Intersections, zeros, max/min, and derivatives on a graph

You plotted a function (or two). Now you need the root, the intersection, the maximum, or the area under the curve. Don't eyeball it. Press 2nd + trace and let the calculator do it exactly.

The menu

2nd + trace opens CALCULATE with these options (firmware varies slightly):

1: value
2: zero
3: minimum
4: maximum
5: intersect
6: dy/dx
7: ∫f(x)dx

1: value (evaluate at x)

Prompts X=?. Type any x-value and press enter. The cursor jumps to that point on the current curve and shows the y-value at the bottom.

Equivalent to Table ASK mode but visually anchored on the graph. Good for "what's the y-value at x=7" questions.

2: zero (find a root)

The calculator asks for a left bound, a right bound, and a guess. Arrow the cursor to the left of the root, enter. Arrow to the right of the root, enter. Arrow near the root, enter. It returns the root to ~12 significant digits.

Rule: the bounds must bracket exactly one root (sign change between them). If the function has two roots close together, tighten the bounds.

3: minimum / 4: maximum

Same three-prompt pattern — left bound, right bound, guess. The calculator returns the local extremum inside that window.

Great for optimization problems. Plot the function, let CALC find the optimum, verify with derivative = 0 by hand for full credit.

5: intersect (the most-used option)

Works only when two or more functions are plotted (both enabled in Y=).

Prompts:

• First curve? — cursor on Y1 (use up/down arrows to switch curves if you have 3+). Press enter.
• Second curve? — cursor on Y2. Press enter.
• Guess? — arrow near the intersection you want. Press enter.

Returns the exact (x, y) of the intersection. If there are multiple intersections, run it again with a different guess.

6: dy/dx (numerical derivative at a point)

Prompts X=?. Returns the slope of the tangent at that x-value — using a centered-difference approximation, accurate to machine precision.

Not a symbolic derivative. If the problem asks "find f'(2)", and f(x) is something messy, this is faster than computing by hand. If the problem asks "find f'(x)", you need the symbolic answer and this won't help.

7: ∫f(x)dx (definite integral)

Prompts lower and upper bounds. Shades the region under the curve and returns the integral.

Pair this with max/min to verify the integral makes sense (area = approximately base × average height, so answers should be in a plausible ballpark).

Combining CALC with multiple functions

If Y1, Y2, Y3 are all enabled when you press CALC, the active curve is whichever one the cursor is on. Use up/down arrows on the graph screen to switch curves before entering CALC. This is how "intersect" can target any pair among three plotted curves.

Gotchas

• "NO SIGN CHNG" on zero: your bounds don't bracket an actual root. Re-plot, re-check where the curve crosses zero, re-try with tighter bounds.
• Intersect returns the same point twice when two curves are tangent (e.g. y=x² and y=1 touch once at x=±1 but are non-tangent; y=x² and y=0 touch once at x=0 with multiplicity 2). Tangent intersections return one point; the calculator doesn't flag multiplicity.
• The graph screen is pixel-resolution. If two curves cross at an irrational point, the cursor lands on the nearest pixel but the returned value is the exact calculated intersection — trust the number, not the pixel position.