1.1 Rectangular Coordinates; Graphing Utilities The coordinate system used here is the Cartesian Coordinate System, also known as rectangular coordinate system. If (x,y) are the coordinates of a point P, then x is called the x-coordinate or abscissa and y is the y-coordinate or ordinate. Distance Formula EX) Find the distance between (-4, 5) and (3, 2). Midpoint Formula EX) Find the midpoint of a line segment from (-5, 5) and (3, 1). HW: pgs. 9-10 1, 15, 21, 29, 39, 49, 57
1.2 Introduction to Graphing Equations Determining Whether a Point is on the Graph of an Equation Any values of x and y that result in a true statement are said to satisfy or be on the graph of the equation. EX) Determine if the following points are on the graph of the equation 2x y 6. A) (2, 3) B) (2, -2) If required to graph an equation that it is not a line by hand, it may be best to simply plot points. In order to graph an equation in the graphing calculator you must solve for y in terms of x. EX) Graph the equation 6x 2 3y 36 using your calculator and complete the table below. x y -3-2 -1 0 1 Intercepts x-intercept the x-coordinate of a point at which the graph crosses or touches the x-axis (also known as zeros or roots). y-intercept the y-coordinate of a point at which the graph crosses or touches the y-axis. Procedure of Finding Intercepts: 1. To find the x-intercept(s) let y = 0 and solve for x. 2. To find the y-intercept(s) let x = 0 and solve for y. EX) Find the x-intercept(s) and y-intercept(s) of the graph of y x 2 4 by hand. EX) Find the x-intercept(s) and y-intercept(s) of the graph of y x 3 16 using your graphing calculator. HW pgs. 19 20 1, 5, 7, 9, 13, 15, 23, 33
1.3 Symmetry; Graphing Key Equations; Circles Symmetry A graph is symmetric with respect to the x-axis if, for every (x, y) on the graph, the point (x, -y) is also on the graph. A graph is symmetric with respect to the y-axis if, for every (x, y) on the graph, the point (-x, y) is also on the graph. A graph is symmetric with respect to the origin if, for every (x, y) on the graph, the point (-x, -y) is also on the graph. To test a graph of an equation for symmetry with respect to the x-axis replace y by y in the equation. If an equivalent equation results, the graph has x-axis symmetry. To test a graph of an equation for symmetry with respect to the y-axis replace x by x in the equation. If an equivalent equation results, the graph has y-axis symmetry. The function is considered an even function. To test a graph of an equation for symmetry with respect to the origin x by x and replace y by y in the equation. If an equivalent equation results, the graph is symmetric to the origin. The function is considered an odd function. EX) Test the equation 2 4x y for symmetry with respect to the x-axis, the y-axis, and the origin. x 2 1 Circles A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point (h, k). The fixed distance r is called the radius, and the fixed point (h, k) is called the center of the circle. EX) Write the standard form of the equation of the circle with radius 5 and center (-3, 6).
2 2 General Form of the Equation of a Circle x y ax by c 0 If an equation of a circle is in general form, we use the method of completing the square to put the equation in standard form so we can easily identify the center and radius. 2 2 EX) Determine the radius and center of the circle, x y 4x 6y 12 0. EX) Find the general form of the equation of a circle with center (1, -2) and whose graph contains the point (4, -2). HW pgs. 30 32 1, 11, 13, 15, 17, 27, 33, 39, 45, 55, 57, 63, 67
1.4 Solving Equations Solving Equations Using Your Calculator 2 methods (Using Zero/Root or Using Intersect) ALWAYS ROUND TO 3 DECIMAL PLACES! 3 EX) Find the solution(s) to the equation x x 1 0 using your calculator. EX) Find the solution(s) to the equation 4x 3 3 2x 1using your calculator. Quadratic equations can be solved by graphing, factoring, or using the quadratic formula. When factoring make sure you set one side of the equation equal to zero. EX) Solve the equation x 2 12 x by graphing and factoring.
Quadratic Formula EX) Solve 3x 2 5x 1 0 using the quadratic formula. EX) Solve 3x 2 2 4x by graphing and using the quadratic formula. Equations Containing Radicals When the variable in an equation occurs in a square root, cube root, or any other radical it is best to undo the radical by isolating the most complicated radical on one side of the equation and then raising both sides of the equation to a power equal to the index of the radical. Be sure to check for extraneous solutions after solving a radical equation. EX) Find the real solutions to the equation 3 2x 4 2 0
EX) Find the real solutions to the equation 2x 3 x 2 2 Equations Involving Absolute Value If a is a positive real number and if u is any algebraic expression, then u a is equivalent to u a or u a EX) Solve the equation x 4 13 HW pgs. 43-44 1, 3, 15, 17, 29, 31, 41, 51, 53, 69, 81, 83, 89, 91
1.5 Solving Inequalities Interval Notation EX) Write each inequality using interval notation A) 1 x 3 B) 4 x 0 C) x 5 D) x 1 EX) Write each interval as an inequality involving x. A) 1,4 B) 2, C),3 2 D),3] Solving Inequalites When solving inequalities flip the sign if you divide or multiply by a negative. Otherwise, inequalities are solved just like equations. EX) Solve 4x 7 2x 3. Write your answer in inequality form and interval notation. EX) Solve 5 3x 2 1. Write your answer in inequality form and interval notation.
Solving Inequalities Involving Absolute Value Less than means use AND and Greater than means use OR. u a is equivalent to a u a which is the same as u a AND u a. u a is equivalent to u a OR u a. EX) Solve 2x 4 3 EX) Solve 2x 5 3 Solving Quadratic Inequalities To solve a quadratic inequality it is best to set one side of the inequality equal to 0. Then sketch a graph of the function and determine which intervals satisfy the inequality. 2 EX) x 6x 5 2 EX) x 7x 12 0 HW pgs. 56-57 1, 3, 5, 13, 15, 17, 21, 23, 25, 45, 51, 57, 61, 63, 67, 69, 77, 81, 83, 85, 91
1.6 Lines EX) Find the slope of the line containing the points (1, 2) and (5, -3).
EX) Write an equation for a line that contains the point (3, 2) and has a slope of ¾. EX) Find an equation of a horizontal line and of a vertical line containing the point (3, 2). EX) Find an equation of a line containing (2, 3) and (-4, 5). EX) Find the slope and y-intercept of the equation 2x + 4y = 8. EX) Determine if the lines 2x +3y = 6 and 4x + 6y = 0 are parallel, perpendicular, or neither. EX) Find an equation for the line that contains the point (2, -3) and is parallel to the line 2x + y = 6. EX) Find an equation for the line that contains the point (1, -2) and is perpendicular to the line x + 3y =6. HW pgs. 71-74 1, 7, 13, 17, 19, 27, 31, 33, 35, 39, 43, 47, 53, 57, 65, 71, 103