algebra problems

number fitting

vegetable relationships

trial and improvement

not a nice question but it does highlight a need to work to two decimal places if asked to give an answer to 1 d.p.

between 2 and 3 find a solution to 1 d.p. to:

substituting,

x = 2.1 gives y = -3.969

x = 2.2 gives y = -3.872

x = 2.1 is just slightly closer to the required y value (of -3.921)
being 0.048 off it compared with 0.049
since it is nearer then it might be tempting to say the solution to 1 d.p. is 2.1

but, the solution to 1 d.p is actually 2.2 and not 2.1

substituting x = 2.15 (as required in an exam question) gives y = -3.929125 so the answer is bigger than 2.15 (actually around 2.158...) which rounds to 2.2

moral: don't jump to conclusions when you are dealing with non-linear functions

exam boards sensibly steer clear of cubic functions with an x squared term

for (another un-nice question) between 3 and 4 find a solution to 1 d.p. to:

then you can obtain two solutions
x = 3.3 and x = 3.4 are both correct

(actual values are 3.29188523551653 and 3.3744064395661)

greatest value

use a spreadsheet or a calculator or zoom in using graphing software to work out the greatest value of y if:

this work is intended to highlight a need to sometimes look beyond initial appearances....

the first question is relatively straightforward,
with a maximum value that seems to be (for integers) 2
but is actually a maximum y = 2.25 (when x = 2.5, as some students might know)

for the second question, a quartic, the value 24 seems to be the highest value for y

the y-values are (designed to be) 24 for x = 1, 2, 3 and 4:

the intention is that students learn not to accept these initial highest values but to dig deeper

 to find the highest value is actually 25













exploring the shape of the quartic function, it comes up through (0,0) then has two peaks, like a tooth, symmetrical about and with a minimum at x = 2.5

then goes back down to negative values through (5,0)

the first maximum is at x = 1.38196601125... and the second is at 5 minus this value, both giving the same y value = 25

you get to this first highest value of y around x = 1.38966 (using a spreadsheet set to 30 decimal places)