Unit 3 Seminar Agenda The Rectangular Coordinate System  The Vocabulary of Graphing  Ordered Pairs  Finding the Midpoint Graphing Lines  Types of Lines  Intercepts Linear Inequalities  Solving  Graphing and Interval Notation

René Descartes’ Rectangular Coordinate System (Cartesian Coordinate System) x - axis y - axis origin (0,0) I II III IV

Quadrants & Finding Coordinates Coordinates like (2,3) are called ordered pairs and are of the form (x,y) x is the x-coordinate (left or right) y is the y-coordinate (up or down) Graphs can be divided into 4 quadrants  Quadrant I => both coordinates are positive  Quadrant II => 1st-coordinate negative / 2nd-coordinate positive  Quadrant III => both coordinates are negative  Quadrant II => 1st-coordinate positive / 2nd-coordinate negative I II III IV

Quadrants & Finding Coordinates x is the x-coordinate (left or right) y is the y-coordinate (up or down)  Quadrant I => both coordinates are positive  Quadrant II => 1st-coordinate negative / 2nd-coordinate positive  Quadrant III => both coordinates are negative  Quadrant II => 1st-coordinate positive / 2nd-coordinate negative  For example the points (ordered pairs) …. (1,2); (-2,-3); (3,-1) I II III IV (1,2) (-2,-3) (3,-1)

Quadrants & Finding Coordinates x is the x-coordinate (left or right) y is the y-coordinate (up or down)  Quadrant I => both coordinates are positive  Quadrant II => 1st-coordinate negative / 2nd-coordinate positive  Quadrant III => both coordinates are negative  Quadrant II => 1st-coordinate positive / 2nd-coordinate negative R: (, ) B: (, ) G: (, ) P: (, )

The Midpoint The midpoint of a line segment is the point on the segment that is IN THE MIDDLE OF THE SEGMENT. If given two ordered pairs, to find the point in the middle between them (THE MIDPOINT) all we do is find the AVERAGE of the X-COORDINATES and the AVERAGE of the Y-COORDINATES. The midpoint between the points (x 1, y 1 ) and (x 2, y 2 ) is This IS VERY EASY ARITHMETIC … LET IT BE EASY! x 1 + x 2 2 ( ) y 1 + y 2 2,

Find the midpoint between the points (–2, 3) and (4, 2) Substitute the given values into the formula Perform the arithmetic (reduce fractions if possible) x 1 + x 2 2 ( ) y 1 + y 2 2, Remember … the midpoint formula is :

Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result. Graphically …. This is what it looks like! (1, ) 5252 (–2, 3) (4, 2)

There are 3 types of lines Horizontal Lines Vertical Lines Diagonal Lines

Horizontal Lines Look like y = b (NOTICE … NO X) From top to bottom y = 5 y = 5/2 y = -1.2 y = -7 NEVER cross the x – axis They have NO x - intercept

Vertical Lines Look like x = a (NOTICE … NO Y) From left to right x = -15/2 x = -3 x = 2.2 x = 6 NEVER cross the y – axis They have NO y - intercept

Diagonal Lines Look like y = mx + b y = -2x – 6 y = 7x – 8 y = 5/2x + 4 y = -1/8x + 2 Cross the x – axis & y – axis They have x –intercepts & y - intercepts

Intercepts The y–intercept of a line is the point where the line CROSSES the y–axis  ALWAYS has order pair (0, b)  We can find the y-intercept of an equation by letting the x = 0 and solving the equation for y The x-intercept of a line is the point where the line CROSSES the x-axis  ALWAYS has order pair (x, 0)  We can find the x-intercept of an line by letting y = 0 and solving the equation for x

Graphing a Line Using Intercepts Given an equation … we can graph its line by finding the x – and y – intercepts. Graph these two ordered pairs. Draw the line YES … this will work EVERY TIME!

Graphing a Line Using Intercepts Graph the given equation using it’s intercepts. Find the y – intercept (let x = 0, solve for y) Find the x – intercept (let y = 0, solve for x) Graph these two ordered pairs. Draw the line YES … this will work EVERY TIME!

Graphing a Line Using Intercepts Graph the given equation using it’s intercepts. Find the y – intercept (let x = 0, solve for y) Find the x – intercept (let y = 0, solve for x) Graph these two ordered pairs. Draw the line YES … this will work EVERY TIME!

Symbols INEQUALITIES are comparisons of two items which are not of the same amount. INEQUALITY SYMBOLS INCLUDE: < less than (use the shift and comma buttons) > greater than (use the shift and period buttons) ≤ less than or equal to (Insert-Symbol-select this symbol, insert, close) ≥ greater than or equal to (Insert-Symbol-select this symbol, insert, close) ≠ not equal to (Insert-Symbol-select this symbol, insert, close)

Looks familiar.... doesn't it? Solving Inequalities There are FOUR BASIC ideas … and they all work together to help us!  Solving an inequality …. means to rearrange an inequality to get the UNKNOWN all alone.  To get the variable all alone …. we UNDOING all the arithmetic attached to the UNKNOWN.  To UNDO all the arithmetic … we will use the OPPOSITE arithmetic.  We can do whatever we want to an inequality as long as we apply the exact same arithmetic TO BOTH SIDES OF THE INEQUALITY SYMBOL! These are our four guiding principles! Do NOT deviate from them!

Looks familiar.... doesn't it?! Steps For Solving Inqualities STEP1: Clear the grouping symbols using the distribution property. STEP2: Clear the fractions by multiplying EVERY term by a common denominator (it does NOT have to be the least common denominator). STEP3: Move all the variables to one side of the inequality symbol using the addition or subtraction property of equality. STEP4: Move all the plain boring numbers to the other side of the inequality symbol using the addition or subtraction property of equality. STEP5: Isolate the variable using the multiplication or division property of equality. STEP6: Substitute your solution into the ORIGINAL inequality to see if a true statement results. STEP1, 2, and 5: If you multiply by or divide by a negative number, you MUST flip the direction of the inequality symbol I use these steps … in this order … EVERYTIME! Doing them in this order REDUCES the number of arithmetic errors made!

Solving Inequalities Solving inequalities is EXACTLY the same as solving equalities with ONE small exception …. When solving inequalities, if you divide by OR multiply by a negative number … you MUST flip the direction of the inequality symbol

Giving the Answer to an Inequality  A parenthesis ( or ) is used to indicate a number is not an element of a set ( ).  A bracket [ or ] is used to indicate a number is a member of a set ≤ or ≥. Example 1: Write in interval notation and graph: x  3 Solution: Interval Notation (- ,3) Example 2: Write in interval notation and graph: x  0 Solution: Interval Notation [0,  ) ) [