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Algebra I – Coordinate Geometry

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    Study Guide Algebra I – Coordinate Geometry 1. Coordinate Graphs Introduction to Coordinate Geometry Coordinate geometry helps us draw and study points, lines, and shapes on a grid called the coordinate plane (or coordinate graph). You may already know that every point on a number line represents a number. In the same way, every point on a coordinate plane is represented by two numbers . These two numbers show exactly where a point is located on the graph. To understand how this works, we use two special lines called coordinate axes . • The horizontal line is called the x - axis . • The vertical line is called the y - axis . These two lines cross each other at a point called the origin . The origin represents the coordinate (0, 0) .

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    Study Guide Figure 1 An x - y coordinate graph Understanding Coordinates Every point on the coordinate plane is described using an ordered pair of numbers , called coordinates . An ordered pair looks like this: [(x, y)] This means: • The first number (x) shows the horizontal position . • The second number (y) shows the vertical position . Important Terms x - coordinate (Abscissa) The first number in the ordered pair. It tells how far a point moves left or right . y - coordinate (Ordinate) The second number in the ordered pair. It tells how far a point moves up or down . Example For the point (1, 4) : • Move 1 unit right from the origin. • Then move 4 units up . That location marks the point (1, 4) on the graph.

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    Study Guide Figure 2 Graphing or plotting coordinates • (1, 4) • (0, 2) • (6, 2) • (3, − 1) • ( − 3, 2) • ( − 4, − 2) Positive and Negative Directions On a coordinate graph, numbers can be positive or negative depending on their position. On the x - axis • Numbers to the right of 0 are positive • Numbers to the left of 0 are negative On the y - axis • Numbers above 0 are positive

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    Study Guide • Numbers below 0 are negative This system helps us find the exact location of any point on the graph. Quadrants of the Coordinate Plane The coordinate plane is divided into four sections called quadrants . These quadrants are labeled using Roman numerals : • Quadrant I • Quadrant II • Quadrant III • Quadrant IV Each quadrant contains points with different combinations of positive and negative coordinates. Figure 3 Coordinate graph with quadrants labeled Place this image right after introducing quadrants .

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    Study Guide Signs of Coordinates in Each Quadrant Remember these patterns to quickly identify the quadrant of a point. Quadrant I • x is positive • y is positive Example : (3, 4) Quadrant II • x is negative • y is positive Example : ( − 3, 4) Quadrant III • x is negative • y is negative Example : ( − 3, − 4) Quadrant IV • x is positive • y is negative Example : (3, − 4) A helpful trick students use is: Start in the top - right and move counterclockwise: I → II → III → IV Graphing Equations on the Coordinate Plane

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    Study Guide Sometimes we want to graph an equation on a coordinate plane. To do this: 1. Choose a value for one variable (usually x ). 2. Substitute that value into the equation. 3. Solve to find the value of y . 4. Write the solution as an ordered pair (x, y) . 5. Plot that point on the graph. 6. Repeat with different values. For example, you might try: • (x = 0) • (x = 1) • (x = 2) Each value gives you another point to plot. After plotting several points, you can connect them to form the graph of the equation . Example 1: Graphing an Equation Let’s learn how to graph an equation on the coordinate plane . Problem Graph the equation: [x + y = 6] To graph this equation, we need to find several pairs of values for x and y that satisfy the equation. These pairs will become points on the graph .

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    Study Guide Step 1: Choose a Value for (x) Start by choosing a value for x , then solve the equation to find y . When (x = 0) Substitute into the equation: [0 + y = 6] [y = 6] So the coordinate is: [(0, 6)] When (x = 1) Substitute into the equation: [1 + y = 6] Subtract 1 from both sides: [y = 5] So the coordinate is: [(1, 5)] When (x = 2) Substitute into the equation: [2 + y = 6] Subtract 2 from both sides: [y = 4] So the coordinate is:

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    Study Guide [(2, 4)] Step 2: Organize the Values in a Table Making a small table can help keep the values organized. Each row in the table gives us a point to plot on the graph . Step 3: Plot the Points Plot the following points on the coordinate plane: • (0, 6) • (1, 5) • (2, 4)

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    Study Guide Figure 4 Plotting of coordinates (0,6), (1,5), (2,4) Step 4: Draw the Line After plotting the points, draw a straight line that passes through all of them . This line represents the graph of the equation (x + y = 6) .

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    Study Guide Figure 5 The line passing through the plotted points Why Do These Points Form a Line? When the solutions of an equation form a straight line when graphed , the equation is called a linear equation . In this example, the points: • (0, 6) • (1, 5) • (2, 4) all lie on the same straight line. This shows that (x + y = 6) is a linear equation . What Are Nonlinear Equations?

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