Lesson 1 · from §97 of the 1910 text
Functions and the Graph of a Linear Function
Functions
An algebraic expression involving one or more letters is a function of the letter or letters involved.
Thus 2x + 3 and x^2 + 5x - 6 are functions of one letter, x; x^2 - 2xy + y^2 and x^3 + y^3 are functions of two letters, x and y.
The letters of a function are usually referred to as variables. A function is called linear, quadratic, or cubic according as its degree with respect to the variable (or variables) is first, second, or third respectively.
After a function of any variable, say x, has once been given, it is convenient to refer to it later in the same discussion by the symbol f(x), which is read the function of x, or more briefly, f of x.
Graph of a linear function
The numerical value of a linear function of x changes with every change in the variable. If x = 1, the linear function 2x + 3 equals 2 + 3, or 5; if x = 2, 2x + 3 equals 4 + 3, or 7. The following table illustrates the change further.
| x = | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| f(x), 2x + 3 = | -3 | -1 | 1 | 3 | 5 | 7 | 9 |
The relation between x and the function 2x + 3 may be represented graphically if the x-axis is kept and the vertical axis becomes the function axis, the axis of f(x). Beginning with (-3, -3) and plotting the points that correspond to the numbers in the table, we locate the points A, B, C, D, E, G, and H. The graph of the function f(x) = 2x + 3 is evidently the straight line AH:
This process of plotting the relation between a function and the variable contained in it is called graphing the function. Writing y for f(x), it is the graph of the equation
y = 2x + 3
Care should always be taken to join the plotted points by a smooth curve or by a straight line, as the case may be. If the graph is not regular and graceful, it is almost certain that an error has been made in plotting the points. No equations in this book have graphs that are part straight line and part curve, or that change their curvature erratically.
Reading roots from the graph
The graph answers questions about equations in x:
- Where the line crosses the x-axis, f(x) = 0. The line crosses at x = -1\tfrac12, so -1\tfrac12 is the root of 2x + 3 = 0.
- To solve 2x + 3 = 4, find the point of the line at the height 4 and read its x-distance.
Readings from a graph are only approximate; small differences are accepted in the answers, and you can always check a reading by substitution.
Check your graph with the calculator. Open the calculator's Graph tab and type the function with y in place of f(x), for example y = 2x + 3, or press the Graph button beside an equation in the lesson. Zoom in where the curve meets the x-axis to read a root more closely. Draw the graph on paper first, though: the point is to learn to read it yourself.
Give a root as a number, like -3/2 or -1.5. Try one:
Quick check
From the graph of f(x) = 2x + 3, read the value of x when f(x) = 7.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Exercises 1–8 refer to the graph of f(x) = 2x + 3 in the lesson.
Read from the graph the value of x when f(x) is zero.
Set 2x + 3 equal to zero, and solve.
Compare the results of Exercises 1 and 2. Are they the same?
Read from the graph the value of x when f(x) is 4.
Set 2x + 3 equal to 4, and solve.
Compare the results of Exercises 4 and 5. Are they the same?
Can the value of x in 2x + 3 = 6 be read from the graph? If so, read it.
Read from the graph the root of 2x + 3 = -2.
Graph the function f(x) = 5 - 2x. First give the values of the function for the table, then plot the points and draw the line.
Can the root of 5 - 2x = 0 be read from the graph just obtained? If so, read it.
From the graph of Exercise 9 read the roots of: (a) 5 - 2x = 9; (b) 5 - 2x = 5; (c) 5 - 2x = -3; (d) 5 - 2x = -7.
Check by substitution the roots obtained in Exercise 11: find the value of 5 - 2x for each root.
What kind of a line do you expect the graph of any linear function of x to give? Is it always a straight line?