Below are graphs of the functions y = x² − 6x + 8 and y − 3x + 6 = 0. a) Coordinates of A. b) X-intercepts of the graph of the function y = x² − 6x + 8. c) Coordinates of point C. d) Average rate of change for the function y = x² − 6x + 8 between points B and C.

Here are the calculations for each part of question 7.

a) Coordinates of A
Point A is an intersection of the parabola $y = x^2 - 6x + 8$ and the line $y - 3x + 6 = 0$.
First, rewrite the equation of the line in the form $y = mx + c$:

$$y = 3x - 6$$

Now, set the two equations equal to each other to find the x-coordinates of the intersection points:

$$x^2 - 6x + 8 = 3x - 6$$ $$x^2 - 6x - 3x + 8 + 6 = 0$$ $$x^2 - 9x + 14 = 0$$

Factor the quadratic equation:

$$(x - 2)(x - 7) = 0$$

This gives two possible x-coordinates: $x = 2$ or $x = 7$.
From the graph, point A is the intersection with the smaller x-coordinate. So, $x_A = 2$.
Substitute $x_A = 2$ into the line equation $y = 3x - 6$ to find $y_A$:

$$y_A = 3(2) - 6$$ $$y_A = 6 - 6$$ $$y_A = 0$$

The coordinates of A are $(2, 0)$.

b) X-intercepts of the graph of the function $y = x^2 - 6x + 8$
X-intercepts occur when $y = 0$. Set the function equal to zero:

$$x^2 - 6x + 8 = 0$$

Factor the quadratic equation:

$$(x - 2)(x - 4) = 0$$

This gives two x-intercepts: $x = 2$ or $x = 4$.
The x-intercepts are $(2, 0)$ and $(4, 0)$.

c) Coordinates of point C
Point C is the other intersection point of the parabola $y = x^2 - 6x + 8$ and the line $y = 3x - 6$.
From part a), the x-coordinates of the intersection points are $x = 2$ and $x = 7$.
Since point A has $x=2$, point C must have $x_C = 7$.
Substitute $x_C = 7$ into the line equation $y = 3x - 6$ to find $y_C$:

$$y_C = 3(7) - 6$$ $$y_C = 21 - 6$$ $$y_C = 15$$

The coordinates of C are $(7, 15)$.

d) Average rate of change for the function $y = x^2 - 6x + 8$ between points B and C.
The function is $f(x) = x^2 - 6x + 8$.
From part c), point C is $(7, 15)$.
Point B is the vertex of the parabola $y = x^2 - 6x + 8$.
The x-coordinate of the vertex of a parabola $y = ax^2 + bx + c$ is given by $x = \frac{-b}{2a}$.
For $y = x^2 - 6x + 8$, $a=1$ and $b=-6$.

$$x_B = \frac{-(-6)}{2(1)}$$ $$x_B = \frac{6}{2}$$ $$x_B = 3$$

Substitute $x_B = 3$ into the parabola equation to find $y_B$:

$$y_B = (3)^2 - 6(3) + 8$$ $$y_B = 9 - 18 + 8$$ $$y_B = -1$$

So, point B is $(3, -1)$.
The average rate of change between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\frac{y_2 - y_1}{x_2 - x_1}$.
Using points B $(3, -1)$ and C $(7, 15)$:

$$\text{Average rate of change} = \frac{15 - (-1)}{7 - 3}$$ $$\text{Average rate of change} = \frac{15 + 1}{4}$$ $$\text{Average rate of change} = \frac{16}{4}$$ $$\text{Average rate of change} = 4$$

Final Answers:

a) The coordinates of A are $\boxed{(2, 0)}$.

b) The X-intercepts are $\boxed{(2, 0) \text{ and } (4, 0)}$.

c) The coordinates of C are $\boxed{(7, 15)}$.

d) The average rate of change is $\boxed{4}$.