https://file.aiquickdraw.com/mathbot/file/172680622311819gd3dl2.jpg

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:

| \( x \)  | \( y = -x^2 + 2x + 1 \) |
|---------|------------------------|
| -1      | -2                     |
| 0       | 1                      |
| 0.5     | 1.75                   |
| 1       | 2                      |
| 2       | 1                      |
| 3       | -2                     |

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.

The variables x and y are connected by the equation y = -x^2 + 2x + 1. Some corresponding values of x and y are given in the table. The tasks involve calculating values of p, graphing, and solving related quadratic equation problems.

Learn to solve and graph the quadratic function y = -x^2 + 2x + 1. Calculate key points like p and use the quadratic formula to find solutions. This guide covers graphing, symmetry, and more for students in Grades 9-12.

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