8.2: Ordered Pairs and Graphing Relations by Point-Plotting
- Page ID
- 176595
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definitions and Theorems
The rectangular coordinate system lets you turn an equation in two variables into a picture. Every ordered pair that satisfies the equation corresponds to a point in the plane, and the collection of all such points is the equation's graph.
An ordered pair \((a,b)\) is a pair of objects in which \(a\) is designated the first coordinate and \(b\) the second coordinate. In the coordinate plane, the ordered pair \((x,y)\) names the point reached by moving \(x\) units horizontally and \(y\) units vertically from the origin; \(x\) is the \(x\)-coordinate (abscissa) and \(y\) is the \(y\)-coordinate (ordinate).
Two ordered pairs are equal if and only if their corresponding coordinates are equal:\[(a,b) = (c,d) \quad \text{if and only if} \quad a = c \text{ and } b = d.\nonumber\]
The pair \((a,b)\) is generally not the same as \((b,a)\). For example, \((3,5)\) and \((5,3)\) name two different points. Reversing the coordinates reverses the horizontal and vertical instructions.
A solution of an equation in the variables \(x\) and \(y\) is an ordered pair \((x,y)\) that produces a true statement when its coordinates are substituted into the equation. The solution set is the collection of all such ordered pairs.
The graph of an equation in two variables is the set of all points in the coordinate plane whose coordinates satisfy the equation. The graph is the visual representation of the equation's solution set: a point lies on the graph if and only if its coordinates are a solution.
Any equation that can be written in the form\[ax + by = c,\nonumber\]where \(a\) and \(b\) are not both zero, has a graph that is a straight line. Consequently, two distinct solutions determine the entire graph.
To graph an equation by point-plotting:
- Choose several convenient values for one variable.
- Substitute each chosen value into the equation and solve for the other variable, recording the results as ordered pairs in a table of values.
- Plot the ordered pairs in the coordinate plane.
- Connect the points with a smooth curve, following the pattern they suggest. If the equation is linear, the points fall on a straight line.
When an equation is already solved for \(y\), it is easiest to choose \(x\)-values. When it is solved for \(x\), choose \(y\)-values instead.
A curved graph cannot be reconstructed from too few points. Plot enough ordered pairs to reveal the shape, and connect them with a smooth curve rather than a series of straight segments. Choosing input values that yield convenient (often integer) outputs makes plotting more accurate.
Examples
Plot each ordered pair and state the quadrant in which it lies, or the axis on which it falls: \((3,2)\), \((-4,1)\), \((-2,-3)\), \((5,-4)\), and \((0,3)\).
- Solution
-
The sign pattern of the coordinates determines the location. A point with a zero coordinate lies on an axis and is not in any quadrant.
- \((3,2)\): both coordinates positive, so the point lies in Quadrant I.
- \((-4,1)\): pattern \((-,+)\), so the point lies in Quadrant II.
- \((-2,-3)\): both coordinates negative, so the point lies in Quadrant III.
- \((5,-4)\): pattern \((+,-)\), so the point lies in Quadrant IV.
- \((0,3)\): the \(x\)-coordinate is \(0\), so the point lies on the \(y\)-axis.
Figure \(\PageIndex{1}\): The five plotted points, each in its quadrant or on an axis.
Determine whether each ordered pair is a solution of \(3x - 2y = 8\): \((4,2)\), \((2,-1)\), and \((0,4)\).
- Solution
-
Substitute each pair into the equation and check whether the result is a true statement.
For \((4,2)\):\[3(4) - 2(2) = 12 - 4 = 8.\nonumber\]This is true, so \((4,2)\) is a solution.
For \((2,-1)\):\[3(2) - 2(-1) = 6 + 2 = 8.\nonumber\]This is true, so \((2,-1)\) is a solution.
For \((0,4)\):\[3(0) - 2(4) = 0 - 8 = -8 \neq 8.\nonumber\]This is false, so \((0,4)\) is not a solution.
Graph \(y = -2x + 3\) by point-plotting.
- Solution
-
The equation is solved for \(y\), so choose \(x\)-values and compute \(y\). For instance, at \(x = -1\), \(y = -2(-1) + 3 = 5\).
\(x\) \(y = -2x + 3\) \((x,y)\) \(-1\) \(5\) \((-1,5)\) \(0\) \(3\) \((0,3)\) \(1\) \(1\) \((1,1)\) \(2\) \(-1\) \((2,-1)\) Because the equation is linear, the points fall on a straight line. Plot the ordered pairs and draw the line through them, extending it in both directions.
Figure \(\PageIndex{2}\): The line \(y=-2x+3\) through the four plotted solutions.
Graph \(y = x^2 - 4\) by point-plotting.
- Solution
-
Choose \(x\)-values symmetrically about \(0\), since the squaring produces matching outputs for opposite inputs. For example, at \(x = -3\), \(y = (-3)^2 - 4 = 5\).
\(x\) \(y = x^2 - 4\) \((x,y)\) \(-3\) \(5\) \((-3,5)\) \(-2\) \(0\) \((-2,0)\) \(-1\) \(-3\) \((-1,-3)\) \(0\) \(-4\) \((0,-4)\) \(1\) \(-3\) \((1,-3)\) \(2\) \(0\) \((2,0)\) \(3\) \(5\) \((3,5)\) The points trace a U-shaped curve with its lowest point at \((0,-4)\). Connect them with a smooth curve rather than straight segments.
Figure \(\PageIndex{3}\): The parabola \(y=x^2-4\), with vertex \((0,-4)\).
Graph \(y = \sqrt{x}\) by point-plotting.
- Solution
-
The expression \(\sqrt{x}\) is defined only for \(x \geq 0\), so the graph exists only on the interval \([0,\infty)\). Choose perfect-square inputs so the outputs are whole numbers.
\(x\) \(y = \sqrt{x}\) \((x,y)\) \(0\) \(0\) \((0,0)\) \(1\) \(1\) \((1,1)\) \(4\) \(2\) \((4,2)\) \(9\) \(3\) \((9,3)\) Plot the points and connect them with a smooth curve that rises steeply near the origin and flattens as \(x\) increases. The graph begins at \((0,0)\) and has no portion to the left of the \(y\)-axis.
Figure \(\PageIndex{4}\): The graph of \(y=\sqrt{x}\), beginning at the origin.
Graph \(x = y^2 - 1\) by point-plotting.
- Solution
-
Here the equation is solved for \(x\), so choose \(y\)-values and compute \(x\). For instance, at \(y = -2\), \(x = (-2)^2 - 1 = 3\).
\(y\) \(x = y^2 - 1\) \((x,y)\) \(-2\) \(3\) \((3,-2)\) \(-1\) \(0\) \((0,-1)\) \(0\) \(-1\) \((-1,0)\) \(1\) \(0\) \((0,1)\) \(2\) \(3\) \((3,2)\) The points trace a sideways U-shaped curve opening to the right, with its leftmost point at \((-1,0)\). Connect them smoothly. Notice that most \(x\)-values here correspond to two different points, so this graph is not the graph of \(y\) as a function of \(x\); point-plotting nonetheless applies to any equation in two variables.
Figure \(\PageIndex{5}\): The graph of \(x=y^2-1\), opening to the right from \((-1,0)\).
Sources
Several parts of this text use modifications from the following sources:
- Wikipedia article: "Ordered pair"
- Wikipedia article: "Cartesian coordinate system"
- Wikipedia article: "Graph of a function"
All of these sources are released under the Creative Commons Attribution-Share-Alike License 4.0.