https://file.aiquickdraw.com/mathbot/file/1728186431919uyc8xbnw.jpg

Let's first address part (a) of the problem:

The given equation is:
\[
y = -x^2 - x + 2
\]
You are asked to graph this function for values of \( x \) from -3 to 2.

### Step-by-step approach:

1. **Find key points for the graph**:
   - We can substitute values of \( x \) from -3 to 2 into the equation to find corresponding \( y \) values.

2. **Determine the shape of the graph**:
   - This is a quadratic equation, and since the coefficient of \( x^2 \) is negative (\( -1 \)), the parabola opens downwards.

I'll calculate the key points and describe how the graph behaves over the interval \([-3, 2]\).

### Part (b) asks you to find approximate solutions to three different equations using the graph:

1. \( -x^2 - x + 2 = 0 \)
2. \( -x^2 - x + 2 = 1 \)
3. \( -x^2 - x + 2 = -2 \)

These are the values where the graph intersects the horizontal lines \( y = 0 \), \( y = 1 \), and \( y = -2 \).

### I'll now compute these points:

#### Part (a) values:
For \( y = -x^2 - x + 2 \):
- \( x = -3 \), \( y = -(-3)^2 - (-3) + 2 = -9 + 3 + 2 = -4 \)
- \( x = -2 \), \( y = -(-2)^2 - (-2) + 2 = -4 + 2 + 2 = 0 \)
- \( x = -1 \), \( y = -(-1)^2 - (-1) + 2 = -1 + 1 + 2 = 2 \)
- \( x = 0 \), \( y = -(0)^2 - 0 + 2 = 2 \)
- \( x = 1 \), \( y = -(1)^2 - (1) + 2 = -1 - 1 + 2 = 0 \)
- \( x = 2 \), \( y = -(2)^2 - (2) + 2 = -4 - 2 + 2 = -4 \)

With these values, we can plot the points \( (-3, -4), (-2, 0), (-1, 2), (0, 2), (1, 0), (2, -4) \) and connect them smoothly to form a downward-opening parabola.

### Part (b):
Use the graph to estimate the \( x \)-coordinates where the graph intersects the lines for each given value of \( y \):
1. For \( y = 0 \): Find approximate \( x \)-values where the graph crosses the x-axis.
2. For \( y = 1 \): Find \( x \)-values where the graph reaches \( y = 1 \).
3. For \( y = -2 \): Estimate the points where the graph crosses \( y = -2 \).

Would you like me to create this graph for you, or provide more detailed analysis? Let me know if you'd like to see the solutions to part (b) visually.

Here are 5 related questions to consider:
1. How do you find the vertex of a quadratic function?
2. What is the axis of symmetry for this parabola?
3. How do the solutions of the equation \( y = 0 \) relate to the x-intercepts of the graph?
4. How would the graph change if the coefficient of \( x^2 \) were positive?
5. What role do discriminants play in solving quadratic equations?

**Tip**: When graphing quadratics, always check if the vertex is a maximum or minimum based on the coefficient of \( x^2 \).

Draw the graph of y = -x^2 - x + 2 for values of x from -3 to 2. Use the graph to find the approximate solutions to the equations: (i) -x^2 - x + 2 = 0, (ii) -x^2 - x + 2 = 1, (iii) -x^2 - x + 2 = -2.

Learn how to graph the quadratic equation y = -x^2 - x + 2 and solve equations using the graph for values between x = -3 and x = 2. This problem involves key algebraic concepts including graphing parabolas and finding approximate solutions to quadratic equations. Suitable for students in Grades 9-11.

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