Answer :
Certainly! Let's break down each part of the question and work through the solutions step-by-step.
### Part (a) Draw the Graph
To draw the graph of [tex]\( y = x^2 - 6x + 7 \)[/tex] for [tex]\( 0 \leq x \leq 6 \)[/tex], we need to:
1. Create a table of values for [tex]\( x \)[/tex] and the corresponding [tex]\( y \)[/tex] values.
2. Plot these points on a coordinate plane.
3. Draw a smooth curve that fits the plotted points.
Here is the table of values:
[tex]\[ \begin{array}{c|c} x & y = x^2 - 6x + 7 \\ \hline 0 & 7 \\ 1 & 1^2 - 6 \cdot 1 + 7 = 2 \\ 2 & 2^2 - 6 \cdot 2 + 7 = -1 \\ 3 & 3^2 - 6 \cdot 3 + 7 = -2 \\ 4 & 4^2 - 6 \cdot 4 + 7 = -1 \\ 5 & 5^2 - 6 \cdot 5 + 7 = 2 \\ 6 & 6^2 - 6 \cdot 6 + 7 = 7 \\ \end{array} \][/tex]
Now, plot these points and draw a smooth parabola through them. The graph is a parabola that opens upward.
### Part (b) Use the Graph to Find:
(i) The value of [tex]\( y \)[/tex] when [tex]\( x = 5.2 \)[/tex]
To find the value of [tex]\( y \)[/tex] when [tex]\( x = 5.2 \)[/tex]:
- Substitute [tex]\( x = 5.2 \)[/tex] in the equation [tex]\( y = x^2 - 6x + 7 \)[/tex].
[tex]\[ y = (5.2)^2 - 6(5.2) + 7 \][/tex]
Calculate step-by-step:
[tex]\[ y = 27.04 - 31.2 + 7 \][/tex]
[tex]\[ y = 34.04 - 31.2 \][/tex]
[tex]\[ y = 2.84 \][/tex]
Therefore, [tex]\( y \)[/tex] is approximately [tex]\( 2.84 \)[/tex] when [tex]\( x = 5.2 \)[/tex].
(ii) The value of the y-intercept of the graph
The y-intercept occurs when [tex]\( x = 0 \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] in the equation [tex]\( y = x^2 - 6x + 7 \)[/tex]:
[tex]\[ y = 0^2 - 6(0) + 7 \][/tex]
[tex]\[ y = 7 \][/tex]
Thus, the y-intercept is [tex]\( 7 \)[/tex].
(iii) The minimum value of [tex]\( y \)[/tex] and the value of [tex]\( x \)[/tex] at which this occurs
The equation [tex]\( y = x^2 - 6x + 7 \)[/tex] is a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 7 \)[/tex]. The vertex of a parabola given by this form occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{-6}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{6}{2} \][/tex]
[tex]\[ x = 3 \][/tex]
Now, substitute [tex]\( x = 3 \)[/tex] back into the equation to find the minimum value of [tex]\( y \)[/tex]:
[tex]\[ y = (3)^2 - 6(3) + 7 \][/tex]
[tex]\[ y = 9 - 18 + 7 \][/tex]
[tex]\[ y = -2 \][/tex]
Therefore, the minimum value of [tex]\( y \)[/tex] is [tex]\( -2 \)[/tex] and it occurs at [tex]\( x = 3 \)[/tex].
### Summary
1. Graph has been plotted for [tex]\( y = x^2 - 6x + 7 \)[/tex] for [tex]\( 0 \leq x \leq 6 \)[/tex].
2. Values Found:
- [tex]\( y \)[/tex] when [tex]\( x = 5.2 \)[/tex] is approximately [tex]\( 2.84 \)[/tex].
- The y-intercept is [tex]\( 7 \)[/tex].
- The minimum value of [tex]\( y \)[/tex] is [tex]\( -2 \)[/tex] and it occurs at [tex]\( x = 3 \)[/tex].
### Part (a) Draw the Graph
To draw the graph of [tex]\( y = x^2 - 6x + 7 \)[/tex] for [tex]\( 0 \leq x \leq 6 \)[/tex], we need to:
1. Create a table of values for [tex]\( x \)[/tex] and the corresponding [tex]\( y \)[/tex] values.
2. Plot these points on a coordinate plane.
3. Draw a smooth curve that fits the plotted points.
Here is the table of values:
[tex]\[ \begin{array}{c|c} x & y = x^2 - 6x + 7 \\ \hline 0 & 7 \\ 1 & 1^2 - 6 \cdot 1 + 7 = 2 \\ 2 & 2^2 - 6 \cdot 2 + 7 = -1 \\ 3 & 3^2 - 6 \cdot 3 + 7 = -2 \\ 4 & 4^2 - 6 \cdot 4 + 7 = -1 \\ 5 & 5^2 - 6 \cdot 5 + 7 = 2 \\ 6 & 6^2 - 6 \cdot 6 + 7 = 7 \\ \end{array} \][/tex]
Now, plot these points and draw a smooth parabola through them. The graph is a parabola that opens upward.
### Part (b) Use the Graph to Find:
(i) The value of [tex]\( y \)[/tex] when [tex]\( x = 5.2 \)[/tex]
To find the value of [tex]\( y \)[/tex] when [tex]\( x = 5.2 \)[/tex]:
- Substitute [tex]\( x = 5.2 \)[/tex] in the equation [tex]\( y = x^2 - 6x + 7 \)[/tex].
[tex]\[ y = (5.2)^2 - 6(5.2) + 7 \][/tex]
Calculate step-by-step:
[tex]\[ y = 27.04 - 31.2 + 7 \][/tex]
[tex]\[ y = 34.04 - 31.2 \][/tex]
[tex]\[ y = 2.84 \][/tex]
Therefore, [tex]\( y \)[/tex] is approximately [tex]\( 2.84 \)[/tex] when [tex]\( x = 5.2 \)[/tex].
(ii) The value of the y-intercept of the graph
The y-intercept occurs when [tex]\( x = 0 \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] in the equation [tex]\( y = x^2 - 6x + 7 \)[/tex]:
[tex]\[ y = 0^2 - 6(0) + 7 \][/tex]
[tex]\[ y = 7 \][/tex]
Thus, the y-intercept is [tex]\( 7 \)[/tex].
(iii) The minimum value of [tex]\( y \)[/tex] and the value of [tex]\( x \)[/tex] at which this occurs
The equation [tex]\( y = x^2 - 6x + 7 \)[/tex] is a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 7 \)[/tex]. The vertex of a parabola given by this form occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{-6}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{6}{2} \][/tex]
[tex]\[ x = 3 \][/tex]
Now, substitute [tex]\( x = 3 \)[/tex] back into the equation to find the minimum value of [tex]\( y \)[/tex]:
[tex]\[ y = (3)^2 - 6(3) + 7 \][/tex]
[tex]\[ y = 9 - 18 + 7 \][/tex]
[tex]\[ y = -2 \][/tex]
Therefore, the minimum value of [tex]\( y \)[/tex] is [tex]\( -2 \)[/tex] and it occurs at [tex]\( x = 3 \)[/tex].
### Summary
1. Graph has been plotted for [tex]\( y = x^2 - 6x + 7 \)[/tex] for [tex]\( 0 \leq x \leq 6 \)[/tex].
2. Values Found:
- [tex]\( y \)[/tex] when [tex]\( x = 5.2 \)[/tex] is approximately [tex]\( 2.84 \)[/tex].
- The y-intercept is [tex]\( 7 \)[/tex].
- The minimum value of [tex]\( y \)[/tex] is [tex]\( -2 \)[/tex] and it occurs at [tex]\( x = 3 \)[/tex].
