8: Geometry and Graphing
- Page ID
- 46207
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Which cyclist will win the race? What will the winning time be? How many seconds will separate the winner from the runner-up? One way to summarize the information from the race is by creating a graph. In this chapter, we will discuss the basic concepts of graphing. The applications of graphing go far beyond races. They are used to present information in almost every field, including healthcare, business, and entertainment.
- 8.1: Solve Geometry Applications- Circles and Irregular Figures
- In this section, we will work on geometry applications for circles and irregular figures. To solve applications with circles, we use the properties of circles from Decimals and Fractions. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. To find the area of one of these irregular figures, we split it into figures whose formulas we know and then add the areas of the figures.
- 8.2: Use the Rectangular Coordinate System (Part 1)
- Just as maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.
- 8.3: Solve Geometry Applications- Volume and Surface Area (Part 1)
- The surface area is a square measure of the total area of all the sides of a rectangular solid. The amount of space inside the rectangular solid is the volume, a cubic measure. The volume, V, of any rectangular solid is the product of the length, width, and height. To find the surface area of a rectangular solid, find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.
- 8.4: Use the Rectangular Coordinate System (Part 2)
- Equations with two variables can be written in the general form Ax + By = C. An equation of this form is called a linear equation in two variables. Linear equations in two variables have infinitely many solutions. For every number that is substituted for x, there is a corresponding y value. This pair of values is a solution to the linear equation and is represented by the ordered pair (x, y).
- 8.5: Graphing Linear Equations (Part 1)
- 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. The method we used at the start of this section to graph a linear equation is called plotting points, or the Point-Plotting Method. You can use two points to graph a line, but if you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work.
- 8.6: Solve Geometry Applications- Volume and Surface Area (Part 2)
- A sphere is the shape of a basketball, like a three-dimensional circle. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height h of a cylinder is the distance between the two bases. In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.
- 8.7: Graphing Linear Equations (Part 2)
- In this section, we will graph equations with only one variable. That is, there is just x and no y, or just y without an x. 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). A horizontal line is the graph of an equation that can be written in the form y = b. The line passes through the y-axis at (0, b).
- 8.8: Use the Rectangular Coordinate System
- Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.
- 8.9: Graphing with Intercepts (Part 1)
- Every linear equation has a unique line that represents all the solutions of the equation. At first glance, two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. To graph a linear equation by plotting points, you can use the intercepts as two of your three points.
- 8.11: Graphing with Intercepts (Part 2)
- We can use the form of equation to choose the most convenient method to graph its line. If the equation has only one variable, it is a vertical or horizontal line. 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.
- 8.12: Graph with Intercepts
- When graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points. At first glance, their two lines might not appear to be the same, but if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x- axis and the y- axis. These points are called the intercepts of the line.
- 8.13: Understand Slope of a Line (Part 1)
- The steepness of the slant of a line is called the slope of the line. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. The slope formula states that slope is equal to y of the second point minus y of the first point over x of the second point minus x of the first point.
- 8.14: Understand Slope of a Line (Part 2)
- In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.
Figure 11.1 - Cyclists speed toward the finish line. (credit: ewan traveler, Flickr)
Contributors
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."