Tangents and normals

The tangent at a point has the same gradient as the curve there. The normal is perpendicular to it, through the same point. Slide the point and watch both equations rewrite themselves.

x = 3.00

(3, 6)the point

4.00gradient

−0.25normal gradient

tangent: y = 4x − 6

normal: y = −0.25x + 6.75

The vertical axis is compressed, so the normal will not look perpendicular on screen. The gradients 4 and −¼ are what make it so.

The method, and the step everyone skips

Find the tangent to y = x² − 2x + 3 at x = 3.

y= 3² − 2(3) + 3 = 9 − 6 + 3 = 6 the step people skip: find the y value dydx= 2x − 2 m= 2(3) − 2 = 4 the gradient at that point y − 6= 4(x − 3) y − y₁ = m(x − x₁) y= 4x − 6

You are given x. You are never given y. The question says "at x = 3" and most people write the point as (3, 3). The y value has to be worked out from the curve, and here it is 6.

Press the mistake button above to see what that does. The line comes out as y = 4x − 9: exactly the right gradient, perfectly parallel to the real tangent, and it misses the parabola completely. A tangent that touches nothing.

The normal

Perpendicular gradients multiply to give −1, so the normal gradient is −1 divided by the tangent gradient.

m_normal= −14 = −0.25 not −4, and not ¼ y − 6= −14(x − 3) same point, new gradient y= −0.25x + 6.75 or x + 4y = 27 if they want it tidy

Steep tangent, shallow normal. It is worth checking your answer against that. If the tangent has gradient 4 and your normal has gradient −4, you have flipped the sign without taking the reciprocal. Slide the point above and watch the two numbers move in opposite directions.

Your turn

1. For y = 5x², find the gradient of the tangent at x = 2.

2. A curve has a tangent of gradient 4 at the point (3, 6). What is the equation of the normal there?

y = −4x + 18 y = −0.25x + 6.75 y = 0.25x + 5.25 y = 4x − 6

3. Someone finds the tangent to y = x² − 2x + 3 at x = 3 and gets y = 4x − 9. Their gradient is right. What went wrong?

They differentiated incorrectly. They used the point (3, 3) instead of (3, 6). They used the normal gradient by mistake. Nothing; y = 4x − 9 is also correct.

Where the marks go

Three marks, usually: the y value, the gradient, the equation. The gradient is the one everybody gets, and the y value is the one that quietly loses two of them, because a wrong point carries through the rest of the question.

Leave your answer in whatever form was asked for. If it says "in the form ax + by + c = 0", then y = −0.25x + 6.75 is not finished.

Sanity check at the end: put the x value back into your line and see whether you get the y value of the point. If you do not, the line does not pass through the point and nothing else can rescue it.

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