The problem consists of three parts: (a) Complete a table of values for the equation $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$. (b) Draw the graph of $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$ using the specified scales. (c) 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:

(a) Complete a table of values for the equation $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$.

(b) Draw the graph of $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$ using the specified scales.

(c) 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) Complete the table of values:

We are given the equation $y = 2x^2 - 7x - 9$. We need to calculate the values of $y$ for $x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6$. We are given the values for $x = -2, 0, 1, 3, 5$. We need to calculate 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 $x = 6$:

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

So, the complete 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) Draw the graph of $y = 2x^2 - 7x - 9$.

Use the values from the table above to plot the graph of $y = 2x^2 - 7x - 9$ for $-3 \le x \le 6$.

The x-axis scale is 2 cm to 1 unit, and the y-axis scale is 2 cm to 4 units.

You will need to plot the points (-3, 30), (-2, 13), (-1, 0), (0, -9), (1, -14), (2, -15), (3, -12), (4, -5), (5, 6), (6, 21).

(c) Use the graph to estimate:

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

Rewrite the equation as $2x^2 - 7x - 26 = 0$. However, we graphed $y = 2x^2 - 7x - 9$, so we must subtract 17 from both sides to make it $y = 2x^2 - 7x - 9 - 17 = 0$, i.e. $y=17$. We need to find the x-values where $2x^2 - 7x - 9 = 17$. The roots of the equation $2x^2 - 7x - 26=0$ are the x-coordinates of the points of intersection of $y = 2x^2 - 7x - 9$ and the line $y = 17$.

From the graph (not shown here, as I cannot draw graphs), we estimate the x-values of intersection to be approximately $x = -2$ and $x = 5.5$.

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

By observing or calculating, you can find the vertex of the parabola $y = 2x^2 - 7x - 9$. To find the x coordinate of the vertex, we use $x = -b/(2a)$ where $a=2$ and $b=-7$. So $x = 7/4 = 1.75$. To find the y coordinate, we substitute this x value into the equation.

$y = 2(1.75)^2 - 7(1.75) - 9 = 2(3.0625) - 12.25 - 9 = 6.125 - 12.25 - 9 = -15.125$.

From the graph (not shown), the minimum point is approximately $(1.75, -15.125)$.

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

Rewrite the inequality as $2x^2 - 7x - 9 < 0$, or $y < 0$. We need to find the range of x-values where the graph of $y = 2x^2 - 7x - 9$ is below the x-axis (i.e. $y < 0$).

From the graph (not shown), the parabola intersects the x-axis at $x = -1$ and $x = 4.5$.

The range of x-values for which $2x^2 - 7x < 9$ is $-1 < x < 4.5$.

3. Final Answer

(a) 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) The graph is not shown here.

(c)

(i) Roots of $2x^2 - 7x = 26$ are approximately $x = -2$ and $x = 5.5$.

(ii) Coordinates of the minimum point of $y$ are approximately $(1.75, -15.125)$.

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