Outro W4

4.1 Use the Rectangular Coordinate System

• Sign Patterns of the Quadrants
            Quadrant I            Quadrant II           Quadrant III             Quadrant VI 

                    $\left(x,y\right)$                 $\left(x,y\right)$                    $\left(x,y\right)$                      $\left(x,y\right)$

                    $\left(+,+\right)$               $\left(-,+\right)$                  $\left(-,-\right)$                    $\left(+,-\right)$

• Points on the Axes
  
  • On the x-axis, y=0. Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates (a,0).
  • On the y-axis, x=0. Points with an x-coordinate equal to 0 are on the y-axis, and have coordinates (0,b).
• Solution of a Linear Equation
  
  • An ordered pair (x,y) is a solution of the linear equation Ax+By=C if the equation is a true statement when the x- and y- values of the ordered pair are substituted into the equation.

4.2 Graph Linear Equations in Two Variables

• Graph a Linear Equation by Plotting Points
  1. Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
  2. Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
  3. Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

4.3 Graph with Intercepts

• Find the x- and y- Intercepts from the Equation of a Line
  • Use the equation of the line to find the x- intercept of the line, let y=0 and solve for x.
  • Use the equation of the line to find the y- intercept of the line, let x=0 and solve for y.
• Graph a Linear Equation using the Intercepts
  1. Step 1. Find the x- and y- intercepts of the line.
     Let y=0 and solve for x.
     Let x=0 and solve for y.
  2. Step 2. Find a third solution to the equation.
  3. Step 3. Plot the three points and then check that they line up.
  4. Step 4. Draw the line.

• Strategy for Choosing the Most Convenient Method to Graph a Line:
  • Consider the form of the equation.
  • If it only has one variable, it is a vertical or horizontal line.
    x=a is a vertical line passing through the x- axis at a
    y=b is a horizontal line passing through the y- axis at b.
  • If y is isolated on one side of the equation, graph by plotting points.
  • Choose any three values for x and then solve for the corresponding y- values.
  • If the equation is of the form ax+by=c, find the intercepts. Find the x- and y- intercepts and then a third point.

4.4 Understand Slope of a Line

• Find the Slope of a Line from its Graph using $m=\frac{rise}{run}$
  1. Step 1. Locate two points on the line whose coordinates are integers.
  2. Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
  3. Step 3. Count the rise and the run on the legs of the triangle.
  4. Step 4. Take the ratio of rise to run to find the slope.

  
  
  

• Graph a Line Given a Point and the Slope
  1. Step 1. Plot the given point.
  2. Step 2. Use the slope formula $m=\frac{rise}{run}$ to identify the rise and the run.
  3. Step 3. Starting at the given point, count out the rise and run to mark the second point.
  4. Step 4. Connect the points with a line.
     

  
  
  

• Slope of a Horizontal Line
  • The slope of a horizontal line, y=b, is 0.
• Slope of a vertical line
  • The slope of a vertical line, x=a, is undefined