### Key Concepts

#### 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.

- Points with a

#### 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 $ax+by=c$ 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

**Intercepts**- The
*x-*intercept is the point, $(a,0)$, where the graph crosses the*y-*axis. The*x-*intercept occurs when y is zero. - The
*y-*intercept is the point, $(0,b)$, where the graph crosses the*x-*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.

- The
**Find the***x*and*y*intercepts from the equation of a line- To find the
*x-*intercept of the line, let $y=0$ and solve for*x*. - To find the
*y-*intercept of the line, let $x=0$ and solve for*y*.**x****y**0 0

- To find the
**Graph a line using the intercepts**- 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.*

- Let $y=0$ and solve for
- 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.
**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.

$x=a$ is a vertical line passing through the*x-*axis at*a*.

$y=b$ is a vertical 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 $Ax+By=C$, 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, $m=\frac{\text{rise}}{\text{run}}$

**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.

**Slope Formula**- The slope of the line between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

**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.