By Harley Flanders; Justin J Price
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Extra info for Calculus with analytic geometry
L s x s 0. )' - -2x2 - 6x + I . 37 Show that the graph of y = ax2 + bx passes through the origin for all choices of a and b. 38 For what value of c does the lowest point of the graph of y = x2 + 6x + c fall on the x-axis? 39 Under what conditions is the lowest point of the graph of y = x2 + bx + c on the y-axis? 40 Show that the rectangle of given perimeter p and largest area is a square. 41 Show that for 0 s x s l, the product x(l - x) never exceeds ! 42 A farmer will make a rectangular pen with 100 ft of fencing.
Consider Fig. 2. For each point (x, y) on the curve y = g(x), there corresponds a point (x - c, y) on the curve y = /(x). The values of y are the same. But on the first curve y = g(x� on the second, y =f (x - c). Conclusion : g(x) =f (x c). This makes sense. If x represents time, then the value of g "now" is the same as the value off at c time units ago. - 28 1 . \' Fla. 2 Horizontal shift (c > 0) The same reasoning shows that the graph or y = f(x + c) is the graph or y = /(x) shirted c units to the lert.
X - r,,r· y= (x - s,r•(x - Szr2 • · ' (x - Str• . We assume this expression is in lowest terms, hence none of the numbers same as any of the numbers sJ . Suppose r is one of the zeros of the numerator. Write r; is the y = g(x)(x - rr. where g(x) is composed of all the other factors of the numerator and denominator lumped together. Note that g(r) + 0. If g(r) = c, then near x = r the graph is like that of y = c(x - rr. Similarly, near a zero s of the denominator, the graph is like that of y d/(x - sr.