Summary: Review Topics

Key Concepts

  • Sign Patterns of the Quadrants
    Quadrant I Quadrant II Quadrant III Quadrant IV
    [latex](x,y)[/latex] [latex](x,y)[/latex] [latex](x,y)[/latex] [latex](x,y)[/latex]
    [latex](+,+)[/latex] [latex](−,+)[/latex] [latex](−,−)[/latex] [latex](+,−)[/latex]
  • Coordinates of Zero
    • Points with a [latex]y[/latex]-coordinate equal to [latex]0[/latex] are on the x-axis, and have coordinates [latex](a, 0)[/latex].
    • Points with a [latex]x[/latex]-coordinate equal to [latex]0[/latex] are on the y-axis, and have coordinates [latex](0, b)[/latex].
    • The point [latex](0, 0)[/latex] is called the origin. It is the point where the x-axis and y-axis intersect.
  • Intercepts
    • The [latex]x[/latex]-intercept is the point, [latex]\left(a,0\right)[/latex] , where the graph crosses the [latex]x[/latex]-axis. The [latex]x[/latex]-intercept occurs when [latex]y[/latex] is zero.
    • The [latex]y[/latex]-intercept is the point, [latex]\left(0,b\right)[/latex] , where the graph crosses the [latex]y[/latex]-axis. The [latex]y[/latex]-intercept occurs when [latex]y[/latex] is zero.
    • The [latex]x[/latex]-intercept occurs when [latex]y[/latex] is zero.
    • The [latex]y[/latex]-intercept occurs when [latex]x[/latex] is zero.
  • Find the x and y intercepts from the equation of a line
    • To find the [latex]x[/latex]-intercept of the line, let [latex]y=0[/latex] and solve for [latex]x[/latex].
    • To find the [latex]y[/latex]-intercept of the line, let [latex]x=0[/latex] and solve for [latex]y[/latex].
  • Graph a line using the intercepts
    1. Find the x- and y- intercepts of the line.
      • Let [latex]y=0[/latex] and solve for [latex]x[/latex].
      • Let [latex]x=0[/latex] and solve for [latex]y[/latex].
    2. Find a third solution to the equation.
    3. Plot the three points and then check that they line up.
    4. Draw the line.
  • Choose the most convenient method to graph a line
  1. Determine if the equation has only one variable. Then it is a vertical or horizontal line.
    • [latex]x=a[/latex] is a vertical line passing through the [latex]x[/latex]-axis at [latex]a[/latex].
    • [latex]y=b[/latex] is a vertical line passing through the [latex]y[/latex]-axis at [latex]b[/latex].
  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.
  3. Determine if the equation is of the form [latex]Ax+By=C[/latex] , find the intercepts.Find the x- and y- intercepts and then a third point.

Find the slope from a graph

  1. Locate two points on the line whose coordinates are integers.
  2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
  3. Count the rise and the run on the legs of the triangle.
  4. Take the ratio of rise to run to find the slope, [latex]m={\Large\frac{\text{rise}}{\text{run}}}[/latex]
  • Slope of a Horizontal Line
    • The slope of a horizontal line, [latex]y=b[/latex] , is [latex]0[/latex].
  • Slope of a Vertical Line
    • The slope of a vertical line, [latex]x=a[/latex] , is undefined.
  • Slope Formula
    • The slope of the line between two points [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex] is [latex]m={\Large\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[/latex]
  • Graph a line given a point and a slope.
    1. Plot the given point.
    2. Use the slope formula to identify the rise and the run.
    3. Starting at the given point, count out the rise and run to mark the second point.
    4. Connect the points with a line.

Key Equations

  • The slope of the line between two points [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex] is [latex]m={\Large\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[/latex].

Glossary

intercepts of a line
Each of the points at which a line crosses the [latex]x[/latex]-axis and the [latex]y[/latex]-axis is called an intercept of the line.
linear equation
An equation of the form [latex]Ax+By=C[/latex], where [latex]A[/latex] and [latex]B[/latex] are not both zero, is called a linear equation in two variables.
ordered pair
An ordered pair [latex]\left(x,y\right)[/latex] gives the coordinates of a point in a rectangular coordinate system. The first number is the [latex]x[/latex] -coordinate. The second number is the [latex]y[/latex] -coordinate.
origin
The point [latex]\left(0,0\right)[/latex] is called the origin. It is the point where the the point where the [latex]x[/latex] -axis and [latex]y[/latex] -axis intersect.
quadrants
The [latex]x[/latex] -axis and [latex]y[/latex] -axis divide a rectangular coordinate system into four areas, called quadrants.
slope of a line
The slope of a line is [latex]m={\Large\frac{\text{rise}}{\text{run}}}[/latex] . The rise measures the vertical change and the run measures the horizontal change.
solution to a linear equation in two variables
An ordered pair [latex]\left(x,y\right)[/latex] is a solution to the linear equation [latex]Ax+By=C[/latex], if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.
x-axis
The x-axis is the horizontal axis in a rectangular coordinate system.
y-axis
The y-axis is the vertical axis on a rectangular coordinate system.
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