11.1 Use the Rectangular Coordinate System
- Sign Patterns of the Quadrants
Quadrant I Quadrant II Quadrant III Quadrant IV (x,y) (x,y) (x,y) (x,y) (+,+) (−,+) (−,−) (+,−)
- Coordinates of Zero
- Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates ( a, 0).
- Points with a x-coordinate equal to 0 are on the y-axis, and have coordinates ( 0, b).
- The point (0, 0) is called the origin. It is the point where the x-axis and y-axis intersect.
11.2 Graphing Linear Equations
- Graph a linear equation by plotting points.
- Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Step 2. Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Step 3. Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line.
- Graph of a Linear Equation:The graph of a linear equation is a straight line.
- Every point on the line is a solution of the equation.
- Every solution of this equation is a point on this line.
11.3 Graphing with Intercepts
- The x-intercept is the point, , where the graph crosses the x-axis. The x-intercept occurs when y is zero.
- The y-intercept is the point, , where the graph crosses the y-axis. The y-intercept occurs when x is zero.
- The x-intercept occurs when y is zero.
- The y-intercept occurs when x is zero.
- Find the x and y intercepts from the equation of a line
- To find the x-intercept of the line, let and solve for x.
- To find the y-intercept of the line, let and solve for y.
x y 0 0
- Graph a line using the intercepts
- Step 1.
Find the x- and y- intercepts of the line.
- Let and solve for x.
- Let and solve for y.
- Step 2. Find a third solution to the equation.
- Step 3. Plot the three points and then check that they line up.
- Step 4. Draw the line.
- Step 1. Find the x- and y- intercepts of the line.
- Choose the most convenient method to graph a line
- Step 1.
Determine if the equation has only one variable. Then it is a vertical or horizontal line.
is a vertical line passing through the x-axis at a.
is a horizontal line passing through the y-axis at b.
- Step 2.
Determine if y is isolated on one side of the equation. The graph by plotting points.
Choose any three values for x and then solve for the corresponding y- values.
- Step 3.
Determine if the equation is of the form , find the intercepts.
Find the x- and y- intercepts and then a third point.
- Step 1. Determine if the equation has only one variable. Then it is a vertical or horizontal line.
11.4 Understand Slope of a Line
- Find the slope from a graph
- Step 1. Locate two points on the line whose coordinates are integers.
- Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
- Step 3. Count the rise and the run on the legs of the triangle.
- Step 4. Take the ratio of rise to run to find the slope,
- Slope of a Horizontal Line
- The slope of a horizontal line, , is 0.
- Slope of a Vertical Line
- The slope of a vertical line, , is undefined.
- Slope Formula
- The slope of the line between two points and is
- Graph a line given a point and a slope.
- Step 1. Plot the given point.
- Step 2. Use the slope formula to identify the rise and the run.
- Step 3. Starting at the given point, count out the rise and run to mark the second point.
- Step 4. Connect the points with a line.