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29 Functions and their Graphs
The concept of a function was introduced and studied in Section 7 of these notes. In this section we explore the graphs of functions. Of particular in- terest, we consider the graphs of linear functions, quadratic functions, cubic functions, square root functions, and exponential functions. These graphs are represented in a coordinate system known as the Cartesian coordi- nate system which we explore next.
The Cartesian Plane The Cartesian coordinate system was developed by the mathematician Ren´e Descartes in 1637. The Cartesian coordinate system, also known as the rectangular coordinate system or the xy-plane, consists of two number scales, called the x-axis and the y-axis, that are perpendicular to each other at point O called the origin. Any point in the system is associated with an ordered pair of numbers (x, y) called the coordinates of the point. The number x is called the abscissa or the x-coordinate and the number y is called the ordinate or the y-coordinate. The abscissa measures the dis- tance from the point to the y-axis whereas the ordinate measures the distance of the point to the x-axis. Positive values of the x-coordinate are measured to the right, negative values to the left. Positive values of the y-coordinate are measured up, negative values down. The origin is denoted as (0, 0). The axes divide the coordinate system into four regions called quadrants and are numbered counterclockwise as shown in Figure 29. To plot a point P (a, b) means to draw a dot at its location in the xy-plane.
Example 29. Plot the point P with coordinates (5, 2).
Solution. Figure 29.1 shows the location of the point P (5, 2) in the xy-plane.
Figure 29.
Example 29. Complete the following table of signs of the coordinates of a point P (x, y).
x y Quadrant I Quadrant II Quadrant III Quadrant IV Positive x-axis Negative x-axis Positive y-axis Negative y-axis
Figure 29.
Solution. By the vertical line test, (b) represents a function whereas (a) and (c) fail to represent functions since one can find a vertical line that intersects the graph more than once.
The domain of a function is the collection of all possible x-coordinates that can be used in the formula of the function. For example, x = 1 is in the domain of f (x) = x + 1 since f (1) = 1 + 1 = 2 whereas x = 1 is not in the domain of f (x) = (^) x−^11 since f (1) = 10 which is undefined. The collection of all values of y-coordinates that correspond to the x-coordinates is called the range of the function. For example, the range of f (x) =
x − 1 is the interval [0, ∞) whereas that of the function f (x) = (^) x^1 − 1 is the set R − { 0 }.
Practice Problems
Problem 29. Plot the points whose coordinates are given on a Cartesian coordinate system.
(a) (2, 4), (0, −3), (− 2 , 1), (− 5 , −3). (b) (− 3 , −5), (− 4 , 3), (0, 2), (− 2 , 0).
Problem 29. Plot the following points using graph papers. (a) (3,2) (b) (5,0) (c) (0,-3) (d) (-3,4) (e) (-2,-3) (f) (2,-3)
Problem 29. Complete the following table.
(x,y) x > 0 , y > 0 x < 0 , y > 0 x > 0 , y < 0 x < 0 , y < 0 Quadrant
Problem 29. In the Cartesian plane, shade the region consisting of all points (x, y) that satisfy the two conditions
− 3 ≤ x ≤ 2 and 2 ≤ y ≤ 4
Problem 29. Determine which of the following graphs represent a function.
Problem 29. Consider the function f whose graph is given below.
The graph of the function C is obtained by plotting the data in the above table. See Figure 29.3. The formula that describes the relationship between C and p is given by
C(p) = 1. 06 p.
Figure 29.
Graphs of Quadratic Functions You recall that a linear function is a function that involves a first power of x. A function of the form
f (x) = ax^2 + bx + c, a 6 = 0
is called a quadratic function. The word ”quadratus” is the latin word for a square. Quadratic functions are useful in many applications in mathematics when a linear function is not sufficient. For example, the motion of an object thrown either upward or downward is modeled by a quadratic function. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in ”width” or ”steepness”, but they all have the same basic ”U” shape. All parabolas are symmetric with respect to a line called the axis of sym- metry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola.
Many quadratic functions can be graphed easily by hand using the techniques of stretching/shrinking and shifting (translation) the parabola y = x^2.
Example 29. Sketch the graph of y = x 22.
Solution. Starting with the graph of y = x^2 , we shrink by a factor of one half. This means that for each point on the graph of y = x^2 , we draw a new point that is one half of the way from the x-axis to that point. See Figure 29.
Figure 29.
When a quadratic function is in standard form, then it is easy to sketch its graph by reflecting, shifting, and stretching/shrinking the parabola y = x^2.
The quadratic function f (x) = a(x − h)^2 + k, a not equal to zero, is said to be in standard form. If a is positive, the graph opens upward, and if a is negative, then it opens downward. The line of symmetry is the vertical line x = h, and the vertex is the point (h, k). Any quadratic function can be rewritten in standard form by completing the square. Note that when a quadratic function is in standard form it is also easy to find its zeros by the square root principle.
Example 29. Write the function f (x) = x^2 − 6 x + 7 in standard form. Sketch the graph of f and find its zeros and vertex.
Figure 29.
For a > 1 the function is increasing. In this case, we say that the function represents an exponential growth. If 0 < a < 1 then the function represents an exponential decay.
Remark 29. Why a is restricted to a > 0 and a 6 = 1? Since t is allowed to have any value then a negative a will create meaningless expressions such as
a (if t = 12 ). Also, for a = 1 the function P (t) = b is called a constant function and its graph is a horizontal line.
Miscellaneous Functions We consider some of the graphs of some important functions.
Square Root Function The square root function is the function f (x) =
x. To get the graph well just plug in some values of x and then plot the points.
x 0 1 4 9 f(x) 0 1 2 3
The graph is given in Figure 29.
Figure 29.
Absolute Value Function We’ve dealt with this function several times already. It’s now time to graph it. First, let’s remind ourselves of the definition of the absolute value function.
f (x) =
x if x ≥ 0 −x if x < 0
Finding some points to plot we get
x -2 -1 0 1 2 f(x) 2 1 0 1 2
The graph is given in Figure 29.
Solution. (a) The graph is given in Figure 29.10.
Figure 29.
(b) A formula of C(x) is
C(x) =
- 50 if 0 ≤ x ≤ (^15)
- 75 if 15 < x ≤ (^25)
- 00 if 25 < x ≤ (^35)
- 25 if 35 < x ≤ (^45)
- 50 if 45 < x ≤ 1.
(c) The cost for a 45 ride is C(^45 ) = $2.25.
Practice Problems
Problem 29. Make a table of five values of the function f (x) = 2x + 3 and then use the points to sketch the graph of f (x).
Problem 29. Make a table of five values of the function f (x) = 12 x^2 + x and then use the points to sketch the graph of f (x).
Problem 29. Make a table of five values of the function f (x) = 3x^ and then use the points to sketch the graph of f (x).
Problem 29. Make a table of five values of the function f (x) = 2 − x^3 and then use the points to sketch the graph of f (x).
Problem 29. Which type of function best fits each of the following graphs: linear, quadratic, cubic, exponential, or step?
Problem 29. Suppose that a function f is given by a table. If the output changes by a fixed amount each time the input changes by a constant then the function is linear. Determine whether each of the following functions below are linear.
x 0 2 4 6 f(x) 20 40 80 160
x 10 20 30 40 g(x) 6 12 18 24
Problem 29. Show how to solve the equation 2x + 3 = 11 using a calculator.
Problem 29. In the linear function f (x) = mx + b the parameter m is called the slope. The slope of the line determines whether the line rises, falls, is vertical or horizontal. Classify the slope of each line as positive, negative, zero, or undefined.
Problem 29. Graph each equation by plotting points that satisfy the equation.
(a) x − y = 4. (b) y = − 2 |x − 3 |. (c) y = 12 (x − 1)^2.
Problem 29. Find the x- and y-intercepts of each equation.
(a) 2x + 5y = 12. (b) x = |y| − 4. (c) |x| + |y| = 4.
