13.3PureStretch

Finding Functions

If you know a curve’s gradient function and one point it passes through, you can recover the original function exactly — by integrating and then using the point to pin down the constant of integration.

25 min Video by Zeeshan Zamurred Integration

Edexcel AS Level Maths: 13.3 Finding Functions (Integrating)Watch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Integrate a gradient function to find the original
Use a known point to find the constant c
Write the full equation of the curve
Apply this to rate-of-change problems

Integrate to undo the derivative

Given $\frac{d y}{d x}$ , integrating gives $y$ — but with an unknown $+ c$ . You need extra information to find $c$ .

Use the point to find c

Substitute the coordinates of a known point on the curve into your integrated expression and solve for $c$ .

y = \int \frac{d y}{d x} d x + c, then use a point to find c

One point fixes the constant.

1Integrate:

y = x^{2} + c

2Use

(1, 4)

4 = 1 + c \Rightarrow c = 3

Answer

y = x^{2} + 3

Tip — Integrate first (keep the + c), THEN substitute the point — not the other way round.

In context

The same idea recovers a quantity from its rate of change plus an initial value — for example, finding displacement from a velocity model and a starting position.

Formula recap

y = \int \frac{d y}{d x} d x + c

Integrate the gradient function.

substitute a point \Rightarrow c

Pin down the constant.

Common mistakes to avoid

Leaving the answer as “+ c” without finding c when a point is given.

Use the point to solve for c and give the exact function.

Substituting the point before integrating.

Integrate first, then substitute to find c.

Key takeaways

Integrate the gradient function to get y (with + c).
Substitute a known point to find c.
State the full equation of the curve.

Test yourself

Ready to lock in Finding Functions? Pick a mode and earn XP & Dobloons.