The problem consists of three parts. Part (a) asks to complete a table of values for the equation $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$. Part (b) asks to draw the graph of the equation using given scales. Part (c) asks to use the graph to estimate the roots of $2x^2 - 7x = 26$, the coordinates of the minimum point of $y$, and the range of values for which $2x^2 - 7x < 9$.

1. Problem Description

The problem consists of three parts. Part (a) asks to complete a table of values for the equation $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$. Part (b) asks to draw the graph of the equation using given scales. Part (c) asks to use the graph to estimate the roots of $2x^2 - 7x = 26$, the coordinates of the minimum point of $y$, and the range of values for which $2x^2 - 7x < 9$.

2. Solution Steps

(a) Completing the table of values:

We need to calculate the values of $y$ for $x = -3, -1, 2, 4, 6$.

For $x = -3$:

$y = 2(-3)^2 - 7(-3) - 9 = 2(9) + 21 - 9 = 18 + 21 - 9 = 30$

For $x = -1$:

$y = 2(-1)^2 - 7(-1) - 9 = 2(1) + 7 - 9 = 2 + 7 - 9 = 0$

For $x = 2$:

$y = 2(2)^2 - 7(2) - 9 = 2(4) - 14 - 9 = 8 - 14 - 9 = -15$

For $x = 4$:

$y = 2(4)^2 - 7(4) - 9 = 2(16) - 28 - 9 = 32 - 28 - 9 = -5$

For $x = 6$:

$y = 2(6)^2 - 7(6) - 9 = 2(36) - 42 - 9 = 72 - 42 - 9 = 21$

Completed table:

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6

--|----|----|----|---|---|---|---|---|---|---

y | 30 | 13 | 0 | -9| -14| -15| -12| -5| 6 | 21

(b) Drawing the graph:

Using the table of values, plot the points on a graph with a scale of 2 cm to 1 unit on the x-axis and 2 cm to 4 units on the y-axis. Draw a smooth curve through the points.

Since I cannot draw the graph, I will proceed with part (c) assuming we have the graph.

(c) Using the graph to estimate:

(i) Roots of the equation $2x^2 - 7x = 26$:

Rewrite the equation as $2x^2 - 7x - 26 = 0$. We want to find the values of $x$ when $y = 2x^2 - 7x - 9 = 26 - 9 = 17$. Find the points on the graph where $y = 17$. The x-coordinates of these points are the roots. From the graph (which I cannot provide), let's assume these roots are approximately $x \approx -2.2$ and $x \approx 5.7$.

(ii) Coordinates of the minimum point of $y$:

The minimum point of the graph is the vertex of the parabola. Visually estimate the coordinates of the lowest point on the curve. From the graph (which I cannot provide), let's assume this point is approximately $(1.75, -15.1)$.

(iii) Range of values for which $2x^2 - 7x < 9$:

Rewrite the inequality as $2x^2 - 7x - 9 < 0$, which is equivalent to $y < 0$. Find the points on the graph where $y=0$. These are the roots of the equation $2x^2 - 7x - 9 = 0$. These roots are $x=-1$ and $x=4.5$. The range of values for $x$ for which $y < 0$ is between these roots. So $-1 < x < 4.5$.

3. Final Answer

(a) Completed table:

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6

--|----|----|----|---|---|---|---|---|---|---

y | 30 | 13 | 0 | -9| -14| -15| -12| -5| 6 | 21

(b) Graph (cannot be drawn).

(c) Estimates from the graph (approximate values):

(i) Roots of $2x^2 - 7x = 26$: $x \approx -2.2$ and $x \approx 5.7$

(ii) Coordinates of the minimum point of $y$: $(1.75, -15.1)$

(iii) Range of values for which $2x^2 - 7x < 9$: $-1 < x < 4.5$