(a) Copy and complete the table of values for the relation \(y = 3x^{2} - 5x - 7\). x -3 -2 -1 0 1 2 3 4 y 35 -7 -9 5 (b) Using scales of 2 cm to 1 unit on ...
Question 1 Report
(a) Copy and complete the table of values for the relation \(y = 3x^{2} - 5x - 7\).
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 35 | -7 | -9 | 5 |
(b) Using scales of 2 cm to 1 unit on the x- axis and 2 cm to 5 units on the y- axis, draw the graph of \(y = 3x^{2} - 5x - 7, -3 \leq x \leq 4\).
(c) From the graph : (i) find the roots of the equation \(3x^{2} - 5x - 7 = 0\) ; (ii) estimate the minimum value of y ; (iii) calculate the gradient of the curve at the point x = 2.
Answer Details
(a) For \(y=3x^{2}-5x-7\):
| \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 35 | 15 | 1 | -7 | -9 | -5 | 5 | 21 |
For example, \(y\) when \(x=-2\) is \(3(-2)^2-5(-2)-7=15\), while when \(x=2\), \(y=3(2)^2-5(2)-7=-5\).
(b) The plotted graph of \(y=3x^2-5x-7\), using the stated scales, is shown below.
(c)(i) The roots are the \(x\)-coordinates where the curve cuts the \(x\)-axis:
\[\boxed{x\approx-0.9\text{ and }x\approx2.6}\]
(c)(ii) The minimum ordinate, read at the turning point, is
\[\boxed{y\approx-9.1}\]
(c)(iii) Using two convenient points on the tangent at \(x=2\), approximately \((1.5,-8.5)\) and \((3.3,4.0)\),
\[\text{gradient}=\frac{4.0-(-8.5)}{3.3-1.5}=\frac{12.5}{1.8}=6.94\approx\boxed{6.9}.\]
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