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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

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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

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Coordinate plane:

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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.
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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

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

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Figure 1.22. Choose a line and a point

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

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

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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.
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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).

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

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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.

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

{(x,f(x)):x∈domain of f}.
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Example 1.3.3. Find a graph.

Find a graph of the function

f(x)=x2,  x∈{−2,−1,0,1,2}.
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 \(f(x)=x^2\text{,}\) \(x\in \{-2,-1,0,1,2\}\)
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Example 1.3.4. Graph of a piecewise defined function.

Graph the function

g(x)={−1 if x∈[−2,0)1 if x=03 if x∈(0,2]
Solution
Figure 1.28. Graph of \(g(x)\)

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Which of these curves represents a graph of a function?

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Figure 1.29. A function is a rule that assigns to each element in a set A exactly one element in a set B.

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Which of these curves represents a graph of a function?

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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
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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).

  1. Find the domain of f.

  2. Find the range of f.

  3. Find f(−1), f(1), and f(3).

  4. Estimate f(0) and f(2).

  5. 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 \(f(x)\text{.}\)

  1. Domain of \(f=\color{red}{[-1,2.7]}\text{.}\)

  2. Range of \(f=\color{blue}{[-1.6,2.6]}\text{.}\)

  3. We read from the graph that \(f(-1)=f(1)=0\text{,}\) and that \(f(3)\) is not defined.

  4. We use the graph to estimate \(f(0)\approx 2.5\) and \(f(2)\approx -1.5\text{.}\)

  5. 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