(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.
Explanation
(a)
x
-3
-2
-1
0
1
2
3
4
y
35
15
1
-7
-9
-5
5
21
(b) (c) (i) The roots of the equation is where the graph cuts the x- axis = -0.9, 2.6. (ii) Minimum value of y = -9.21 (iii) At x = 2, gradient = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\) = \(\frac{4 - (-8.5)}{3.3 - 1.5}\) = \(\frac{12.5}{1.8}\) = 6.9444 \(\approxeq\) 6.9
