11.S: Graphs (Summary)
- Page ID
- 5045
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)KEY TERMS
| horizontal line | The graph of an equation that can be written in the form y = b whose line passes through the y-axis at (0, b). |
| intercepts of a line | Each of the points at which a line crosses the x-axis and the y-axis is called an intercept of the line. |
| linear equation | An equation of the form Ax + By = C, where A and B are not both zero, is called a linear equation in two variables |
| ordered pair | An ordered pair (x, y) gives the coordinates of a point in a rectangular coordinate system. The first number is the x-coordinate. The second number is the y-coordinate. |
| origin | The point (0, 0) is called the origin. It is the point where the point where the x-axis and y-axis intersect. |
| quadrants | The four areas of a rectangular coordinate system that has been divided by the x-axis and y-axis. |
| slope of a line | The slope of a line is m = \(\dfrac{rise}{run}\). The rise measures the vertical change and the run measures the horizontal change. |
| solution to a linear equation in two variables | An ordered pair (x, y) is a solution to 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. |
| vertical line | A vertical line is the graph of an equation that can be written in the form x = a. The line passes through the x-axis at (a, 0). |
| x-axis | The horizontal axis in a rectangular coordinate system. |
| y-axis | The vertical axis on a rectangular coordinate system. |
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.
11.2 Graphing Linear Equations
- Graph a linear equation by plotting points.
- Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- 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 x-axis. The x-intercept occurs when y is zero.
- The y-intercept is the point, (0, b), 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 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 |
- Graph a line using the intercepts
- Find the x- and y-intercepts of the line.
- Let y = 0 and solve for x.
- Let x = 0 and solve for y.
- Find a third solution to the equation.
- Plot the three points and then check that they line up.
- Draw the line.
- Find the x- and y-intercepts of the line.
- 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.
- 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.
- 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 Ax + By = C, find the intercepts. Find the x- and y-intercepts and then a third point.
- 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
- 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, m = \(\dfrac{rise}{run}\).
- 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 (x1, y1) and (x2, y2) is m = \(\dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}\).
- 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.
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."