The problem requires us to complete a table of values for the equation $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$. Then, we need to draw the graph of the equation and 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 requires us to complete a table of values for the equation $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$. Then, we need to draw the graph of the equation and 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 $y$ values for $x = -1, 2, 4, 6$.

For $x = -1$:

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

For $x = 2$:

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

For $x = 4$:

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

For $x = 6$:

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

The completed table is:

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

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

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

(b) Drawing the graph:

Using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 4 units on the y-axis, we can plot the points from the completed table and draw a smooth curve through them to represent the graph of $y = 2x^2 - 7x - 9$. Since the graph is not provided here, it's impossible to give an exact answer to part (c). I can only provide a general method.

(c) Estimating from the graph:

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

This is equivalent to solving $2x^2 - 7x - 26 = 0$.

To solve this graphically, we can rewrite the equation as $2x^2 - 7x - 9 = 17$. This means we want to find the $x$ values where the graph $y = 2x^2 - 7x - 9$ intersects the line $y = 17$.

Find the points of intersection of the curve and the horizontal line $y=17$, and the $x$-coordinates of these points are the roots.

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

The minimum point is the vertex of the parabola. Find the coordinates of the lowest point on the graph.

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

This is equivalent to $2x^2 - 7x - 9 < 0$, which means we are looking for the $x$ values where the graph of $y = 2x^2 - 7x - 9$ is below the x-axis ($y=0$). The range of $x$ values between the two roots of the equation $2x^2-7x-9 = 0$ satisfy this inequality.

The roots of $2x^2-7x-9=0$ can be read from the graph where $y=0$. From the completed table in (a) we have x = -1 and x = 4.5 are approximately the roots.

Therefore, $-1 < x < 4.5$.

3. Final Answer

(a) Completed table:

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

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

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

(b) Graph: Not provided.

(c) Estimates from the graph (approximate):

(i) Roots of $2x^2 - 7x = 26$: The x values where y=

1

7. These would need to be read off the actual graph to give a numerical answer.

(ii) Coordinates of the minimum point of $y$: Approximate value would need to be read off the actual graph to give a numerical answer.

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