Suppose that f:A -> B is a function from A to B. The graph of f is the set of points in the plane described by (x,f(x)), where x is in the domain of f. If we plot (x,f(x)) for a sufficient number of values of x, we find that the points (x,f(x)), plotted in the Cartesian plane, are ordered along a curve.
In this Workout, we shall be looking at graphs of functions that come from linear expressions, that is where the variable x is not raised to a power. For example,
f(x) = 2x - 6.
The graph of such an expression will be a set of points (x,f(x)) which are ordered along a straight line. Indeed, if we look at a table of values for (x,f(x)), we find
If we plot these points on the plane,
the points by straight lines, we find that all the points lie
on the single
straight line illustrated below.
Since graphs of functions are plotted with reference to an x-axis and a y-axis, it is often useful to label the function y. We say that y = f(x), or that y is a function of x. Notice that the value of f(x) is measured along the vertical y axis. For the same reason, we may write equations, such as y = 2x - 6, when we mean f(x) = 2x - 6.
Recall 2: (The x and y intercepts of a straight line.)
Notice that any straight line that you can draw in the plane must intersect one of the axes, either the x-axis or the y-axis. Therefore, straight lines can be classified into three categories:
1. Lines that intersect only the
2. Lines that intersect only the y-axis.
3. Lines that intersect both the x-axis and the y-axis.
Lines of type 1 cannot be the graph of any function. They must be vertical lines. Therefore, such lines are completely determined by where they cross the x-axis. For example if a vertical line crosses the x-axis at x = 9, then the x value of its graph must always be 9. In other words, the graph must be the set of all points (9,y), and it does not matter what y is. (y can be any real number.) This is a peculiar case, since such a vertical straight line cannot be the graph of any function. Recall that a function must have a unique value of f(x) for each value of x.
Lines of type 2 are horizontal lines. Therefore they have the same y value for every x value. In this case it is the graph of a function that takes on the same value for all values of the independent variable x. It is the graph of the form (x, b), where b is the value at which the line crosses the y-axis. For example, a horizontal line of height 5 is the set of points of the form (x,5). This means that it is the graph of the function f(x) = 5. A function for which the value of the function is the same for every value of x is called a constant function. The graph of any constant function is always horizontal and it crosses the y-axis at the value of the constant.
Now we come to the last type. Lines of type 3 intersect the x-axis exactly one time and intersect the y-axis exactly one time. The point at which the line intersects the x-axis is called the x-intercept. The point at which the line intersects the y axis is called the y-intercept.
How do we find the intercepts?
Let y=0 to find the
x-intercept; let x=0 to find
Recall 3: (The x- and y-intercepts determine the line)
Euclid says that two points determine a unique line. We can see this geometrically. With two points, we are able to draw an unambiguous, well-defined, unique line connecting those two points and extending indefinitely into space. How do we determine a line of type 3? The easiest two points to find are the x-intercept and the y-intercept. These points will help us graph the line, if we already know the function. Later we will learn how to find the function, if we already know the points.
The x-intercept is a point of the form (a,0), where a is some fixed real number. The y-intercept is a point of the form (0,b), where b is some fixed real number. These two points can easily be obtained from the function by setting y equal to 0 and x equal to 0, respectively. For example, if the function is f(x) = 2x - 6, then, by setting y = f(x) = 0, we have 0 = 2x - 6. Solving for x gives x = 3. So the x-intercept for the graph of this function is the point (3,0). Set x = 0 to find the y-intercept. Doing so yields (0,-6). We therefore have two points on our graph, (3,0) and (0,-6). Draw the line connecting these two points to sketch the graph of f(x) = 2x - 6. Notice that we get the same straight line that appears in Recall 1; this time we simply plotted two points instead of 13.
Recall 4: (The slope of a line)
If you take another look at the line illustrated in Recall 1, (the graph of f(x) = 2x - 6) you may notice that the y value increases 2 units every time the x value increases by 1 unit. All lines of type 3 have the property that their y value increases or decreases each time the x value increases by one unit. The amount that the y value increases or decreases for each unit of increase of the x value is called the slope of the line. The line given by the function f(x) =2x - 6 has a slope of 2--that is, it increases by 2 units every time x increases by 1 unit. To see this, take the point (0,-6); by increasing x one unit you come to the point on the line given by (1,-4). The difference between -4 and -6 is 2 units. The point (x,y) on the curve has moved 2 units higher.
It is no accident that the coefficient of the x
the function f(x) = 2x - 6 is the same as the slope
line. In general the graph of a function
will be a straight line of slope a. The form of the function above is called the standard form of a line function. In this form a is the slope and b is called y-intercept of the line.
Another way of finding the slope is to pick any two points on the graph, say (x0,y0) and (x1,y1). Then the difference between the abscissa values x1 and x0 is the horizontal change, called the run. The difference between the ordinate values y1 and y0 is the vertical change, called the rise. The slope is then calculated as the ratio
For example, if you picked the x-intercept (3,0) and the y-intercept (0,-6) as the two points, then
which is 2, as expected.
Recall 5: (The y-intercept)
The y-intercept has a special use in determining the function from the graph. If the y-intercept and the slope are known, then the function is also known. All linear functions may be written as
f(x) = ax + b.
We have seen (in the last Recall) that the coefficient of the x term is the slope of the straight line that is the graph of the function. What is the significance of the constant term b? To answer, simply set x equal to 0 and notice that we have f(0) = b. Therefore, b is the value of y at which the line crosses the y axis--in other words, b is the y value of the y-intercept. This means that by knowing two quantities--the slope a and the y-intercept b--we may write the function that describes the line. The function is
f(x) = ax + b.
For example, if a line has a slope of 2 and a y-intercept of (0,-6), then the line is the graph of the function
f(x) = 2x - 6.
Recall 6: (How to find the line, knowing the slope and a point)
The slope of a line may be computed knowing any two points (x0,y0) and (x1,y1). We have seen that slope is given by the ratio
Therefore, if we know the slope--call it m--and if we know one point--say (x0,y0), then take any point (x,y) on the line and compute the slope once more. We get
This is the equation of the line with slope m and passing through the point (x0,y0)
Thus, we have the function f(x) = m(x - x0)+y0 whose graph is the straight line of slope m passing through (x0,y0).
Recall 7: (How to find the line knowing only two points)
If we know two points (x0,y0) and (x1,y1), then we can find the slope of the line going through these two poins by
m =(y1-y0)/(x1-x0) .
From the last Recall, we know how to find the equation of the line by knowing the slope and one point. Now we know the slope and the point (x0,y0), and we may proceed as we did in Recall 6.
Example 1. Find the x- and y-intercepts of the graph of the function
f(x) = 4x + 12.
Solution: The y-intercept
point (x,f(x)) having the x value equal to 0.
Therefore, we are looking for (0,f(0)). We must compute f(0). It is f(0) = 4(0) + 12 = 12. The y-intercept is (0,12).
The x-intercept is the point (x,f(x)) having the y value equal to 0. Set f(x) = 0 to get 0 = 4x + 12, or x = -3. The x-intercept is, therefore, (-3,0).
Example 2. Find the slope of the function
f(x) = -4x - 3.
Solution: We know that any function that can be put in the form
f(x) = mx + b
must have a graph that is a straight line with slope m. Therefore, the slope of f(x) is m = -4.
To see that the line with the equation given by the function f(x) = mx + b has slope m, check the rise per unit run ratio. Consider (x,f(x)) and (x+1,f(x+1)) and find
which is equal to
which equals m.
Example 4. Sketch the graph of the function
f(x) = (x-16)/2.
Solution: Rewrite this function in standard form as
f(x) = x/2 - 8.
Notice that this function has no higher powers of x than 1. Therefore, the graph of this function is a straight line. Compare the form of this function to the form
f(x) = mx + b,
where m is the slope of the
line and b is the y-intercept.
We find that m =1/2 and b =
-8. Knowing that the slope is 1/2
and that the y-intercept is -8, we
sketch the line below. We find the x-intercept
solving the equation
x/2-8=0. We get the point (16, 0). So the graph is
the line that goes
through the points (0, -8) and (16, 0)
Example 4. Find the function whose graph is the straight line passing through the two points (-12,-8) and (5,9). Where does this curve cross the y- axis? Where does it cross the x-axis?. Sketch the line through the two points (-12,-8) and (5,9) to be sure that it does cross the axes at the computed x and y intercepts.
Solution: First let's find the slope with the formula given in Recall 4, ; we get m=1.
Now we have a
slope and a point, namely slope 1 and point
(5,9). So the equation of the
y = x + 4.
To complete the example, notice that it crosses the y-axis
(0,4) -- (the y-intercept).
It crosses the x-axis when y = 0, or at (-4,0) (the x-intercept)
Notice that by comparing the form of the equation with
the form of
a function whose graph is the line with slope m and y-intercept
b, we may
check that our result is correct.
1. Sketch the graph of 6(x +3) = 2y - 4 by plotting the points (-4,f(-4)), (-3,f(-3)), (-2,f(-2)), (-1,f(-1)), (0,f(0)), (1,f(1)), (2,f(2)), 3,f(3)), and (4,f(4)).
2. Find the equation of the line that has slope 3 and y-intercept -8.
16. Sketch the line x = 8.
2. y = 3x -
3. y = 3x +
4. y = -2x -
5. y = -3x -
6. y = x -
7. (-7,0) and (0,28)