A) Sketch the graph y = 8 - (x-2)2. b) Hence sketch inverse of this graph.

The problem asks to sketch the graph of a given function and then sketch its inverse. Since I cannot draw graphs directly, I will describe the key features of each graph that would be used to sketch them.

Part a) Sketch the graph of $y = 8 - (x-2)^2$

This is a quadratic function in the form $y = a(x-h)^2 + k$, where $a = -1$, $h = 2$, and $k = 8$.

Step 1: Identify the vertex and direction of opening. The vertex of the parabola is at $(h, k) = (2, 8)$. Since $a = -1$ (which is negative), the parabola opens downwards.

Step 2: Find the y-intercept. Set $x = 0$: $y = 8 - (0-2)^2 = 8 - (-2)^2 = 8 - 4 = 4$ The y-intercept is $(0, 4)$.

Step 3: Find the x-intercepts. Set $y = 0$: $0 = 8 - (x-2)^2$ $(x-2)^2 = 8$ $x-2 = \pm\sqrt{8}$ $x-2 = \pm 2\sqrt{2}$ $x = 2 \pm 2\sqrt{2}$ The x-intercepts are $(2 - 2\sqrt{2}, 0)$ and $(2 + 2\sqrt{2}, 0)$. Approximately, these are $(-0.83, 0)$ and $(4.83, 0)$.

To sketch the graph: Plot the vertex $(2, 8)$. Plot the y-intercept $(0, 4)$. Plot the x-intercepts $(2 - 2\sqrt{2}, 0)$ and $(2 + 2\sqrt{2}, 0)$. Draw a downward-opening parabola passing through these points, symmetric about the line $x=2$.

Part b) Hence Sketch inverse of this graph

To sketch the inverse of a graph, we reflect the original graph across the line $y=x$. This means swapping the x and y coordinates of all points on the original graph.

Step 1: Identify key points of the inverse graph by swapping coordinates. Original vertex: $(2, 8) \implies$ Inverse vertex: $(8, 2)$ Original y-intercept: $(0, 4) \implies$ Inverse x-intercept: $(4, 0)$ Original x-intercepts: $(2 - 2\sqrt{2}, 0)$ and $(2 + 2\sqrt{2}, 0)$ $\implies$ Inverse y-intercepts: $(0, 2 - 2\sqrt{2})$ and $(0, 2 + 2\sqrt{2})$

Step 2: Determine the shape and orientation of the inverse graph. The original graph is a parabola opening downwards. Its inverse will be a parabola opening to the left. The equation of the inverse can be found by swapping $x$ and $y$ in the original equation and solving for $y$: $x = 8 - (y-2)^2$ $(y-2)^2 = 8 - x$ $y-2 = \pm\sqrt{8-x}$ $y = 2 \pm\sqrt{8-x}$ This shows that the inverse consists of two branches, $y = 2 + \sqrt{8-x}$ and $y = 2 - \sqrt{8-x}$. The domain of the inverse is $x \le 8$.

To sketch the inverse graph: Draw the line $y=x$. Plot the inverse vertex $(8, 2)$. Plot the inverse x-intercept $(4, 0)$. Plot the inverse y-intercepts $(0, 2 - 2\sqrt{2})$ and $(0, 2 + 2\sqrt{2})$. Draw a parabola opening to the left, passing through these points, symmetric about the line $y=2$. This graph will be the reflection of the original parabola across the line $y=x$.

Summary of graphs:

• Original Graph ($y = 8 - (x-2)^2$):
  • Shape: Parabola opening downwards.
  • Vertex: $\boxed{\mathbf{(2, 8)}}$
  • Y-intercept: $\boxed{\mathbf{(0, 4)}}$
  • X-intercepts: $\boxed{\mathbf{(2 - 2\sqrt{2}, 0) \text{ and } (2 + 2\sqrt{2}, 0)}}$
• Inverse Graph ($y = 2 \pm\sqrt{8-x}$):
  • Shape: Parabola opening to the left.
  • Vertex: $\boxed{\mathbf{(8, 2)}}$
  • X-intercept: $\boxed{\mathbf{(4, 0)}}$
  • Y-intercepts: $\boxed{\mathbf{(0, 2 - 2\sqrt{2}) \text{ and } (0, 2 + 2\sqrt{2})}}$