Lesson 2 · from §98 of the 1910 text
Graph of a Quadratic Function
Graph of a quadratic function
The quadratic function f(x) = 4x^2 - 4x - 15 may be graphed as follows. When x = 1, 4x^2 - 4x - 15 = 4 - 4 - 15 = -15, or f(x) = -15. In like manner the other numbers in this table can be obtained:
| x = | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|
| f(x), 4x^2 - 4x - 15 = | 33 | 9 | -7 | -15 | -15 | -7 | 9 | 33 |
To represent these numbers conveniently, it is necessary to use different scales for x and f(x): one space on the x-axis is one unit, but one space on the function axis is four units. If we begin with (-3, 33) and plot the points, we get the points A, B, C, D, E, G, H, and I. Drawing a smooth curve through them gives the graph below. This curve is called a parabola.
It is often best to represent the values of f(x) on a different scale from the values of x. Always inspect the table of values and decide what scale to use before plotting a single point. The scale should be as large as possible and yet show enough of the curve to indicate its shape clearly. It will often be convenient not to put the intersection of the axes in the center of the page.
Solving a quadratic equation graphically
The curve crosses the x-axis at x = -1\tfrac12 and x = 2\tfrac12. There f(x) = 0, so these are the roots of 4x^2 - 4x - 15 = 0. Factoring gives the same roots: (2x + 3)(2x - 5) = 0.
To solve 4x^2 - 4x - 15 = 9, read the x-distances of the points of the curve at the height 9; or transpose first, 4x^2 - 4x - 24 = 0, and graph that function, reading where it crosses the x-axis. Either way, a quadratic equation can have two roots.
Some quadratic equations, like x^2 - 2x - 4 = 0, cannot be solved by the factoring you have learned so far. Their roots are irrational, but the graph still gives their approximate values. Answers read from a graph are accepted if they are within about a tenth of the true value.
Writing y for f(x), the parabola above is the graph of
y = 4x^2 - 4x - 15
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.
Type the roots separated by commas, like -3/2, 5/2. Try one:
Quick check
From the graph of f(x) = 4x^2 - 4x - 15, read the roots of 4x^2 - 4x - 15 = 33.
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, 4, 7, and 9 refer to the graph of f(x) = 4x^2 - 4x - 15 in the lesson.
Read from the graph the values of x for which 4x^2 - 4x - 15 equals zero.
Set 4x^2 - 4x - 15 equal to zero, and solve by factoring.
Compare the answers to Exercises 1 and 2. Are they the same?
Read from the graph the values of x for which f(x) equals 20.
What, then, are the roots of 4x^2 - 4x - 15 = 20?
Solve 4x^2 - 4x - 15 = 20 by factoring: write 4x^2 - 4x - 35 as a product of two factors.
Read from the graph the roots of: (a) 4x^2 - 4x - 15 = 9; (b) 4x^2 - 4x - 15 = -7.
Check your answers to Exercise 7 by solving the equations (a) and (b) by factoring. Transposed, they are 4x^2 - 4x - 24 = 0 and 4x^2 - 4x - 8 = 0. Factor each first member.
Can you read from the graph the value of x which makes f(x) equal -25? equal -20? Explain: what is the least value f(x) reaches?
Graph the function x^2 - 2x - 4. First fill out the table (the book gives f(-1) = -1, f(0) = -4, f(1) = -5); then plot the eleven points and draw through them as smooth a curve as possible.
Is the curve obtained in Exercise 10 similar in shape to that of the graph in the lesson?
Read from the graph of Exercise 10 the approximate values of x which make the function x^2 - 2x - 4 equal zero.
What are the roots of the equation x^2 - 2x - 4 = 0? (Give them approximately, from your graph.)
Can you solve the equation x^2 - 2x - 4 = 0 by factoring (with whole numbers)? graphically?
If the terms of a quadratic equation are transposed so that the second member is zero, and then the function in the first member is graphed, can the roots of the original equation be read from the curve thus obtained?
Solve Exercises 16 and 17 graphically and check each by substituting in the original equation; or solve by factoring and compare.
x^2 - 3x = 4
x^2 - 4x + 4 = 0
What peculiarity has the curve obtained in Exercise 17? How many roots has x^2 - 4x + 4 = 0? What values of x make x^2 - 4x + 4 equal zero? equal to +4? to -1? to -10?
Exercises 19 and 20 cannot be solved by the factoring previously explained. They have irrational roots, but the approximate values of the roots can be obtained graphically. Solve them graphically.
x^2 - 2x = 2
x^2 + 2x = 5
Graph the linear function 3x + 4, using the same scale for x and f(x); then graph the function using different scales. Compare the two lines obtained and the values of x and f(x) where each graph crosses the x-axis and the function axis.
Proceed as in Exercise 21 with the quadratic function x^2 + 3x + 1: where does its graph cross the x-axis, and where the function axis?