Which of these points lies on the
curve 𝑦 equals four 𝑥 squared? Circle your answer. The possibilities are a half,
negative one; two, 16; one, 16; or negative three, negative 36.
To answer this question, we could
just substitute each possible 𝑥-value into the equation of the curve and work out
the corresponding 𝑦-value to see which pairs of coordinates do satisfy the correct
We do need to be a little bit
careful with the equation of this curve though. And we need to remember the order
of operations BIDMAS which tells us that indices or powers come before
multiplication. This means that if we’re
calculating four 𝑥 squared, we have to square 𝑥 first and then multiply the result
by four. A common mistake would be to
multiply 𝑥 by four first and then square our answer.
We can see that these two processes
will give different answers if we consider what happens when 𝑥 is equal to one. If we apply BIDMAS correctly, then
we’re squaring one first and then multiplying by four. One squared is equal to one. So we have four multiplied by one
which is equal to four.
If, however, we applied BIDMAS
incorrectly, so we multiplied the one by four first and then squared it, we’d have
four multiplied by one all squared which is equal to four squared which is equal to
Notice that one, 16 is one of the
possibilities we’ve been given but it’s a red herring because remember this was an
incorrect application of BIDMAS. The correct value of 𝑦 when 𝑥 is
one is four not 16. So we can eliminate this point. We can actually eliminate two other
points without any calculation if we consider what the graph of 𝑦 equals four 𝑥
squared looks like. It’s a quadratic curve. And as it has a positive
coefficient of 𝑥 squared, it’s a U-shaped curve.
When we square an 𝑥-value whether
it’s positive or negative, we always get a positive answer or zero. And when we multiply this by four,
we’ll still get a number that’s positive or zero which means that 𝑦 is always
greater than or equal to zero.
So we can eliminate the coordinates
a half, negative one and negative three, negative 36 because they each have negative
𝑦-coordinates. So there’s no way that they can lie
on this curve. This only leaves the point two,
16. But let’s confirm that this does
indeed lie on the curve by substitution.
We can substitute two into the
equation of the curve for 𝑥 and we have four multiplied by two squared. Remember we must square two
first. So we have four multiplied by four
which is equal to 16. This is the correct 𝑦-value for
the pair of coordinates we’ve been given.
So we can conclude that the only
one of the four points which lies on the curve 𝑦 equals four 𝑥 squared is the
point two, 16.