The Coordinate Plane and Graphs

Section 1.3 The Coordinate Plane and Graphs

With me, everything turns into mathematics.

Figure 1.18. René Descartes, French philosopher, mathematician, and scientist, 1596 – 1650

Problem. How to connect geometry and algebra?

René Descartes: “Any problem in geometry can easily be reduced to such terms that a knowledge of the length of certain straight lines is sufficient for construction.”

Figure 1.19. Rene Descartes, *Discours de la méthode*– 1637

Coordinate plane:

Figure 1.20. Each point in the plane is associated with a unique pair of real numbers representing its position in the grid of vertical and horizontal lines.

Example 1.3.1. City map.

A city map as a grid of streets and avenues:

Figure 1.21. Each point in the city is associated with a unique pair of real numbers representing its position in the grid of streets and avenues

The Coordinate plane in slow motion: See Figure 1.22 –. Figure 1.25.

Figure 1.22. Choose a line and a point

Figure 1.23. Note that by choosing the point \(1\) we divide all other points on the line, excluding \(0\text{,}\) in two sets: Those that are on the same side of \(0\) as the point \(1\text{,}\) and those that are not. This is how we get “positive” and “negative” rays with \(0\) as their initial point.

Figure 1.24. The \(x\)- and \(y\)-axis are two mutually perpendicular number lines with the same origin.

Figure 1.25. The coordinate plane establishes a one-to-one correspondence between the points in the plane and the set of all ordered pairs \((x,y)\) of real numbers.

Example 1.3.2. Locate points.

Locate on the coordinate plane the following points: \((0,0), \ (1.5,2), \ (-2, 2.5), \ (-1,-2.5), \ (1.5,-2), \ (2,0), \ (0,2), \ (-2,0), \ (0,-2)\text{.}\)

Solution

Figure 1.26. Points \((0,0), \ (1.5,2), \ (-2, 2.5), \ (-1,-2.5), \ (1.5,-2), \ (2,0), \ (0,2), \ (-2,0), \ (0,-2)\text{.}\)

Must Know! The graph of a function \(f\) is the set of all ordered pairs \((x,f(x))\) as \(x\) varies over the domain of \(f\text{.}\)

The same, just a little bit different: The graph of a function \(f\) is the set

\begin{equation*} \{(x,f(x)): x\in \text{domain of }f\}. \end{equation*}

Example 1.3.3. Find a graph.

Find a graph of the function

\begin{equation*} f(x)=x^2, \ \ x\in \{-2,-1,0,1,2\}. \end{equation*}

Solution

Solution 1:

Graph of \(f=\{(-2,4),(-1,1),(0,0),(1,1),(2,4)\}\text{.}\)

Solution 2:

Graph of \(f\text{:}\)

Figure 1.27. Graph of \(f(x)=x^2\text{,}\) \(x\in \{-2,-1,0,1,2\}\)

Example 1.3.4. Graph of a piecewise defined function.

Graph the function

\begin{equation*} g(x)=\left\{ \begin{array}{rcc} -1\amp \text{ if }\amp x\in [-2,0)\\ 1\amp \text{ if }\amp x=0\\ 3\amp \text{ if }\amp x\in (0,2]\\ \end{array} \right. \end{equation*}

Solution

Figure 1.28. Graph of \(g(x)\)

Which of these curves represents a graph of a function?

Figure 1.29. A function is a rule that assigns to each element in a set \(A\) exactly one element in a set \(B\text{.}\)

Which of these curves represents a graph of a function?

Figure 1.30. A set of points in the coordinate plane is the graph of a function if and only if every vertical line intersects the set in at most one point

Example 1.3.5. Putting everything together.

Let the function \(f\) be given by its graph, as shown in Figure 1.31:

Figure 1.31. Graph of \(f(x)\text{.}\)

Find the domain of \(f\text{.}\)
Find the range of \(f\text{.}\)
Find \(f(-1)\text{,}\) \(f(1)\text{,}\) and \(f(3)\text{.}\)
Estimate \(f(0)\) and \(f(2)\text{.}\)
How many solutions does the equation \(f(x)=1\) have?

Solution

We use Figure 1.32 to answer all questions.

Figure 1.32. Graph of \(f(x)\text{.}\)

Domain of \(f=\color{red}{[-1,2.7]}\text{.}\)
Range of \(f=\color{blue}{[-1.6,2.6]}\text{.}\)
We read from the graph that \(f(-1)=f(1)=0\text{,}\) and that \(f(3)\) is not defined.
We use the graph to estimate \(f(0)\approx 2.5\) and \(f(2)\approx -1.5\text{.}\)
Observe that the line \(y=1\) intersects the graph of the function \(f\) at three points. This implies that the equation \(f(x)=1\) has three solutions