CSEC Math Jan 2012 P2 Q6

CSEC Math Jan 2012 Paper 2 Question 6

CSEC Math Jan 2012 Paper 2 Question 6

Updated on April 16th, 2012: Added a video explanation

The table below shows corresponding values of \(x\) and \(y\) for the function \(y = x^2 – 2x – 3\), for integer values of \(x\) from -2 to 4.

A table showing the corresponding values of x and y for the given function

A table showing the corresponding values of x and y for the given function

For part (a), we need to find the value of \(y\) when \(x = -1\) and when \(x = 2\).

When \(x= -1\),

$$
\begin{eqnarray}
y &=& (-1)^2 – 2(-1) – 3 \\
&=& 1 + 2 – 3 \\
&=& 0.
\end{eqnarray}
$$

When \(x= 2\),

$$
\begin{eqnarray}
y &=& (2)^2 – 2(2) – 3 \\
&=& 4 + 4 – 3 \\
&=& -3.
\end{eqnarray}
$$

In part (b) we are asked plot the points whose \(x\) and \(y\) values are recorded in the table above, and to draw a smooth curve through the points. The graph below uses a scale of 2 cm to represent 1 unit on the \(x\)-axis, and 1 cm to represent 1 unit on the \(y\)-axis.

A graph showing a smooth curve through the points in the table

A graph showing a smooth curve through the points in the table

For part (c), using the graph we estimate that the value of \(y\) when \(x = 3.5\) is approximately \(2.2\) (N.B. The exact answer is \(y = 2.25\)).

Using the graph to estimate the value of y at x = 3.5

Using the graph to estimate the value of y at x = 3.5

Finally, in part (d) we make the following observations:

  1. The equation of the axis of symmetry of the graph is \(x = 1\).
  2. \(-4\) is the minimum value of the function \(y\).
  3. When \(x^2 – 2x – 3 = 0\), i.e. \(y = 0\) then from the table we see that \(x = -1\) or \(x = 3\).