Theory question 113
Graphs of FunctionsBelow is the graph of the relation y = ax2 + bx + c, where a, b, c, are constants. Use the graph to: (a) Find the roots of the quadratic equation ax2 + bx + c = 0; (b) Determine the values of constants a, b, and c in the relation using the values of the coordinates P, Q and hence write down the relation illustrated in the graph; (c) Find the maximum value of y and the corresponding value of x at this point (d) Find the values of x when y = 2.
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(a) The curve cuts the x-axis at P(−1.5, 0) and Q(2, 0): roots x = −1.5 and x = 2.
(b) y = a(x + 1.5)(x − 2). The curve meets the y-axis at (0, 6), so 6 = a(1.5)(−2) and a = −2.
y = −2(x + 1.5)(x − 2) = −2x2 + x + 6: a = −2, b = 1, c = 6.
(c) Maximum at x = −b/2a = 0.25: y = 6.125 (about 6.1 from the graph).
(d) −2x2 + x + 6 = 2 gives 2x2 − x − 4 = 0, x ≈ 1.7 or x ≈ −1.2.
Known roots let you write y = a(x − r₁)(x − r₂); one more point gives a.