A student is investigating the curve \(y = 2x^2 - 5\) before drawing its graph. The table below shows some values of \(x\) and the corresponding values of \...
Question 1 Report
A student is investigating the curve \(y = 2x^2 - 5\) before drawing its graph. The table below shows some values of \(x\) and the corresponding values of \(y\).
| \(x\) | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 3 | -5 | -3 | 13 |
- Complete the table of values. (2)
- Write down the two values of \(x\) in the table for which \(y = 3\). (1)
- Give a reason why \(y\) has the same value when \(x = -2\) and when \(x = 2\). (1)
Answer Details
This question tests completing a table of values for a quadratic function and using the symmetry of \(x^2\) to explain why two different \(x\)-values give the same \(y\)-value.
| \(x\) | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| \(y\) | 3 | -3 | -5 | -3 | 3 | 13 |
- Substituting into \(y=2x^2-5\): when \(x=-1\), \(y=2(1)-5=\) -3; when \(x=2\), \(y=2(4)-5=\) 3 [2].
- From the completed table, \(y=3\) when \(x=-2\) and \(x=2\) [1].
- Squaring removes the sign of a number, so \((-2)^2\) and \(2^2\) are both equal to 4; since only \(x^2\) appears in the formula, \(x=-2\) and \(x=2\) give the same value of \(y\) [1].
Any function built only from even powers of \(x\) (like \(x^2\) here) is symmetrical about the \(y\)-axis; recognising this saves recalculating: once \(y\) is known for a positive \(x\), the same \(y\) automatically holds for \(-x\).
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