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You provided the equation \( y = 2 - x^2 \) and the point \((3, -7)\). Let's determine if the point lies on the curve defined by the equation.

To check if the point \((3, -7)\) is on the curve, substitute \( x = 3 \) into the equation and see if the corresponding \( y \)-value equals \(-7\).

\[
y = 2 - x^2
\]

Substitute \( x = 3 \):

\[
y = 2 - (3)^2 = 2 - 9 = -7
\]

So, when \( x = 3 \), \( y = -7 \), which matches the given point \((3, -7)\). Therefore, the point \((3, -7)\) lies on the curve.

Would you like more details or have any questions? Here are five additional related questions you might find interesting:

1. How would you find the slope of the tangent line to the curve at the point \((3, -7)\)?
2. What is the derivative of the function \( y = 2 - x^2 \)?
3. How do you find the equation of the tangent line to the curve at a given point?
4. What are the critical points of the function \( y = 2 - x^2 \)?
5. Can you determine the vertex of the parabola defined by \( y = 2 - x^2 \)?

**Tip:** When verifying if a point lies on a curve, always substitute the coordinates into the equation to check if both sides of the equation are equal.

Verify if the point (3, -7) lies on the quadratic curve defined by the equation y = 2 - x^2. This problem illustrates key algebraic concepts, including substitution and quadratic functions. Suitable for students in Grades 9-10.

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