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Algebra II - Segments Lines and Inequalities

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    Study Guide Algebra II – Segments Lines and Inequali Ɵ es 1. Rectangular Coordinate System A rectangular coordinate system helps us locate points on a graph using numbers. Every point on a graph is described by two coordinates , written as an ordered pair : (x, y) . These coordinates are also called Cartesian coordinates , named after the French mathematician René Descartes , who developed this system. When we know: • the slope and intercept of a line, or • the coordinates of two points on the line we can determine the equation of that line . Key Terms You Should Know Let’s go through some important terms that are used when working with coordinate systems. 1. Coordinates of a Point On a number line , each point corresponds to a single number. Similarly, on a coordinate plane , each point is identified by two numbers . These two numbers together are called the coordinates of the point . Example: The point (3, 4) means: • move 3 units along the x - direction • then 4 units along the y - direction

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    Study Guide 2. x - axis and y - axis To locate points on a plane, we use two perpendicular lines : • The x - axis → a horizontal line • The y - axis → a vertical line These two axes intersect at the center of the plane. 3. Coordinate Plane The coordinate plane is the entire flat surface that includes: • the x - axis • the y - axis • all the points on the graph It is the system we use to plot and locate points. 4. Ordered Pairs Every point on a coordinate plane is written as an ordered pair : The order matters . • The first number (x) tells how far to move left or right . • The second number (y) tells how far to move up or down . Example: (2, 5) means: • move 2 units right • move 5 units up

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    Study Guide 5. Origin The point where the x - axis and y - axis intersect is called the origin . The coordinates of the origin are: It is the starting point for locating all other points. 6. x - Coordinate The x - coordinate is the first number in an ordered pair. It shows movement parallel to the x - axis . • Positive x - value → move right • Negative x - value → move left Example: • (4, 2) → move 4 units right • ( - 3, 2) → move 3 units left 7. y - Coordinate The y - coordinate is the second number in an ordered pair. It shows movement perpendicular to the x - axis . • Positive y - value → move up • Negative y - value → move down Example: • (2, 5) → move 5 units up • (2, - 4) → move 4 units down

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    Study Guide 8. Quadrants The x - axis and y - axis divide the coordinate plane into four regions called quadrants . Figure 1. The quadrants. The quadrants are numbered counterclockwise , starting from the upper - right. Quadrant I • x is positive • y is positive Quadrant II • x is negative • y is positive

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    Study Guide Quadrant III • x is negative • y is negative Quadrant IV • x is positive • y is negative Graph of an Ordered Pair When we place a point on the coordinate plane using its ordered pair, that plotted point is called the graph of the ordered pair . For example: • The graph of (2,3) is the point located 2 units right and 3 units up from the origin. Summary • A rectangular coordinate system is used to locate points on a graph using ordered pairs (x, y) . • The x - axis is horizontal and the y - axis is vertical. • Their intersection point is called the origin (0,0) . • The x - coordinate tells how far to move left or right . • The y - coordinate tells how far to move up or down . • The coordinate plane is divided into four quadrants :

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    Study Guide o Quadrant I → (+,+) o Quadrant II → ( - ,+) o Quadrant III → ( - , - ) o Quadrant IV → (+, - ) • Plotting an ordered pair on the plane gives the graph of that point . 2. Quiz: Rectangular Coordinate System 1. In which quadrant is the point with coordinates ( – 3, – 21)? Answer Choices • quadrant II • quadrant III • quadrant IV Correct Answer quadrant III Why This Is Correct The quadrant of a point depends on the signs of the x - and y - coordinates . • If x is negative and y is negative , the point lies in Quadrant III . For the point ( – 3, – 21) : • x = – 3 (negative) • y = – 21 (negative) Since both coordinates are negative , the point is located in Quadrant III .

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    Study Guide 2. In which quadrant is the point with coordinates (21, – 3)? Answer Choices • quadrant II • quadrant III • quadrant IV Correct Answer quadrant IV Why This Is Correct To find the quadrant, look at the signs of x and y . • If x is positive and y is negative , the point lies in Quadrant IV . For (21, – 3) : • x = 21 (positive) • y = – 3 (negative) A positive x and negative y place the point in Quadrant IV . 3. Which describes the order of the signs of an ordered pair in the second quadrant? Answer Choices • positive, negative • negative, positive • positive, positive Correct Answer negative, positive

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    Study Guide Why This Is Correct In the second quadrant , the coordinates follow this pattern: • x is negative • y is positive So ordered pairs in Quadrant II look like ( – , +) . Example: ( – 4, 7) lies in Quadrant II because the x - coordinate is negative and the y - coordinate is positive. 4. Which describes the order of the signs of an ordered pair in the fourth quadrant? Answer Choices • positive, negative • negative, positive • positive, positive Correct Answer positive, negative Why This Is Correct In the fourth quadrant , the coordinates follow this pattern: • x is positive • y is negative So ordered pairs in Quadrant IV look like (+, – ) . Example: (5, – 2) is in Quadrant IV because x is positive and y is negative.

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    Study Guide 5. Which describes the order of the signs of an ordered pair in the third quadrant? Answer Choices • positive, negative • negative, positive • negative, negative Correct Answer negative, negative Why This Is Correct In the third quadrant , both coordinates are negative. • x is negative • y is negative So ordered pairs in Quadrant III look like ( – , – ) . Example: ( – 6, – 4) lies in Quadrant III because both coordinates are negative. 3. Distance Formula In coordinate geometry, we often need to find the distance between two points on a graph. The distance formula helps us calculate this distance using the coordinates of the points. To understand where this formula comes from, let’s look at an example. Example with Three Points

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    Study Guide Figure 1. A right triangle. Suppose we have three points on a coordinate plane: • Point A = (2, 2) • Point B = (5, 2) • Point C = (5, 6) These points form a right triangle . Finding AB To find the length of AB , we only need to look at the x - coordinates because both points have the same y - value (2) . So, the length of AB is 3 units .

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