(a) Figure 1(c) in the text shows a graph of the equation y = x2 + 3x − 5. Use a graphing utility to reproduce the graph. [Use the same viewing rectangle that is used in Figure 1(c).]
(b) Add the graph of the equation y = x2 − 3x − 5 to the picture that you obtained in part (a). (This new equation is the same as the one in part (a) except that the sign of the coefficient of x has been reversed.) Note that the x-intercepts of the two graphs appear to be negatives of one another.
(c) Use the quadratic formula to determine exact expressions for the roots of the two equations x2 + 3x − 5 = 0 and x2 − 3x − 5 = 0. You’ll find that the roots of one equation are the opposites of the roots of the other equation. [In general, the graphs of the two equations y = ax2 + bx + c and y = ax2 − bx + c are symmetric about the y-axis. Thus, the x-intercepts (when they exist) will always be opposites of one another.]
Figure 1(c) the graph of y = x2 + 3x − 5 has two x-intercepts; the equation x2 + 3x − 5 = 0 has two real roots.
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