Alright, let's walk through the steps to solve this problem!
### Part (a) Completing the Table
We need to calculate the values of [tex]\( y \)[/tex] for the quadratic equation [tex]\( y = 2x^2 - 5x + 1 \)[/tex] at each given [tex]\( x \)[/tex] from [tex]\(-3\)[/tex] to [tex]\(5\)[/tex].
Here is the completed table for the values of [tex]\( y \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y = 2x^2 - 5x + 1 \\
\hline
-3 & 34 \\
-2 & 19 \\
-1 & 8 \\
0 & 1 \\
1 & -2 \\
2 & -1 \\
3 & 4 \\
4 & 13 \\
5 & 26 \\
\hline
\end{array}
\][/tex]
### Part (b) Drawing the Graph
To draw the graph, we would typically use a graph paper with the given scales.
- X-axis scale: [tex]\(2 \text{ cm}\)[/tex] per unit.
- Y-axis scale: [tex]\(2 \text{ cm}\)[/tex] for [tex]\(10 \text{ units}\)[/tex], simplifying to [tex]\(0.2 \text{ cm}\)[/tex] per unit.
By plotting each point from the table above on the graph paper and connecting them smoothly, we obtain the parabola representing [tex]\( y = 2x^2 - 5x + 1 \)[/tex].
### Part (c) Drawing the Graph of [tex]\( y = x + 6 \)[/tex]
Next, we plot the graph of the linear equation [tex]\( y = x + 6 \)[/tex].
Here is the table of values for [tex]\( y \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y = x + 6 \\
\hline
-3 & 3 \\
-2 & 4 \\
-1 & 5 \\
0 & 6 \\
1 & 7 \\
2 & 8 \\
3 & 9 \\
4 & 10 \\
5 & 11 \\
\hline
\end{array}
\][/tex]
Plot these points on the same graph and draw a straight line through them to represent [tex]\( y = x + 6 \)[/tex].
### Part (d) Estimating the Least Value of [tex]\( y \)[/tex] and Corresponding [tex]\( x \)[/tex]
From the quadratic graph [tex]\( y = 2x^2 - 5x + 1 \)[/tex], we can see that the minimum value of [tex]\( y \)[/tex] is:
[tex]\[
\text{Minimum } y = -2 \quad \text{when } x = 1
\][/tex]
### Solving the Equation [tex]\( 2x^2 - 5x + 1 = x + 6 \)[/tex]
To solve the equation for points of intersection:
[tex]\[
2x^2 - 5x + 1 = x + 6
\][/tex]
Rearrange to form a quadratic equation:
[tex]\[
2x^2 - 6x - 5 = 0
\][/tex]
The solutions for [tex]\( x \)[/tex] (i.e., the roots of the quadratic equation) are:
[tex]\[
x \approx 3.679 \quad \text{and} \quad x \approx -0.679
\][/tex]
Thus, the solutions to the equation [tex]\( 2x^2 - 5x + 1 = x + 6 \)[/tex] are approximately [tex]\( x = 3.679 \)[/tex] and [tex]\( x = -0.679 \)[/tex].
This completes the detailed step-by-step solution.
