Theory question 110
Graphs of Functions(a) Copy and complete the table of values for the relation y = −x2 + x + 2 for −3 ≤ x ≤ 3.
x: −3, −2, −1, 0, 1, 2, 3
y: __, −4, __, 2, __, __, −4
(b) Using scales of 2 cm to 1 unit on the x-axis and 2cm to 2 units on the y-axis, draw a graph of the relation y = −x2 + x + 2.
(c) From the graph, find the: (i) minimum value of y; (ii) roots of the equation x2 − x − 2 = 0; (iii) gradient of the curve at x = −0.5.
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(a) x = −3: −10; −1: 0; 1: 2; 2: 0 (given: −2: −4, 0: 2, 3: −4).
(b) Plot the points and draw a smooth curve (it opens downwards).
(c)(i) The curve turns at the top: maximum y = 2.25 at x = 0.5. (Over −3 ≤ x ≤ 3 the least value of y is −10, at x = −3.)
(ii) x2 − x − 2 = 0 is the same as y = 0: x = −1 and x = 2.
(iii) Draw the tangent at x = −0.5: gradient ≈ 2.
x2 − x − 2 = 0 is −(the relation) = 0, so read where the curve cuts the x-axis.