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A. Find two points on the line B. Find the equation of the line F(x) =x^2

Let's start by analyzing the image and the problem.

### Part A: Finding Two Points on the Line
The parabola given is \( f(x) = x^2 \), and the line intersects the parabola at two points, \( x = -1 \) and \( x = 2 \).

1. **At \( x = -1 \)**:
   \[
   f(-1) = (-1)^2 = 1
   \]
   So, one point is \( (-1, 1) \).

2. **At \( x = 2 \)**:
   \[
   f(2) = (2)^2 = 4
   \]
   So, another point is \( (2, 4) \).

Thus, the two points on the line are \( (-1, 1) \) and \( (2, 4) \).

### Part B: Finding the Equation of the Line
To find the equation of the line passing through the points \( (-1, 1) \) and \( (2, 4) \), we use the **point-slope form** of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is one of the points.

#### Step 1: Find the Slope \( m \)
The slope \( m \) between the two points \( (-1, 1) \) and \( (2, 4) \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{2 - (-1)} = \frac{3}{3} = 1
\]

#### Step 2: Use Point-Slope Form
We can use either point, say \( (-1, 1) \). Substituting \( m = 1 \), \( x_1 = -1 \), and \( y_1 = 1 \), the equation becomes:
\[
y - 1 = 1(x + 1)
\]
Simplifying:
\[
y - 1 = x + 1
\]
\[
y = x + 2
\]

Thus, the equation of the line is \( y = x + 2 \).

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### Summary of Solutions:
- Two points on the line are \( (-1, 1) \) and \( (2, 4) \).
- The equation of the line is \( y = x + 2 \).

Would you like further clarification or additional details?

Here are some related questions to consider:
1. How do you verify if a point lies on a line?
2. What happens if the slope between two points is zero?
3. Can we find the intersection points of a line with any other function?
4. How do we determine the area between a parabola and a line?
5. What is the geometric interpretation of the slope?

**Tip:** When dealing with points and lines, always calculate the slope first, as it provides the rate of change and direction of the line.

Find two points on the line and the equation of the line intersecting the parabola f(x) = x^2 at x = -1 and x = 2.

Learn how to find two points on a line that intersects the parabola f(x) = x^2 at x = -1 and x = 2, and determine the equation of the line using the point-slope formula. Suitable for Grades 9-11, this problem covers key algebraic concepts such as linear equations, slope calculation, and quadratic functions.

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