The problem is divided into three parts. Part (a) requires completing a table of values for the equation $y = 2x^2 - 7x - 9$ for $x$ ranging from -3 to 6. Part (b) asks to draw the graph of this equation using given scales on the x and y axes. Part (c) asks to use the graph to estimate the roots of the equation $2x^2 - 7x = 26$, the coordinates of the minimum point of y, and the range of values for which $2x^2 - 7x < 9$.

1. Problem Description

The problem is divided into three parts. Part (a) requires completing a table of values for the equation $y = 2x^2 - 7x - 9$ for $x$ ranging from -3 to

6. Part (b) asks to draw the graph of this equation using given scales on the x and y axes. Part (c) asks to use the graph to estimate the roots of the equation $2x^2 - 7x = 26$, 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 find the values of $y$ for $x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6$. We already have some values. Let's calculate the missing ones:

* $x = -3$: $y = 2(-3)^2 - 7(-3) - 9 = 2(9) + 21 - 9 = 18 + 21 - 9 = 30$. Given as not correct. Recalculating: $y = 2(-3)^2 - 7(-3) - 9 = 18 + 21 - 9 = 30$. The image must be wrong and the table entry is indeed

3

0. * $x = -2$: $y = 2(-2)^2 - 7(-2) - 9 = 2(4) + 14 - 9 = 8 + 14 - 9 = 13$. Given as correct.

* $x = -1$: $y = 2(-1)^2 - 7(-1) - 9 = 2(1) + 7 - 9 = 2 + 7 - 9 = 0$.

* $x = 0$: $y = 2(0)^2 - 7(0) - 9 = -9$. Given as correct.

* $x = 1$: $y = 2(1)^2 - 7(1) - 9 = 2 - 7 - 9 = -14$. Given as correct.

* $x = 2$: $y = 2(2)^2 - 7(2) - 9 = 2(4) - 14 - 9 = 8 - 14 - 9 = -15$.

* $x = 3$: $y = 2(3)^2 - 7(3) - 9 = 2(9) - 21 - 9 = 18 - 21 - 9 = -12$. Given as correct.

* $x = 4$: $y = 2(4)^2 - 7(4) - 9 = 2(16) - 28 - 9 = 32 - 28 - 9 = -5$.

* $x = 5$: $y = 2(5)^2 - 7(5) - 9 = 2(25) - 35 - 9 = 50 - 35 - 9 = 6$. Given as correct.

* $x = 6$: $y = 2(6)^2 - 7(6) - 9 = 2(36) - 42 - 9 = 72 - 42 - 9 = 21$.

So the completed table is:

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:

The scale is 2 cm to 1 unit on the x-axis and 2 cm to 4 units on the y-axis. Plot the points from the completed table and draw a smooth curve through them.

(c) Using the graph to estimate:

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

This is equivalent to $2x^2 - 7x - 26 = 0$. Since the graph is $y = 2x^2 - 7x - 9$, we need to find the x-values where $y = 2x^2 - 7x - 9 = 26 - 9 = 17$. From the graph, estimate the x-values where $y = 17$. These values are approximately $x = -2.5$ and $x = 6$.

Therefore, the roots are approximately -2.5 and

6.

(ii) Coordinates of the minimum point of y:

From the graph, estimate the coordinates of the minimum point. The minimum point occurs at approximately $x = 1.75$. The corresponding y-value is approximately $y = -15.1$.

Therefore, the coordinates of the minimum point are approximately (1.75, -15.1).

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

This is equivalent to $2x^2 - 7x - 9 < 0$. Since the graph is $y = 2x^2 - 7x - 9$, we need to find the x-values where $y < 0$. This corresponds to the portion of the graph below the x-axis. The x-intercepts are the roots of $2x^2 - 7x - 9 = 0$. From the table we have $y=0$ when $x=-1$. Then solve for $2x^2-7x-9 =0$ using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \frac{7 \pm \sqrt{49 - 4(2)(-9)}}{4} = \frac{7 \pm \sqrt{49 + 72}}{4} = \frac{7 \pm \sqrt{121}}{4} = \frac{7 \pm 11}{4}$

$x = \frac{7 + 11}{4} = \frac{18}{4} = 4.5$ or $x = \frac{7 - 11}{4} = \frac{-4}{4} = -1$.

The roots are -1 and 4.

5. Therefore, $2x^2 - 7x - 9 < 0$ when $-1 < x < 4.5$.

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 (Not provided, requires plotting)

(c)

(i) Roots of $2x^2 - 7x = 26$: Approximately -2.5 and

6. (ii) Coordinates of the minimum point of y: Approximately (1.75, -15.1).

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