J17

Range of values (Use graph of example 2.18)
Example
Find the range of values of x for which the function x 2 – 3x + 2
(a) is negative,
(b) is less than (x + 2),
(c) is greater than 1.
Solution
(a) x 2 – 3x + 2 is negative (i.e. below the x-axis) for values of x from 1 to 2. The range
is 1 < x < 2.
(b) x 2 – 3x + 2 is less than x + 2 (i.e. below the line y = x + 2) fro x = 0 to
x = 4, i.e. 0 < x < 4.
(c) In the given domain -1 ≤ x ≤ 4, the function x 2 – 3x + 2 is greater than 1 in two
ranges:
From -1 to 0.4 and from 2.6 to 4,
i.e. -1 < x < 0.4 and 2.6 < x < 4.
Intersecting graphs
Example
Draw the graph of y = 1 + x – 2x 2 , taking values of x in the domain -3 ≤ x ≤ 3.
Using the same scale and axes, draw the graph of y = 2x – 5.
Use your graphs to answer the following questions:
(a) What is the maximum value of the function 1 + x – 2x 2 ?
(b) Write down the x-coordinates of the points of intersection of the functions y = 1
+ x – 2x 2 and y = 2x – 5. Show that these values of x satisfy the equation 2x 2 + x
– 6 = 0.
Table of values for y = 1 + x – 2x 2
x -3 -2 -1 0 1 2 3
1 1 1 1 1 1 1 1
x -3 -2 -1 0 1 2 3
-2x 2 -18 -8 -2 0 -2 -8 -18
y -20 -9 -2 1 0 -5 -14
Table for y = 2x – 5
x -3 0 3
y -11 -5 1
When plotted, a curve and a straight line as shown in the figure below are obtained.