lesson
A changing gradient makes a curve
1 · Learn
The graph y = x² is not a straight line. Squaring negative and positive values of equal magnitude gives the same result, so the graph is symmetric about the y-axis. Its rate of change is not constant.
Plot enough points and draw a smooth curve appropriate to the relation. A straight line joining only two selected points would misrepresent the intermediate values.
Read the supplied graph of y = x². To estimate x when y = 2, start at 2 on the vertical axis, follow the dashed horizontal line to BOTH sides of the curve, then read down to the x-axis. The readings are about -1.4 and 1.4. These are approximate: 1.4² = 1.96, not exactly 2. Symmetry gives a negative and a positive input for this height. At y = 0 the curve meets only once, at x = 0; a negative y has no real input here.
2 · Worked example
At x = -3 and x = 3, y = 9 in both cases. The minimum of y = x² occurs at (0,0), since real squares are non-negative.
3 · Your turn
Find y for x = -4 and x = 2 in y = x².
Check your answer
16 and 4.
4 · Apply your learning
Compare equal steps in x with the changing differences in y and explain why the graph is not linear.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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