Let's work through the problem in the image step by step.

(a) Find the value of $p$

The equation given is $y = -x^2 + 2x + 1$.

We are tasked to find $p$, which corresponds to $x = 0.5$ in the table.

Substituting $x = 0.5$ into the equation: $y = -(0.5)^2 + 2(0.5) + 1$ $y = -0.25 + 1 + 1 = 1.75$ Thus, $p = 1.75$.

(b) Draw the graph of $y = -x^2 + 2x + 1$

You are instructed to graph the equation $y = -x^2 + 2x + 1$ for $-1 \leq x \leq 3$.

Here’s how you would plot the points:

Plot these points on the grid and draw the curve.

(c) Use the graph in part (b) to find:

(i) The value of $y$ when $x = 0.7$

From the graph, we can estimate the value of $y$ when $x = 0.7$, or we can use the equation to calculate it:

$y = -(0.7)^2 + 2(0.7) + 1$ $y = -0.49 + 1.4 + 1 = 1.91$

So, $y \approx 1.91$.

(ii) The values of $x$ when $y = 0.5$

To find when $y = 0.5$, we substitute $y = 0.5$ into the equation and solve for $x$:

$0.5 = -x^2 + 2x + 1$ $-x^2 + 2x + 0.5 = 0$ Multiplying by -1 to simplify: $x^2 - 2x - 0.5 = 0$

Using the quadratic formula: $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-0.5)}}{2(1)}$ $x = \frac{2 \pm \sqrt{4 + 2}}{2} = \frac{2 \pm \sqrt{6}}{2}$ $x = 1 \pm \frac{\sqrt{6}}{2} \approx 1 \pm 1.22$ So, $x \approx 2.22$ or $x \approx -0.22$.

(iii) The maximum value of $y$ and the value of $x$ at which this occurs

The maximum value of $y$ for a quadratic equation $y = -x^2 + 2x + 1$ occurs at the vertex, given by:

$x = \frac{-b}{2a}$ Here, $a = -1$ and $b = 2$, so: $x = \frac{-2}{2(-1)} = 1$

Substituting $x = 1$ into the equation: $y = -(1)^2 + 2(1) + 1 = -1 + 2 + 1 = 2$

So, the maximum value of $y$ is 2, and it occurs at $x = 1$.

(d) State the equation of the line of symmetry of the graph

The line of symmetry for a parabola is the vertical line through the vertex. From part (c)(iii), the vertex occurs at $x = 1$.

Thus, the equation of the line of symmetry is: $x = 1$

Would you like further details or clarifications on any part?

Here are some related questions to expand on the concepts:

1. What is the general form of a quadratic equation, and how do we find its vertex?
2. Can you derive the quadratic formula and explain each term?
3. How does the sign of $a$ in $ax^2 + bx + c$ affect the direction of the parabola?
4. What methods are available for solving quadratic equations besides the quadratic formula?
5. How do you determine the domain and range of a quadratic function?

Tip: Always check the vertex and axis of symmetry of a quadratic equation to easily sketch the graph and find maximum or minimum values.