a function has these three points on it, h(-4)=-3, h(0)=7, h(9)=-3. connect these three points and find the area of the created figure

Given the three points \( (-4, -3) \), \( (0, 7) \), and \( (9, -3) \), we can interpret these points as the vertices of a geometric shape, which appears to be a triangle. To find the area, we can apply the **shoelace formula** (also known as the **surveying formula**), which is useful for calculating the area of a polygon given its vertices.

The formula is:

\[
\text{Area} = \frac{1}{2} \left| x_1 y_2 + x_2 y_3 + x_3 y_1 - (y_1 x_2 + y_2 x_3 + y_3 x_1) \right|
\]

Where the points are \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \). For the given points:

- \( (x_1, y_1) = (-4, -3) \)
- \( (x_2, y_2) = (0, 7) \)
- \( (x_3, y_3) = (9, -3) \)

Now, let's plug the values into the formula.

\[
\text{Area} = \frac{1}{2} \left| (-4)(7) + (0)(-3) + (9)(-3) - ((-3)(0) + 7(9) + (-3)(-4)) \right|
\]

I'll compute this.The area of the triangle formed by connecting the points \((-4, -3)\), \((0, 7)\), and \((9, -3)\) is \(65\) square units.

Would you like more details or have any questions?

Here are five related questions to expand on this:

1. How would the formula change if the points were not ordered in a counterclockwise direction?
2. Can this area be verified using a different method, such as splitting the shape into simpler parts?
3. How does the slope of the lines between these points help determine the type of triangle?
4. What if one of the points was shifted? How would it impact the area?
5. Could this method be generalized for shapes with more than three vertices?

**Tip:** The shoelace formula can be extended to calculate the area of any polygon, as long as its vertices are known.

A function has these three points on it, h(-4)=-3, h(0)=7, h(9)=-3. Connect these three points and find the area of the created figure.

Find the area of the triangle formed by the points (-4, -3), (0, 7), and (9, -3) using the shoelace formula. This detailed solution explains how to apply coordinate geometry concepts and the shoelace theorem, suitable for high school students in Grades 10-12.

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