Tangent and Normal Drawn on a Curve

ROLE: Act as a calculus teacher who graphs a curve and draws its tangent and normal at a stated point.

CONTEXT: I teach [CLASS/GRADE] under [BOARD or EXAM e.g. CBSE / ICSE / JEE]. The figure must clearly show the curve, the point of contact, the tangent line, and the perpendicular normal.

YOUR TASK:
1. Plot the curve on labelled x and y axes with a light grid.
2. Mark the given point P on the curve and label its coordinates.
3. Draw the tangent line at P (touching the curve at P) and the normal line at P (perpendicular to the tangent).
4. Write the equation of the tangent and of the normal on the graph.
5. Use distinct line styles for the curve, the tangent, and the normal, with a small key.

INPUT: Here is the curve: [PASTE IT, e.g. y = x^2]. The point: [GIVE IT, e.g. x = 1, or the point P(1, 1)].

HOW TO WORK IT OUT: First find the y-value of the point if only x is given, confirming it lies on the curve. Differentiate to get the slope of the tangent at P, then the normal slope as the negative reciprocal. Write both line equations using point-slope form, showing the steps. VERIFY two ways: confirm the tangent passes through P and has the computed slope, and confirm the normal slope times the tangent slope equals -1. Never guess the slope; never let the tangent cross the curve at P as if it were a secant.

OUTPUT FORMAT:
1. WORKING: y-value at P, derivative, tangent slope, normal slope, both equations.
2. THE GRAPH: one clean labelled image with curve, point P, tangent, normal, equations, and a small line-style key.
3. CHECK: one line confirming slopes multiply to -1.

RULES: Tangent must touch the curve at P, not cut across it. Normal must be visibly perpendicular to the tangent. Axes labelled. Plain white background.

SIGNATURE: Sign the bottom-right of every page with an elegant handwritten signature "[YOUR NAME]" with a small underline flourish.