Below are graphs of the functions y = x^2 - 6x + 8 and y - 3x + 6 = 0. a) Coordinates of A. b) X-intercepts of the graph of the function y = x^2 - 6x + 8. c) Coordinates of point C. d) Average rate of change for the function y = x^2 - 6x + 8 between points B and C.

Question

Below are graphs of the functions y = x^2 - 6x + 8 and y - 3x + 6 = 0. a) Coordinates of A. b) X-intercepts of the graph of the function y = x^2 - 6x + 8. c) Coordinates of point C. d) Average rate of change for the function y = x^2 - 6x + 8 between points B and C.

Asked on March 26, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-26T06:14:17.078Z

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 $\boxed{4}$.

Accepted Answer · 2026-03-26T06:14:17.078Z

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 $\boxed{4}$.