The Slope of a Line (2.2) Find the slope of a line given two points on the line (Objective #1) A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal distance (run). To find the slope of a line given two points on a line use the following formula: (Note: ) EXAMPLE #1: (2 4) (5 1) EXAMPLE #2: 1 2 1 (4 1) Find the slope of a line given an equation of the line (Objective #2) To find the slope of a line given an equation of the line solve the equation for and arrange the terms in the follow form: EXAMPLE #3: x y 6 EXAMPLE #4: 2x 3y 4 Graph a line given its slope and a point on the line (Objective #3) To graph a line given its slope and a point on the line begin by plotting the point on a Cartesian plane. From the point apply the rise and go UP if or go DOWN if then run and go RIGHT if or go LEFT if plot the second point and connect the two points to form a solid line. EXAMPLE #5: Graph. Given slope and point ( ) (refer to figure 1) Use slopes to determine whether two lines are parallel, perpendicular or neither (Objective #4) Apply Objectives #1 and #2. Figure 1
(Objective #1) Find slope given two points. ( )( ) 2.2 The Slope of a Line (Objective #1) Find slope given two slopes. ( ) ( ) (Objective #2) Find the slope given an equation. (Objective #2) Find the slope given an equation. (Objective #3) Graph a line given slope and point. ( ) (Objective #3) Graph a line given slope and point. ( ) (Objective #4) Parallel, perpendicular, or either. (Objective #4) Parallel, perpendicular, or either.
Writing Equations of Lines (2.3) Write an equation of a line given its slope and y-intercept (Objective #1) To write an equation of a line given its slope and y-intercept, apply the following formula and replace slope or m and y-intercept or b with the given values: (Note: b = 2 nd value in a pair when the 1 st value is 0. Only apply the formula if m is defined) EXAMPLE #1: Given slope = m = 2, y int = (0, 7) Graph a line using its slope and y-intercept (Objective #2) To graph a line given its slope and a point on the line, begin by plotting the point on a Cartesian plane. From the given point, apply the rise and go UP if + or go DOWN if then run and go RIGHT if + or go LEFT if plot the second point and connect the two points to form a solid line. EXAMPLE #2: Graph. Given slope = m = 1 and point (0, 2) 3 Write an equation of a line given its slope and a point on the line (Objective #3) To write an equation of a line given its slope and a point on the line, apply the following formula and replace slope or m and the given point (x, y) with the given values and solve the equation for b or y- intercept. Then write the formula again but replace the values for just m and b not x and y. (Note: A different formula can be used to write the equation of a line: y y 1 = m(x x 1 ) ) EXAMPLE #3: Given slope = m = 1 2, Point = ( 2, 7) Write an equation of a line given two points on the line (Objective #4) To write an equation of a line given two points, calculate the slope using the following formula and apply Objective #3 using only one of the given points for x and y. (Note: x 1 x 2 ) EXAMPLE #4: Given two points: ( 2, 9) and (5, 4)
Write equations of horizontal and vertical lines (Objective #5) To write an equation of a horizontal line, use the following formula: y = y value of given point. To write an equation of a vertical line use the following formula: x = x value of given point. (Note: m = 0 is a horizontal line; m = undefined is a vertical line) EXAMPLE #5: a. b. Write an equation of a line parallel or perpendicular to a given line (Objective #6) To write an equation of a line parallel or perpendicular to a given line, solve the given equation for y and arrange the terms in the following form: y = mx + b. Then determine the slope of the line using the given equation and apply the following: To write the equation of the line, apply Objective #3 and replace the m and the given point (x, y) the line passes through. EXAMPLE #6:
2.3 Writing Equations of Lines (Objective #1) Write equation given slope and y-intercept. (Objective #2) Graph a line using slope and y-intercept. (Objective #3) Write equation given slope and point. (Objective #4) Write equation given two points. (Objective #5) Write equation of horizontal line. (Objective #5) Write equation of vertical line. (Objective #6) Write equation of a line parallel or perpendicular. (Objective #6) Write equation of a line parallel or perpendicular.
Introduction to Relations and Functions (2.5) Define and identify relations and functions (Objective #1) A function is a relation or a set of ordered pairs in which every 1 st component or input corresponds to exactly ONE 2 nd component or output. EXAMPLE #1: YES NO If the set of ordered pairs is a function, the ordered pair can be symbolize using ( ) or ( ) where means function but if there is more than one function any other letters in the alphabet can be used such as ( ) ( ). Find the domain and range (Objective #2) The domain is the set of first component or input in each pair. The range is the set of second component or output in each pair. (Note: Never repeat a value when writing the domain and range) EXAMPLE #2: Domain: Range: EXAMPLE #3: Identify functions defined by graphs and equations (Objective #3) The vertical line test can be used to identify if a relation is a function, if the vertical line crosses the graph at any point more than once the relation is not a function. EXAMPLE #4: A. B. An equation is a function only if for every input or x in the domain there is exactly one corresponding output or y in the range. Solve for to determine if there is exactly one for every EXAMPLE #5: A. B.
Basics of Functions and Their Graphs (2.5) (Objective #1) Is the following relation a function? Circle one and explain. (Objective #1) Is the following relation a function? Circle one and explain. Yes *( ) ( ) ( ) ( )+ No Yes *( ) ( ) ( ) ( )+ No (Objective #2) (Objective #2) *( ) ( ) ( ) ( )+ Domain: Domain: Range: Range: (Objective #1 & #2) (Objective #2) Domain: Domain: Range: Range: (Objective #3) Is the following graph a function? Circle one and explain. Yes No (Objective #3) Is the following equation a function? Circle one and explain. Yes No
Systems of Linear Equations in Two Variables (3.1) Decide whether an ordered pair is a solution of a linear system (Objective #1) To decide whether an ordered pair is a solution of a linear system, replace the given ordered pair for and in ALL the given equations. If ALL equations are TRUE then the given point is a solution. If ONE equation is FALSE then the given point is NOT the solution to the linear system. EXAMPLE #1: A system of linear equations and their solutions can be classified as follows: Solve linear systems (with two equations and two variables) by substitution (Objective #2) To solve a system of linear equations using substitution follow the steps below: STEPS: 1. Solve one equation for or. 2. To see if the lines cross, replace or substitute the expression for or in STEP 1 in the other equation and simplify. 3. Solve for the remaining variable. The lines will cross at this particular or. If then the lines DO NOT cross hence the system is inconsistent or has NO SOLUTION. If then the lines are the same hence the system is consistent dependent or has INFINITELY MANY SOLUTIONS. 4. If the lines cross the and values are the same so replace the value in STEP 3 into one or other equation and solve for the missing corresponding pair. EXAMPLE #2: Solve linear systems (with two equations and two variables) by elimination (Objective #3) To solve a system of linear equations using elimination follow the steps below: STEPS: 1. Line up the LIKE terms. 2. Add in order to eliminate at least one variable. If at least one variable DOES NOT eliminate, multiply one equation or both equations by a number so that either the terms or the terms have opposite numbers or opposite coefficients. (Note: Find the lowest common multiple between the two numbers or coefficients of or in order to know what to multiply the equation(s) by) 3. Solve for the remaining variable. The lines will cross at this particular or. If then the lines DO NOT cross hence the system is inconsistent or has NO SOLUTION. If then the lines are the same hence the system is consistent dependent or has INFINITELY MANY SOLUTIONS. 4. If the lines cross the and values are the same so replace the value in STEP 3 into one or other equation and solve for the missing corresponding pair. EXAMPLE #3:
3.1 Systems of Equations in Two Variables (Objective #1) Is the given point a solution to the system of equations? (Objective #1) Is the given point a solution to the system of equations? (Objective #2) Solve the system of equations by substitution. (Objective #2) Solve the system of equations by substitution. (Objective #3) Solve the system of equations by elimination. (Objective #3) Solve the system of equations by elimination.
Integer Exponents and Scientific Notation (4.1) Use the product rule for exponents (Objective #1) To multiply expressions with common bases ADD the exponents and keep the base the same. EXAMPLE #1: Define 0 and negative exponents (Objective #2) EXAMPLE #2: d. Use the quotient rule for exponents (Objective #3) To divide expressions with common bases SUBTRACT the exponents and keep the base the same. EXAMPLE #3: Use the power rule for exponents (Objective #4) Power rule states to multiply exponents, if there is an exponent outside of a parenthesis as follows: EXAMPLE #4: EXAMPLE #5: Simplify exponential expressions (Objective #5) Rearrange the negative exponents to make them positive and get rid of parenthesis by multiplying the exponents, then add or subtract the exponents if the bases are common. No negative exponents in the final answer.
4.1 Integer Exponents and Scientific Notation (Objective #1) Use the product rule for exponents. (Objective #2) Define 0 and negative exponents. (Objective #3) Use the quotient rule for exponents. (Objective #3) Use the quotient rule for exponents. (Objective #4) Use the power rule for exponents. (Objective #4) ) Use the power rule for exponents. (Objective #5) Simplify exponential expressions. (Objective #5) Simplify exponential expressions.
Adding and Subtracting Polynomials (4.2) Know the basic definitions for polynomials (Objective #1) A polynomial is a finite sum of terms where the exponents on the variables are whole numbers (0,1,2,3 ) and can be written in descending (high to low exponents) order as follows: EXAMPLE #1: Add and Subtract Polynomials (Objective #2) To add polynomials combine the coefficients and keep the base and exponent the same on each like term as follows: To subtract polynomials distribute the negative in front of a parenthesis then combine the coefficients and keep the base and exponent the same on each like term as follows: EXAMPLE #2: a. b.
4.2 Adding and Subtracting Polynomials (Objective #1) Which of the following is a polynomial? (Objective #1) Identify polynomial as a monomial, binomial, trinomial. (Objective #3) Add. (Objective #3) Add. (Objective #4) Add. (Objective #4) ) Subtract. (Objective #5) Subtract. (Objective #5) Subtract.
Multiplying Polynomials (4.4) Multiply terms (Objective #1) A term in a polynomial is a product and can be written as follows: a. To multiply two or more terms, multiply the coefficient, add the exponents, and keep the base and exponent the same. EXAMPLE #1: Multiply any two polynomials (Objective #2) To multiply any two polynomials apply the distributive property and multiply each term inside the 1 st parenthesis by each term inside the 2 nd parenthesis. EXAMPLE #2: a. b. EXAMPLE #3: Multiply binomials (Objective #3) Use the F.O.I.L method to multiply binomials Find the product of a sum and difference of two terms (Objective #4) EXAMPLE #4: EXAMPLE #5: Find the square of a binomial (Objective #5) EXAMPLE #6:
4.4 Multiplying Polynomials (Objective #1) Multiply terms. (Objective #2) Multiply any two polynomials (Objective #2) Multiply any two polynomials (Objective #3) Multiply binomials. (Objective #3) Multiply binomials. (Objective #4) ) Find the product of a sum and difference of two squares. (Objective #4) Find the product of a sum and difference of two squares. (Objective #5) Find the square of two binomials.
Dividing Polynomials (4.5) Divide a polynomial by a monomial (Objective #1) Divide: EXAMPLE #1: Divide a polynomial by a polynomial of two or more terms (Objective #2) Steps for long division: EXAMPLE #2:
4.5 Dividing Polynomials (Objective #1) Divide a polynomial by a monomial. (Objective #1) Divide a polynomial by a monomial. (Objective #2) Divide a polynomial by two or more terms. (Objective #2) Divide a polynomial by two or more terms. (Objective #2) Divide a polynomial by two or more terms. (Objective #2) Divide a polynomial by two or more terms.