Theory question 108
InequalitiesUsing a scale of 2cm to 1unit on the x-axis and 1cm to 1unit on the y-axis, draw on the same axes the graphs of y = 3 + 2x − x2 y = 2x − 3 for −3 ≤ x ≤ 4.
Using your graph: (i) Solve the equation 6 − x2 = 0. (ii) find the maximum value of 3 + 2x − x2, (iii) find the range of x for which 3 + 2x − x2 ≤ 1, expressing all your answers correct to one decimal place.
Show the model answer
Table for y = 3 + 2x − x2: x = −3, −2, −1, 0, 1, 2, 3, 4 gives y = −12, −5, 0, 3, 4, 3, 0, −5
Line y = 2x − 3: e.g. (−3, −9), (0, −3), (4, 5)
(i) 6 − x2 = 0 is where the curve meets the line (3 + 2x − x2 = 2x − 3): x ≈ −2.4 or x ≈ 2.4
(ii) Maximum value = 4 (at x = 1)
(iii) Draw y = 1: the curve is on or below it for −3 ≤ x ≤ −0.7 and 2.7 ≤ x ≤ 4
Rearrange the equation to show it is the curve minus the line equal to zero.