Tangent line to y = x^2 - 4x at x = 3 Step by Step

Step by step

The question

Question: find the tangent line to y = x² − 4x at x = 3.

y = x² − 4x

1 Find the point

The line touches the curve at x = 3.

Its y is f(3): put 3 in.

We get −3.

f(3) = 3² − 4(3) = 9 − 12 = −3

So the point is (3, −3).

point = (3, −3)

2 Find the slope

The slope of the curve is the derivative.

Use the power rule on each term.

f′(x) = 2x^(2 − 1) − 4 × 1

Now we have f′(x) = 2x − 4.

= 2x − 4

At x = 3: put 3 in.

The slope is m = 2.

f′(3) = 2(3) − 4 = 6 − 4 = 2

Careful!

The slope of a line is one number.

f′(x) changes with x, so we put x = 3 in.

Careful: not m = 2x − 4 ✗

3 Rise = m × run

We call the slope m.

m = rise over run.

So the rise is m times the run.

rise = m × run

Like 8/2 is 4, so 8 is 4 times 2.

82 = 4 → 8 = 4 × 2

Take any point (x, y) on the line.

We go from (3, −3) to (x, y).

The rise is y − (−3): the change in y.

The run is x − 3: the change in x.

rise = y − (−3)

run = x − 3

Put them in: rise = m × run becomes y − (−3) = 2(x − 3).

y − (−3) = 2(x − 3)

Taking away a negative is the same as adding.

− (−) = +

So it becomes y + 3 = 2(x − 3).

y + 3 = 2(x − 3)

This is called point-slope form.

It is made from a point and the slope.

y − y₁ = m(x − x₁)

4 Write as y = mx + b

The 2 in front of the bracket multiplies every part inside.

So we write 2 times each part.

y + 3 = 2 × x + 2 × (−3)

Work out each multiplication.

Now we have y + 3 = 2x − 6.

y + 3 = 2x − 6

We want y by itself.

The + 3 is in the way.

To remove + 3, we take away 3.

We do it on both sides, so they stay equal.

y ~~+ 3 − 3~~ = 2x − 6 − 3

+ 3 and − 3 make zero.

Cross them out.

−6 minus 3 is −9.

So y = 2x − 9.

y = 2x − 9

5 Check

Let's check on a picture.

The curve y = x² − 4x.

Our line touches it at (3, −3), with the same steepness.

Right!

Let's look back.

First: find the point.

Then: find the slope.

Then: rise = m × run.

Then: write as y = mx + b.

Last: check.

Answer