(a) Copy and complete the following table of values for the relation \(y = x^{2} - 2x - 5\) x -3 -2 -1 0 1 2 3 4 5 y -2 -6 -2 3 10 (b) Draw the graph of the...
Question 1 Report
(a) Copy and complete the following table of values for the relation \(y = x^{2} - 2x - 5\)
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| y | -2 | -6 | -2 | 3 | 10 |
(b) Draw the graph of the relation \(y = x^{2} - 2x - 5\); using a scale of 2 cm to 1 unit on the x- axis, and 2 cm to 2 units on the y- axis.
(c) Using the same axes, draw the graph of \(y = 2x + 3\).
(d) Obtain in the form \(ax^{2} + bx + c = 0\) where a, b and c are integers, the equation which is satisfied by the x- coordinate of the points of intersection of the two graphs.
(e) From your graphs, determine the roots of the equation obtained in (d) above.
Answer Details
(a) Completing the table for \(y=x^2-2x-5\). For example, when \(x=-3\), \(y=9+6-5=10\), and when \(x=2\), \(y=4-4-5=-5\).
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|
| y | 10 | 3 | -2 | -5 | -6 | -5 | -2 | 3 | 10 |
(b) and (c) The plotted curve is the parabola \(y=x^2-2x-5\). The straight line is \(y=2x+3\), which passes through \((0,3)\) and \((3,9)\).
(d) At a point of intersection, both graphs have the same \(y\)-value:
\[ x^2-2x-5=2x+3 \] \[ x^2-4x-8=0 \](e) The roots are the \(x\)-coordinates of the intersections. From the graph, they are approximately \(x=-1.5\) and \(x=5.5\).
The more accurate solutions are \(x=2\pm2\sqrt{3}\), giving \(x\approx-1.46\) and \(x\approx5.46\). The supplied reference answer uses \(y=2x-3\), but this conflicts with the question stem, which states \(y=2x+3\). Therefore \(x^2-4x-8=0\) is the equation consistent with the stated line.
Download The App On Google Playstore
Everything you need to excel in your exams
