given blood pressure medicine at a dosage of 2 grams. The patient's dosage was then decreased to 0.45 grams. By how much was the patient's dosage decreased?

*Practices lining up decimals* 1.55 grams

Explain how to solve systems of two linear equations by elimination. Solve using elimination: x + 6y = 15 3x - 12y = 18

*SHOW WORK* (9.6, 0.9)

Explain how to solve systems of two linear equations by substitution. Solve using substitution: x + 6y = 15 3x - 12y = 18

*SHOW WORK* (9.6, 0.9)

Solve the following system of equations: y = -2x + 2 y = -2x^2 + 4x + 2

*SHOW WORK* (0, 2) and (3, -4)

Find the distance and midpoint between the points (2, 4) and (8, 6)

*WRITE OUT THE FORMULAS AND SHOW WORK* Distance: SQRT(40) or 2sqrt(10) Midpoint: (5, 5)

Convert 42 inches to feet Convert 15 feet to yards

*write conversion factors and use proportions to convert to given units* 3.5 ft 5 yds

At a hotel, 3/4 of the 100 rooms are occupied today. Yesterday, 4/5 of the 100 rooms were occupied. On which day were more of the rooms occupied and by how much?

*write conversion factors and use proportions to convert to given units* Yesterday had the most rooms occupied (80 of 100) by 5.

Discuss equivalent units for converting between U.S. standard and metric equivalent of length

1 in = 2.54 cm 1 ft = 0.305 m (=12 in) 1 yd = 0.914 m (=3 ft) 1 mi = 1.609 km (=5280 ft)

Explain how to solve a quadratic equation by completing the square

1. If a ≠ 1, then divide whole equation by a 2. Make is so that the equation is in the form x^2 + bx = c 3. Add (b / 2)^2 to both sides 4. Factor one side of the equation into a perfect square: (x + b/2)^2 = c + (b/2)^2 5. Square root both sides to solve for x

List the steps used in solving y = 2x^2 + 8x + 4

1. sub 0 in for y in the quadratic equation: 0 = 2x^2 + 8x + 4 2. Try the x and box method (factor the quadratic equation. if a ≠ 1, list the factors of ac) 3. When all else fails, use the quadratic formula x = -2 + sqrt(2) and x = -2 - sqrt(2)

A patient was given 40 mg of a certain medicine. Later, the patient's dosage was increased to 45 mg. What was the percent increase in his medication?

12.5%

Simplify the following: (7/8) - (8/16)

3/8

Simplify the following: (3/2) + ( 4(0.5) - 0.75) + 2

4.75

A woman's age is thirteen more than half of 60. How old is she?

43 (show work)

At a school, for every 20 female students there are 15 male students. This same student ratio happens to exist at another school. If there are 100 female students at the second school, how many male students are there?

75

In a hospital emergency room, there are 4 nurses for every 12 patients. What is the ratio of nurses to patients? If the nurse-to-patient ratio remains constant, how many nurses must be present to care for 24 patients?

8 nurses

Define diagonal matrix, identity matrix, and zero matrix

A diagonal matrix is a square matrix that has a zero for every element in the matrix except the elements on the main diagonal. all the elements on the main diagonal must be nonzero numbers. (i.e. perfect R.R.E.F.) If every element on the main diagonal matrix is equal to one, the matrix is called an identity matrix. the identity matrix is often represented by the letter I. a zero matrix is a matrix that has zero as the value for every element in the matrix. The zero matrix is the identity for matrix addition. This should not be confused with the identity matrix

Explain how to transpose a matrix

A matrix N may be transposed to matrix N^T by changing all rows into columns and columns into rows. The easist way to accomplish this is to swap the positions of the row and column notations for each element. To quickly transpose a matrix by hand, begin with the first column and rewrite a new matrix with those same elements in the same order in the first row. Write the elements in the second column of the original matrix in the second row of the transposed matrix. Continue this process until all columns have been completed. If the original matrix is identical to the transposed matrix, the matrices are symmetric.

Define matrix and the associated terminology

A matrix is a rectangular array of numbers or variables, often called elements, which are arranged in columns and rows. A matrix is generally represented by a capital letter, with its elements represented by the corresponding lowercase letter with two subscripts indicating the row and column of the element. A matrix can be described in terms of the number of rows and columns it contains in the format a x b, where a is the number of rows and b is the number of columns. A vector is a matrix that has exactly one column (column vector) or exactly one row (row vector).

Explain how to use matrices to find the translation of a planar figure

A translation moves a figure along the x-axis, the y-axis, or both axes without changing the size or shape of the figure. To calculate the new coordinates of a planar figure following a translation, set up a matrix of the coordinates and a matrix of the translation values and add the two matrices ie. HV + XY where XY is a matrix of known coordinates (x's in row 1, y's in row 2) and HV is a matrix of the planar shift values (h's in row 1, v's in row 2)

Discuss weighted mean

A weighted mean, or weighted average, is a mean that uses "weighted" values. The formula is weighted mean = (w_1*x_1 + w_2*x_2 + w_3*x_3 + ... + w_n*x_n) / (w_1 + w+2 + w_3 + ... + w_n) Weighted values, such as w_1, w_2, w_3,..., w_n are assigned to each member of the set x_1,x_2, x_3,..., x_n. If calculating weighted mean, make sure a weight value for each member of the set is used

Discuss the Fundamental Theorem of Algebra, the Remainder Theorem, and the Factor Theorem as they apply to functions

According to the Fundamental Theorem of Algebra, every non-constant single variable polynomial has exactly as many roots as the polynomial's highest exponent. For example, if x^4 is the largest exponent of a term, the polynomial will have exactly 4 roots. However, some of these roots may have multiplicity or be non-real numbers. For instance, in the polynomial function f(x) = x^4 - 4x +3, the only real roots are 1 and -1. the root 1 has multiplicity of 2 and there is one non-real root (-1 - sqrt(2)*i) The Remainder Theorem is useful for determining the remainder when a polynomial is divided by a binomial. The Remainder Theorem states that if a polynomial function f(x) is divided by a binomial x-a where a is a real number, the remainder of the division will be the value of f(a). If f(a) = 0, then a is a root of the polynomial The Factor Theorem is related to the Remainder Theorem and states that if f(a) = 0 then (x-a) is a factor of the function

Describe the differences between algebraic functions and transcendental functions

Algebraic functions are those that exclusively use polynomials and roots. These would include polynomial functions, rational functions, square root functions, and all combinations of these functions, such as polynomials as the radicand. These combos may be joined by addition, subtraction, multiplication, or division but may not include variables as exponents Transcendental Functions a re functions that are non-algebraic. Any function that includes logarithms, trig functions, variables as exponents, or any combos that include any of these is not algebraic in nature, even if the function includes polynomials or roots

Discuss conformability for matrix operations

All four basic operations can be used with operations between matrices (although division is usually discard in favor of multiplication by the inverse), but there are restrictions on the situations in which they can be used. Matrices that meet all the qualifications for a given operation are called conformable matrices. However, conformability is specific to the operation; two matrices that are conformable for addition are not necessarily conformable for multiplication

Describe an augmented matrix and explain how it can be used to find the inverse of a matrix

An augmented matrix is formed by appending the entries from one matrix onto the end of another. Augmented matrices can be used to find the inverse of invertible matrices. I don't understand, so look at the card.

Explain one-to-one functions and the purpose of the horizontal line test

Each x value has exactly one value for y and each value of y has exactly one x value. The horizontal line test determines if a function is one-to-one. The function must still pass the vertical line test as well. If a function is one-to-one then it is also invertible

Describe how to solve a system of equations with a linear and quadratic equation with the substitution method.

Generally, the simplest way to solve a system of equations consisting of a linear equation and a quadratic equation algebraically is through the method of substitution. One possible strategy is to solve the linear equation for y and then substitute that expression into the quadratic equation. After expansion and combining like terms, this will result in a new quadratic equation for x which, like all quadratic equations, may have zero, one, or two solutions. Plugging each solution for x back into one of the original equations will then produce the corresponding value of y

Describe writing a function rule using a table

If a given data set, place the corresponding x and y-values into a table and analyze the relationship between them. Consider what you can do to each x-value to obtain the corresponding y-value using one or more of the mathematical operations (addition, subtraction, etc.) If an equation can be constructed that satisfies the pattern of the table, then the table contains a function.

Discuss the following properties of functions: argument, domain of definition, graph, zeros, roots, and intercepts

In functions with notation f(x), the x-value is the argument. The domain is the set of all values for x in a function. Unless said otherwise, assume the domain is the set of real numbers which will yield a range of the real numbers. This is the domain of definition. Thegraph of a function is the set of all ordered pairs (x, y) that satisfy the equation of the function. The points that have zero as the y-value are called the zeros of the function, aka x-intercepts. The points that have a 0 x values are the y-intercepts

Explain how to use matrix multiplication to find the solution to a system of equations

Matrices can be used to represent the coefficients of a system of linear equations and can be very useful in solving those systems. Matrix multiplication provides a simple one-step method for solving a system of any size. Take for instance three equations with three variables where all a, b, c, and d are known constants. To solve this system, define three matrices: A (all coefficients a, b, and c from equations), D (all values on the right of the equal sign), and X (a column vector of x, y, y, the solution to the problem) The three equations in our system can be fully represented by a single matrix equation: A*X = D We know that the identity matrix times X is equal to X, and we know that any matrix multiplied by its inverse is equal to the identity matrix. A^-1 * A * X = I*X; thus X = A^-1 *D Our goal then is to find the inverse of A, or A^-1. Once we have that, we can premultiply matrix D by A^-1 to find matrix X.

Explain monotone, even, and odd functions, and discontinuities in functions

Monotone function: a function whose graph either constantly increases or constantly decreases. Even Function: has a graph that is symmetric with respect to the y-axis and satisfies the equation f(x) = f(-x) Odd Function: has a graph that is symmetric with respect to the origin and satisfies the equation f(x) = -f(-x) Any time there are vertical assymptotes or holes in a graph, such that the complete graph cannot be drawn as one continuous line, a graph is said to have discontinuities. (i.e. hyperbolas, f(x) = tan x

Explain the correct Order of Operations, including a discussion of PEMDAS

Perform the operations in this order: parentheses, exponents, multiplication, division, addition, subtraction Order of Operations is a set of rules that dictates the order in which we must perform each operation in an expression that includes multiple different operations, O.o.O. tells us which operations to do first. Multiplication and division have equal precedence because they have an inverse relationship with one another, so work from left to right. Same logic applies to addition and subtraction.

Explain how to graphically solve a system of equations containing a linear equation and a quadratic equation.

Plot both equations on the same graph. The linear equation will of course produce a straight line, while the quadratic equation will produce a parabola. These two graphs will intersect either once, twice, or never; each point of intersection is a solution of the system

Explain stretch, compression, and reflection as they relate to the graphs of functions

Stretch, compression, and reflection are effects of manipulating parts of a fnuction. If the entire function is multiplied by a real number greater than 1, then the graph is stretched vertically. If the function is multiplied by a value k, 0<k<1, then the graph is compressed vertically. If k<0, then the graph is relfected about the x-axis in addition to being either stretched or compressed The same logic applies when f(k*x) except the effects are horizontal.

A sporting-goods store sells baseballs, volleyballs, and basketballs $3/Baseball, $8/Volleyball, and $15/Basketball Here are the same store's sales numbers for one weekend. Baseballs Volleyballs Basketballs Friday 5 4 4 Saturday 7 3 10 Sunday 4 3 6 Find the total sales for each day by multiplying matrices.

Take the table and matrix M and the prices as matrix N, where N is a 3 x 1 matrix. Multiply M x N to get $107 on Friday, $195 on Saturday and $126 on Sunday See card for more details

Explain how to find the composite of two functions

The composite of two functions is written as (f of g) (x), aka f (g(x)). The output of the second function, g(x), is the input of the first function, f(x). You can either plug in the x value into g(x), solve for x, then plug that value into f(x) and solve for that output OR You can plug in g(x) into the function of f(x) wherever there is an x and create one big composite function f(g(x))

Define determinant and explain how to find it for a 2 x 2 and a 3 x 3 matrix

The determinant of a matrix is a scalar value that is calculated by taking into account all the elements of a square matrix. A determinant only exists for square matrices. Finding the determinant of a 2 x 2 matrix is as simple as remembering a simple equation. For a 2 x 2 matrix M, the determinant is obtained by the equation |M| = (m_11 * m_22) - (m_12 * m_21). The shortcut for 3 x 3 matrices is: add the products of each unique set of elements diagonally left-to-right and subtract the products of each unique set of elements diagonally right-to-left. Calculating the determinant of a matrix larger than 3 x 3 is difficult to type, so google it.

Discuss the commutativity of matrix multiplication

The first thing it is necessary to understand about matrix multiplication is that it is not commutative. In scalar multiplication, the operation is commutative, meaning that a*b = b*a. For matrix multiplication, this is not the case: A x B ≠ B x A. The terminology must be specific when describing matrix multiplication. the operation A x B can be described as A multiplied (or postmultiplied) by B, or B premultiplied by A

Explain how to find the inverse of a 2 x 2 matrix and define nonsingular as it relates to matrix inverses

The inverse of a matrix M is the matrix that, when multiplied by matrix M, yields a product that is the identity matrix. Not all matrices have inverses. If the determinant of a matrix is 0, the matrix is said to be singular, and does not have an inverse. Only a square matrix whose determinant is not zero ( a nonsingular matrix) has an inverse. If a matrix has an inverse, that inverse in unique to that matrix. For any matrix M that has an inverse, the inverse is represented by the symbol M^(-1). To calculate the determinant of a 2 x 2 matrix, let the determinant of M be |M| and M = [m_11 m_12 ... m_21 m_22]. M^(-1) = (1/|M|) * [m_22 -m_12 ... -m_21 m_11]

Describe how to manipulate equations to find missing values

The most important thing to remember is that whatever operation you do to onbe side of the equal side, you must to to the other as well so the equation remains balanced. To find missing values, you must to the inverse operations necessary to "undo" all operations crowding / attached to the unknown value.

Define negative of a matrix and equal matrix

The negative of a matrix is also known as the additive inverse of a matrix. I f matrix N is the given matrix, then matrix -N is its negative. this means that every element n_ab is equal to -n_ab in the negative. To find the negative of a given matrix, change the sign of every element in the matrix and keep all elements in their original corresponding positions in the matrix. If two matrices have the same order and all corresponding elements in the two matrices are the same, then the two are equal matrices.

Discuss the quadratic formula

The quadratic formula is used to solve quadratic equations when other methods are more difficult. To use the quadratic formula to solve a quadratic equation, begin by rewriting the equation in standard form ax^2 + bx + c = 0, where a, b, and c are coefficients. Once you have identified the values of the coefficients, substitute those values into the quadratic formula (sing the song and write it out). Evaluate the equation and simplify the expression. Again, check each root by substituting into the original equation. In the quadratic formula, the portion of the formula under the radical (b^2 - 4ac) is called the discriminant. If the discriminant is positive, there are 2 different real roots. If it is negative, there are not real roots.

Describe solving quadratic equations with graphing. Explain the quadratic formula.

The solution(s) of a quadratic equation are the values of x when y = 0. On the graph, y = 0 is where the parabola crosses the x-axis, or the x-intercepts. This is also referred to as the roots, or the zeros of a function. Given a graph, you can locate the x-intercepts to find the solutions. If there are no x-intercepts, the function has no solution. If the parabola crosses the x-axis at one point, there is one solution and if it crosses at two points then there are two solutions. Since the solutions exist where y = 0, you can also solve the equation by substituting 0 in for y. Then try factoring the equation by finding factors of ac that add up to equal b. You can use the guess and check method, the box method, or grouping. Once you find a pair that works, write them as the product of two binomials and set them equal to zero. Finally, solve for x to find the solutions. See card 143 for quadratic formula

Discuss percentage problems and the process to be used for solving them

There are generally three types of percentage problems: 1. Find what percentage of some number another number is (What percentage of 40 is 8?) 2. Find what number is some percentage of a given number (What number is 20% of 40?) 3. Find what number another number is a given percentage of (What number is 8 20% of?) Solve them and write the steps/process

Review division of polynomials

To divide polynomials, begin by arranging the terms of each polynomial in order of one variable. You may arrange in ascending or descending order, but be consistent with both polynomials. To get the first term of the quotient, divide the first term of the dividend by the first term of the divisor. Multiply the first term of the quotient by the entire divisor and subtract the product from the dividend. Repeat for second and successive terms until you get a remainder of zero or a remainder whose degree is less then the degree of the the divisor. If the quotient has a remainder, write the answer as a mixed expression in the form: quotient + (remainder) / (divisor)

Explain how to solve a quadratic equation by factoring

Vide Code: 336566 To solve a quadratic equation by factoring, begin by rewriting the equation in standard form, if necessary. factor the side with the variable then set each of the factors equal to zero and solve the resulting linear equations. Check your answers by substituting the roots you found into the original equation. If, when writing the equation in standard form, you have an equation in the form x^2 + c = 0 or x^2 = -c, set x^2 = c or x^ = -c and take the square root of c. If c = 0, the only real root is 0. If c is positive, there are 2 real roots: the positive and negative square root values. If c is negative, then there are no real roots because you cannot take the square root of a negative number.

Explain roots in quadratic equations

Video Code: 198376 Use the x and box method The roots of a quadratic equation are the solutions when ax^2 + bx + c = 0. To find the roots of a quadratic equation, first replace y with 0. If 0 = x^2 + 6x -16, the find the values of x, you can factor the equation if possible. When factoring a quadratic equation where a = 1, find the factors of c that add up to b. That is the factors of -16 that add up to 6. the factors of -16 are -4 and 4, -8 and 2, and -2 and 8. Write these factors as the product of two binomials, 0 = (x - 2)(x + 8). Now solve each binomial for x.

Describe the graphs of quadratic functions

Video Code: 317436 A quadratic function has a parabola for its graph. If a is positive, the parabola faces upwards. If a is negative, the parabola faces downwards. The axis of symmetry is a vertical line that passes through the vertex. To check the number of intersections with the x-axis, check the real-roots

Discuss the characteristics of polynomial funstions

Video Code: 351038 A polynomial function is a function with multiple terms and multiple powers of x such as f(x) = a_n*x^n + a_(n-1)*x^(n-1) + a_(n-2)*x^(n-2)+...+ a_1*x + a_0 where n is a non-negative integer that is the highest exponent in the polynomial, and a_n is non-zero. The domain of a polynomial is the set of all real numbers. If the greatest exponent in the polynomial is even, the polynomial is said to be of even degree and the rang of the set of real numbers that satisfy the function. If the greatest exponent in the polynomial is odd, the polynomial is odd and the range is the set of all reals.

Explain rational expressions. Review the operations of rational expressions

Video Code: 351038 Rational expressions are fractions with polynomials in both the numerator and the denominator; the value of the polynomial in the denominator cannot equal zero. To add or subtract rational expressions, first find the common denominator, then rewrite each fraction as a n equivalent fraction with the common denominator. Finally, add or subtract the numerators to get the numerator of the answer, and keep the common denominator as the denominator of the answer. When multiplying rational expression factor each polynomial and cancel like factors (a factor which appears in both the numerator and the denominator). Then, multiply all remaining factors in the numerator to get the numerator of the product, and multiply the denominators to get the denominator of the product. Remember - cancel entire factors, not individual terms. To divide rational expression, take the reciprocal of the divisor )the rational expression you are dividing by) and multiply by the dividend

Define ratio and proportion and give examples

Video Code: 505355 A ratio is a comparison of two quantities in a particular order. A proportion is a relationship between two quantities that dictates how one changes when the other changes. A direct proportion describes a relationship in which a quantity increases by a set amount for every increase in the other quantity, or decrease by that same amount for every decrease in the other quantity. An inverse proportion is a relationship in which an increase in one quantity is accompanied by a decrease in the other, or vice versa.

Solve the following system of equations: x + y = 1 y = (x + 3)^2 - 2

Video Code: 658153 *SHOW WORK* (-1, 2) and (-6, 7)

Describe how to multiply using decimals

Video Code: 731574 First, multiply the numbers as if they were whole (i.e. don't have a decimal). Then, count how many spaces the decimal point had to move to make the numbers whole. Now, on your new product, move the decimal point the number of spaces found in the previous step.

Explain simplifying rational expressions, using (x - 1) / (1 - x^2) as an example

Video Code: 788868 Show work! Factor the numerator and denominator completely. Factors that are the same and appear in the numerator and denominator have a ratio of 1. The denominator, 1 - x^2, is a difference squares. It can be factored as (1 - x)(1 + x). the factor 1 - x and the numerator are opposites and have a ratio of -1. Rewrite the numerator as -1(1 - x). so the rational expression can be simplified as follows: -1 / (1 + x)

Explain reduced-row echelon forms of matrices

When a system of equations has a solution, finding the transformation of the augmented matrix will result in one of the three reduced row-echelon forms. Only one of these forms will give a unique solution to the system of equations, however. The following examples show the solutions indicated by particular results: diagonal of 1s gives the unique solution x = x_0; y = y_0; z = z_0 diagonal of 1s but 0 for the bottom right element gives a non-unique solution x = x_0 - k_1 * z; y = y_0 - k_2*z 1 in the 11 position but zeroes everywhere else gives a non-unique solution x = x_0 - j_1*y - k_1*z

Discuss the Rational Root Theorem as it applies to functions

any rational root of a polynomial function f(x) = a_n*x^n + a_(n-1) * x^(n-1) + a_(n-2) * x^(n-2) + ... + + a_1 * x + a_0 with integer coefficients will, when reduced to its lowest terms, be a positive or negative fraction such that the numerator is a factor of a_0 and the denominator is a factor of a_n. For instance, if the polynomial function f(x) = x^3 + 3x^2 - 4 has any rational roots, the nmuerators of those roots can only be factors of 4 (1, 2, 4), and the denominators can only be factors of 1 (a). The function in this examples has roots of 1 (or 1/1) and -2 (or -2/1)

Ray earns $10 an hour at his job. Write an equation for his earnings as a function of time spent working. How long does Ray have to work in order to earn $360?

f(x) = 10 * x, where x is the number of hours worked f(x) = 360 = 10 * x...inverse operation of 10* to isolate x... x = 36

Define constant of proportionality

When two quantities have a proportional relationship, there exists a constant of proportionality between the quantities; the product of this constant and one of the quantities is equal to the other quantity. AKA a constant rate ($3/ gallon, $10/pound, etc.) used to find total "cost"

Two weeks ago, 2/3 of the 60 customers at a skate shop were male. Last week, 3/6 of the 80 customers were male. During which week were there more male customers and by how much?

*practices writing conversion factors when finding number of males instead of the given ratios* Week 1: 40 males Week 2: 40 males Each week had the same number of males, so there is no difference

At a school, 40% of the teachers teach English. If 20 teachers teach English, how many teachers work at the school?

*write conversion factors and use proportions to convert to given units* 10% = 5 teachers --> 100% = 50 teachers

2 miles to kilometers 5 feet to centimeters

*write conversion factors and use proportions to convert to given units* 3.218 km 152.4 cm

15 kg to pounds 80 ounces to pounds

*write conversion factors and use proportions to convert to given units* 33 lbs 5 lbs

15.14 liters to gallons 8 quarts to liters

*write conversion factors and use proportions to convert to given units* 4 gallons 7.57 liters

13.2 lbs to grams 9 gallons to pints

*write conversion factors and use proportions to convert to given units* 6000 grams 72 pints

Write each decimal in words 0.06 0.6 6.0 0.009 0.113 0.901

0.06: six hundredths 0.6: six tenths 6.0: six 0.009: nine thousandths 0.113: one hundred thirteen thousandths 0.901: nine hundred one-thousandths

It takes Andy 10 minutes to read 6 pages of his book. He has already read 150 pages in his book that is 210 pages long. Find how long it takes Andy to read 1 page. Find how long it will take him to finish his book if he continues to read at the same speed.

1 minute 40 seconds per page 100 more minutes to finish book

Discuss equivalent units for converting between U.S. standard and metric equivalents of weight

1 ounce = 28.35 g 1 lb = 453.6 g (=16 ounces) 1 ton = 907.2 kg (=2,000 lbs)

Discuss equivalent units for converting between U.S. standard and metric equivalents of capacity

1 ounce = 29.573 ml 1 cup = 0.273 liters (=8 ounces) 1 pint = 0.473 liters (=16 ounces) 1 quart = 0.946 liters (=2 pints = 32 ounces) 1 gallon = 3.785 liters (=4 quarts = 8 pints = 128 ounces)

Discuss equivalent units for converting between U.S. standard and metric equivalents of fluids

1 tsp = 5 ml 1 tbsp = 3 tsp = 14.8 ml 2 tbsp = 1 fluid ounce = 30 ml 1 glass/cup = 8 fluid ounces = 240 ml

Round each number: 1. To the nearest ten: 11, 47, 118 2. To the nearest hundred: 78, 980, 248 3. To the nearest thousand: 302, 1274, 3756

1. 10, 50, 120 (anything ending in 5+ goes up, anything ending in 4- goes down) 2. 100, 1000, 200 (anything ending in 50+ up, 49- down) 3. 0, 1000, 4000 (500+ goes up, 499- goes down)

In a bank, the banker-to-customer ratio is 1:2. If seven bankers are on duty, how many customers are currently in the bank?

14

Simplify the following: (2/5) / (4/7)

14 / 20 = 7 / 10

Order the following rational numbers from least to greatest: 0.55, 17%, sqrt(25), (64/4), (25/50), 3

17%, (25/50), 0.55, 3, sqrt(5), (64/4)

A barista used 125 units of coffee grounds to make a liter of coffee. The barista later reduced the amount of coffee to 100 units. By what percentage was the amount of coffee grounds reduced?

20%

Convert 15% to both a fraction and a decimal

3/20 and 0.15

Simply the following: (1/4) + (3/6)

3/4

The McDonalds are taking a family road trip, driving 300 miles to their cabin. It took them 2 hours to drive the first 120 miles. They will drive at the same speed all the way to the cabin. Find the speed at which they are driving and how much longer it will take them.

60 mph. 3 hours left.

Explain fractions, numerators, and denominators

A fraction is a number that is express as one integer written above another integer, dividing line between them (x/y). It represents the quotient of the two numbers "x divided by y." It can also be thought of as x out of y equal parts The top number of the fraction is the numerator and is the number of parts under consideration/division. The bottom number of the fraction is the denominator and represents the total number of equal parts. The 4 in (1/4) means that the whole consists of 4 equal parts. a fraction cannot have a denominator of zero; this is referred to as "undefined"

Discuss functions, including domain, range, independent and dependent variables, and the vertical line test

A function is an equation that has exactly one output per input. The set of all values for the input/independent variable is the domain. The set of all output/dependent variables is the range. The vertical line test determines whether or not a graphed equation is a function. To pass this test, a vertical line parallel to the y-axis must scan horizontally across the x-axis without having more than one point touch it at a time. If the graphed equation passes, then it is a function.

Describe number lines and their use

A number line is a graph to see the distance between numbers. This shows the relationship between numbers. A number line may have a point for zero and may show negative numbers on the left side of the line. Any positive numbers go on the right side.

Name each point on the number line below (on card) *if you don't have the card, find a number line to practice with*

A: 0.25 B: 1.5 C: 2 D: 2.75

In a performance review, an employee received a score of 70 for efficiency and 90 for meeting project deadlines. Six months later, the employee received a score of 65 for efficiency and 96 for meeting project deadlines. What was the percent change for each score on the performance review?

About 7.1% for employee's efficiency About 6.7% for meeting project deadlines

Jane ate lunch at a local restaurant. She ordered a $4.99 appetizer, a $12.50 entrée, and a $1.25 soda. If she wants to tip her server 20%, how much money will she spend in total?

Add all items together, find 20% of that total, add to the first sum, then that will be final total: n = food cost. n*0.2=t. N = total cost = n+t OR N=n*1.2 (accounts for n and for t in the 1.2) N = $22.49

Discuss the terms as they relate to equations: one variable linear equation, root, solution set, empty set, and equivalent equations

Equation: States that two mathematical expressions are equal. One Variable Linear Equation: An equation written in the form ax+b=0, where a is non-zero Root: A solution to a one-variable equation; a number that makes the equation true when it is substituted for the variable Solution Set: The set of all solutions of an equation Empty Set: A situation in which an equation has no true solution Equivalent Equations: Equations with identical solution sets

Define the term "factor" and explain common and prime factors with examples

Factors are numbers that are multiplied together to obtain a product. A prime number has only two factors (1 and itself), but other numbers can have many factors A common factor is a number that divides exactly into two or more other numbers. (i.e. 3 and 5 are common factors of 15 and 30) A prime factor is also a prime number. Therefore the prime factors of 30 are 3 and 5

Explain how to factor a polynomial

First check for a common monomial factor. When the greatest common monomial factor has been factored out, look for patterns of special products: the difference of two squares for trinomial factors. If the factor is a trinomial but not a perfect trinomial square, look for a factorable form, such as x^2+ (a+b)x +ab = (x+a)(x+b) For factors with four terms, look for groups to factor. Once you have found the factors, write the original polynomial as the product of all the factors. Make sure all of the polynomials are prime. Monomial factors may be prime or composite. Check your work by multiplying the factors to make sure you get the original polynomial.

Demonstrate the use of a unit rate as the slope

For example, I set a schedule to do 16 flashcards per day. I then made a table using the function y=16*x where x is the numbered day I am on of my study streak and y is the number of cards I need to have done by the end of the day

Explain inequalities, conditional inequalities, and absolute inequalities

Inequality: A mathematical statement showing that two mathematical expressions are not equal. Inequalities use the > (greater than) and < (less than) symbols rather than the equal sign. Graphs of the solution set of inequalities are represented on a number line. Open circles are used to show that an equation approaches a number but is never equal to that number. Conditional inequality: An inequality that has a certain values for the variable that will make the condition true, and other values for the variable that will make the condition false Absolute inequality: An inequality that can have any real number as the value for the variable to make the condition true, and no real number value for the variable that will make the condition false. To solve an inequality, follow the same rules as solving an equation. However, when multiplying or dividing an inequality by a negative number, you must reverse the direction of the inequality sign Double Inequality: A situation in which two inequality statements apply to the same variable expression. When working with absolute values in inequalities, apply the following rules: |ax+b|<c then -c< ax+b < c |ax+b|>c then ax+b > c or ax+b < -c

Describe monomials and polynomials

Monomial: A single constant, variable, or product of constants and variables, such as2, x, 2x, 2/x, etc. There will never be addition or subtraction symbols in a monomial. Like monomials have like variables, but they may have different coefficients. The degree of a monomial is the sum of the exponents of the variables Polynomial: An algebraic expression which uses addition and subtraction to combine two or more monomials. Two terms make a binomial; three terms a trinomial. The degree of a Polynomial is the highest degree of any individual term

Explain how to solve one-variable linear equations

Multiply all terms by the lowest common denominator to eliminate any fractions. Look for addition or subtraction to undo so you can isolate the variable on one side of the equal side. Divide both sides by the coefficient of the variable. When you have a value for the variable, substitute this value into the original equation to make sure you have a true equation.

A patient was given pain medicine at a dosage of 0.22 grams. The patient's dosage was then increase to 0.80 grams. By how much was the patient's dosage increased?

Note key words/phrases that indicate the necessary operation. (See card for elaboration on this) 0.8-0.22= 0.58 grams

On Monday, Lucy spent 5 hours observing sales, 3 hours working on advertising, and 4 hours doing paperwork. On Tuesday, she spent 4 hours observing your sales, 6 hours working on advertising, and 2 hours doing paperwork. What was the percent change for time spent on each task between two days?

Observing Sales: 20% Advertising: 100% Paperwork: 50%

Discuss converting from smaller units to larger units and from larger units to smaller units of measurement

Smaller to larger units: divide the number of the known by the equivalent amount Larger to smaller: multiply the number of the known amount by the equivalent amount You can also set up conversion fractions. One fraction will be the conversion fraction (i.e. 100 cm/ 1 m) and the other will be the known (4 meters = 4m /1). The known unit will be in the numerator of the known fraction and denominator of the conversion factor. The desired unit will be in the numerator of the conversion factor. When multiplied, the known unit "n" will cancel out and you will be left with the desired unit.

Describe system of equations

System of Equations: A set of simultaneous equations that all use the same variables. A solution to a system of equations must be true for each equation in the system Consistent System: A system of equations that has at least one solution Inconsistent System: A system of equations that has no solution. Systems of equations may be solved using one of four methods: substitution, elimination, transformation of the augmented matrix and using the trace feature on a graphing calculator

Explain the decimal system and define the terms decimal, decimal point, and decimal place

The decimal system (aka base 10 system)is a number system that uses ten different digits (0-9). A nonexample is the binary system which only uses two digits. Decimal: a number that uses a decimal point to show the part of the number that is less than one (i.e. 1.2453) Decimal Point: a symbol used to separate the ones place from the tenths place in decimals or dollars from cents in currency Decimal Place: the position of a number to the right of the decimal point. (see card 4 for example)

Explain how to find the midpoint of two points and discuss distance between two points

To find the midpoint of two points: let a be the point with smaller coordinate values and b be the larger of the two. Find the difference by b-a and divide by 2. That is the midpoint of two points The distance between two points is the same as the length of the hypotenuse of a right triangle with the two given points as endpoints, and the two sides of the right triangle parallel to the x-axis and y-axis. The length of the segment parallel to the x-axis is the difference between the x-coordinates of the two points. The length of the segment parallel to the y-axis is the difference between the y-axis coordinates of the two points. Use the Pythagorean theorem a^2+b^2=c^2 or c=sqrt(a^2+b^2 to find the distance. The formula is: distance = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)

Discuss the use of x and y in functional relationships

Typical variables used in functional relationships. Written as (x, y) coordinates. The x-value is the independent variable and the y-value is the dependent variable. A relation is a set of data in which there is not a unique y-value for each x-value in the dataset. A relation is a relationship between the x and y-values in each coordinate but doesn't apply to the relationship between the values of x and yin the data set. A function is a relation where one quantity depends on the other. In a function, each x-value in the data set has one unique y-value because the y-value depends on the x-value

Explain how to use the trace feature of a graphing calculator to solve systems of equations

Using the trace feature on a calculator requires the you rewrite each equation, isolating the y-variable on one side of the equal sign. Enter both equations in the graphing calculator and plot the graphs simultaneously. Use the trace cursor to find where the two lines cross. Use the zoom feature if necessary to obtain more accurate results. Always check your answer by substituting into the original equations. The trace method is likely to be less accurate than other methods due to the resolution of graphing calculators, but is useful tool to provide an approximate answer.

Describe the equation of a line in standard form, slope-intercept form, point slope form, two-point form, and intercept form

Video Code: 113216 Standard Form: Ax+By=C; the slope is -A/B and the y-intercept is C/B Point-Slope Form: y -y_1 = m(x-x_1), where m is the slope and (x_1, y_1) is a point on the line Slope-Intercept Form: y=mx+b, where m is the slope and b is the y-intercept Two-Point Form: (y_2 - y_1)/(x_2 - x_1) where (x_1, y_1) and (x_2, y_2) are two points on the given line Intercept Form: x/x_1 + y/y_1 = 1, where (x_1, 0) is the point at which a line intersects the x-axis, and (0, y_1)is the point at which the same line intersects the y-axis

Explain the relationship between percentages, fractions, and decimals

Video Code: 141911, 262335, and 837268 Percentages can be thought of as fractions that are based on a whole of 100; one whole is equal to 100%. Fractions can be expressed as percentages by finding equivalent fractions with a denominator of 100. To express a percentage as a fraction, divide the percentage number by 100 and reduce the fraction to its simplest possible terms. To convert from a decimal point to a percentage, move the decimal point two places to the right. To convert from a percentage to a decimal, move it two places to the left. Remember that the percentage number will always be larger than the equivalent decimal number

Describe and discuss unit rate

Video Code: 185363 Unit rate expresses a quantity of one thing in terms of one unit of another. (i.e. 60 miles/ 1 hour) The denominator of a unit rate is always 1 They are used to compare different situations to solve problems. They can also help determine the length of time a given event will take.

Discuss improper fractions and mixed numbers

Video Code: 211077 Proper fractions have a denominator greater than its numerator. Improper fractions are fractions whose numerator's absolute value is greater than that of its denominator's. A mixed number is a number that contains both an integer and a fraction. An improper fraction can be written as a mixed number and vice versa.

Define absolute value and show that |3|=|-3| using a number line

Video Code: 314669 A precursor to working with negative numbers A number's absolute value is simply the distance away from zero a number is on the number line. It is always positive and written as |x|

Describe the process for adding, subtracting, multiplying, and dividing fractions

Video Code: 378080, 300874, and 638849 When adding and subtracting fractions, they must have the same denominators. This can be achieved through common multiples. When multiplying fractions, multiply the numerators by the numerators and the denominators by the denominators. When dividing fractions, multiply the dividend by the reciprocal of the divisor

Describe the process of adding and subtracting decimals

Video Code: 381101 When adding/subtracting decimals, you must make sure the decimals are lined up and each digit is lined up with its correspondent(s) in the same place value. One way to ensure this happens is by drawing columns, creating a visual grouping aid

Define the following terms: slope, horizontal, vertical, parallel, perdendicular

Video Code: 766664 Slope: A ratio of the change in height to the change in horizontal distance. If the value is positive, the slope points to the upper right corner of the graph and is upwards. If the slope is negative, then the slope is downward and points toward the bottom right of the graph. If the slope is 0, the two points have the same y value and the line is horizontal. If the points have the same x values, then the slope is undefined Horizontal: Having a slope of 0. On a graph, a line that has a constant distance from the x-axis. A flat line. Vertical: Having an undefined slope. On a graph, a line that has a constant distance from the y-axis. A straight line. Parallel: When 2+ lines have the same slope and never intersect Perpendicular: Two lines with opposite reciprocal slopes. When they intersect they form a 90 degree angle

Discuss the use of parentheses in operations

Video Code: 978600 Parentheses are used to designate which operations should be done first when there are multiple operations

A runner's heart beats 422 times over the course of 6 minutes. About how many times did the runner's heart beat during each minute?

Video code: 126243 Divide both numbers by 2: 211 and 3. Round 211 to the closest multiple of 3: 210 210 divided by 3 is 70. So about 70 bpm.

Write the place value of each digit in the following number: 14,059.826

Video code: 205433 1: ten-thousands 4: thousands 0: hundreds 5: tens 9: ones 8: tenths 2: hundredths 6: thousandths

List four basic mathematical operations and give examples of each

Video code: 521157 and 643326 addition: 3+2=5, the sum subtraction: 3-2=1, the difference multiplication: 3*2=6, the product division: 6/2=3, the quotient

Describe exponents and parentheses

Video code: 600998 and 947743 An exponent is a superscript number placed next to another number at the top right. It indicates how many times the base number is to be multiplied by itself. Exponents provide a shorthand way to write what would be a longer mathematical expression. A negative exponent is the same as the reciprocal of a positive exponent. Parentheses are used to designate which operation to do first when there are multiple operations in an expression.

List standard units for measurement in metric

gram, liter, meter are base units prefixes for base unit from biggest to smallest: tetra, giga, mega, kilo, hecto, deca, base unit, deci, centi, milli, micro, nano, pico Power of 10 to go from base unit to the prefix: 12, 9, 6, 3, 2, 1, 0, -1, -2, -3, -6, -9, -12 Letter preceding base unit abbreviation: T, G, M, K, H, D, *base unit*, d, c, m, "mew", n, p 1 metric ton = 1,000 kg = 1,000,000 g = 1 megagram