Cha. 16 Algebra and Functions Notes

Rational Equations and Inequalities.

Rational Equations and Inequalities are a rational algebraic expression is the quotient of two expressions. Example 3y + 5 / y - 2. You may be asked to solve equations or inequalities involving such expressions. For example: For what value of x if the following equation true? 3 = x - 1 / 2x + 3 Multiplying both sides by 2x + 3 gives: x = -2

Rules of Exponents

Rules of Exponents examples: x^0=1 b² = b x b p^-2 = 1/p x 1/p = 1/p^2 x^a/b = b√x^a = (b√x)^a y^1/2 = √y

Systems of Linear Equations and Inequalities

Systems of Linear Equations and Inequalities include: Two or more linear equations using the same variables. Two equations with two variables (x and y); the solution will always be an ordered pair that MUST be true for both equations. You may be asked to solve systems of two or more linear equations or inequalities. You can solve this system of equations by eliminating one of the variables.

Solving Equations Involving Radical Expressions

When Solving Equations Involving Radical Expressions, you use, PEMDAS/ both sides of the equal sign. Combine like terms inside the brackets. The equation 5√x + 14 = 29 is a radical equation because it involves a radical expressions. You can solve this equation: x = 9

Solving Quadratic Equations by Factoring

When Solving Quadratic Equations by Factoring, you may be asked to solve quadratic equations that can be factored. ( You will not be expected to know the quadratic formula). Pull all terns on one side of the equal sign, leaving zero on the other side. Move all terms to the L side, combine, and put in ax^2 + bx + c = 0 form.

Solving or One variable in Terms of Another

When solving or One variable in Terms of Another, If 3x + y = z, what is x in terms of y and z? Subtract y from each side of the equation. Divide both sides by 3 to get x. Then x in terms of y and z is z-y/3

Working with "Unsolvable" Equations

When working with "Unsolvable" Equations, at first, some of the equations may look like they can't be solved. You will find that although you can't solve the equation, you can answer the question. Example: If a + b = 5, what is the value of 2a + 2b? The question doesn't ask for the value of a or b. It asks you for the value of the entire quantity 2a + 2b. 2a + 2b can be factored. You are asked what 2 times a + b is. That's 2(a + b) = 2 x 5 = 10

Absolute Value

Absolute Value is the distance a number is from zero on a number line. ALWAYS POSITIVE. , A number's distance from zero on a number line. It is written |a| and is read "the absolute value of a." It results in a number greater than or equal to zero (ex: |4| = 4 and |-4| = 4).

Algebraic Expressions

Algebraic Expressions are expressions consisting of one or more numbers and variable along with one or more arithmetic operations. Example: 4x + 5x = 9x.

Inequalities

An inequality is a mathematical sentence involving <, >, or = plus < or > Cannot multiply or divide an inequality by a variable, unless you kno wthe sign of the number that the variable stands for. Example" 2x + 1 > 11 2x > 11 - 1 2x > 10 x > 5

Translate as You Read

As you read word problems, Translate the Words into mathematical expressions and equations. When you have finished reading the word problem, you will have already translated it into mathematical expressions and equations.

Difference of Two squares

Difference of Two squares examples: a² - b² = (a + b)(a - b)

Direct Translation into Mathematical Expressions

Direct Translation into Mathematical Expressions involve translating sentences of math into an equation. Many word problems require you to translate the verbal description of a mathematical fact or relationship into mathematical terms. Example: "3 times the quantity 4x + 6" translates to 3(4x + 6)

Evaluating Expressions with Exponents and Roots

Evaluating Expressions with Exponents and Roots: Replacing variables with numbers and preforming the math operations to calculate a value, replacing variables with numbers and preforming the math operations to calculate a value, Evaluate −n + 8.9 for n = −2.3. , Insert the value(s) given for the unknown(s) and do the arithmetic, making sure to follow the rules for the order of operations. Examples: y^2/3 = ^3√8^2 = ^3√64 = 4.

Exponent Main points

Exponent Main points: When multiplying expressions with the same base, add the exponents This rule also holds for the exponents that are not positive integers. When dividing expressions with the same base, subtract exponents When a number raised to an exponent is raised to a second exponent, multiply the exponents.

Exponents

Exponents are the number that is small and raised to show how many times to multiply the number by itself. Example: b² = b x b

Factoring

Factoring expresses a polynomial as the product of monomials and polynomials. Types include: Difference of two squares, finding common factors, and factoring quadratics.

Functions as models

Functions can be used as models of real-life situations. Here is an example: The Temperature in City X is W(t) degrees Fahrenheit t hours after sundown at 5:00 p.m. The function W (t) is given by: W(t) = 0.1 (400 - 40t + t^2) for 0 ≤ t ≤ 12 You can draw a graph for this function.

Qualitative Behavior of Graphs and Functions

In Qualitative Behavior of Graphs and Functions, you will need to understand how the properties of a function and its graph are related. For example, the zeros of the function f are given by the points where the graph f(x) in the xy- plane intersects the x-axis.

Direct and Inverse Variation

In direct variation, y=kx, where k is a nonzero constant. In direct variation, the variable y changes directly as x does. Direct Variation- y=kx, where k is constant. As x gets larger, y gets larger. If the number of units of B were to double, the number of units of A would double

Linear Functions: Their Equations and Graphs

Linear Functions: Their Equations and Graphs: You may be asked to answer questions involving linear equations and their graphs. Therefore, you will need to understand the concepts of slope and intercepts: y= mx+b, where m and b are constants, is a linear function and the graph of y= mx+b in the xy- plane is a one with a slope m and a y- intercept b.

Translations

Translations: In mathematics, the movement of a point, line, or shape along a straight line. Translations may be combined with rotations and reflections. A transformation that preserves the size, shape, and orientation of a figure while sliding (moving) to a new location. You may be asked questions of the effects of simple translations on the graph of a function.

Using New Definitions

When you Use New Definitions, please remember: For Functions, especially those involving more than one variable, a special symbol is sometimes introduced and defined. The key to these questions is to make sure that you read the definition carefully. Some questions will ask you to apply the definitions of the symbol to more complicated situations You may be asked to compare two values, each of which required the use of the symbol. You may be asked to evaluate am expression that involves multiplying, dividing, adding, acquiring or subtracting terms that involve the symbol. You may be asked to solve an equation that involves the use of the symbol.

Word Problem Steps

Word Problem Steps: Read and Interpret what is being asked Determine what information you are given Determine what information you need to know Decide what mathematical skills or formula you need to apply to find the answer Work out the answer Double-check to make sure the answer makes sense. When checking word problems, don't substitute your answer into your equations, because they may be wrong. Instead check word problems by checking your answer with the original words

Word Problems

Word Problems are verbal descriptions of mathematical situations that children solve by applying their math skills. Cindy's is preparing for a party. If her recipe calls for 5 pounds of ground beef and 10 pounds of lasagna noodles to serve 20, how many pounds will she need to feed 200 people? This is ratio and proportion problem. Set up proportions. Then, cross multiply. 5/20= x/200. 20x=1000. x=50 pounds. Note: these types of problems may have not always whole number.

Quadratic Functions: Their equations and Graphs

You may be asked to answer questions involving quadratic equations and their graphs. You will beed to identify some of the basic features of the graph of a quadratic equation, such as its highest or lowest point, its zeros and its direction.