## 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.

- Points with a [latex]y[/latex]-coordinate equal to [latex]0[/latex] are on the
**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**- 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].

- Find a third solution to the equation.
- Plot the three points and then check that they line up.
- Draw the line.

- Find the
**Choose the most convenient method to graph a line**

- 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].

- 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. - 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**

- Locate two points on the line whose coordinates are integers.
- Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- 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.**- Plot the given point.
- Use the slope formula to identify the rise and the run.
- Starting at the given point, count out the rise and run to mark the second point.
- 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.

**term**- definition

**term**- definition