Theory question 108
Graphs of FunctionsThe table below is for the relation y = 2 + x − x2
x: −2, −1.5, −1, −0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3
y: −4, −1.75, 0, 1.25, 2, 2.25, 2, 1.25, 0, −1.75, −4
(a) Using a scale of 2cm to 1 unit on each axis, draw the graph of the relation in the interval −2 ≤ x ≤ 3.
(b) From your graph, find the greatest value of y and the value of x for which this occurs,
(c) Using the same scale and axis, draw the graph of y = 1 − x. (d) Use your graph to solve the equation 1 + 2x − x2 = 0.
Show the model answer
(a) Plot the 11 points and join with a smooth curve (an upside-down parabola).
(b) Greatest value y = 2.25, at x = 0.5.
(c) Straight line through (0, 1), (1, 0) and (3, −2).
(d) 2 + x − x2 = 1 − x is the same as 1 + 2x − x2 = 0, so the solutions are the x-values where the curve and the line cross: x ≈ −0.4 and x ≈ 2.4.
Subtract the line's equation from the curve's to see why their crossing points solve the equation.