Practical algebra (revised) prepared for the use of the midshipmen at the United States naval academy - Softcover
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ... or of (x + a)2, a being any given number. For instance, suppose we should construct the graph of (x2 + l) independently. For any value of x, the point representing (a + 1) would be one unit higher than the point representing x2; thus the graph of (z2 + l) would be the graph of x2 raised one unit. We can, however, get the same effect more easily by leaving the graph where it is and lowering the rr-axis one unit. In the same way, the graph of (x2--2) can be obtained from the graph of x2 by raising the z-axis two units. Again, suppose we should construct the graph of (z+1)2; in the table of values of x and (rr + 1)2, we should have the same set of values for (z+1)2 that we had for x2, except that the corresponding value of x would be one less. This means that the graph of (x+1)2 is the graph of x2 shifted one unit to the left, or is the graph of x2 with the y-axis shifted one unit to the right. In the same way, the graph of (x--2)2 can be obtained from the graph of x2 by shifting the «/-axis two units to the left. These considerations are perfectly general, and may be applied to any function: The graph of f(x) +a can be obtained from the graph of f(x) by shifting the x-axis a units downward; and The graph of f(x + a) can be obtained from the graph of f(x) by shifting the y-axis a units to the right. If a is negative, the shift is in the opposite direction. Examples. Draw the graphs of the following functions: 1. z2-3. 2. 2x2 + l. 3.-$x-. 4. x1. 5.-2x2. 6. x2 + x3. 7. x-and 2x + 3 on the same axes. How do the graphs show the values of x for which these two functions are equal? 8. 3? and 2x--l on the same axes. How do the graphs show the values of x for which x3--2x +1 = 0? 83. Slope of the Graph.--The steepness or slope of a graph at any point, which is t...
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