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

Step by step

The question

Question: find the tangent line to y = −x² + 2x + 3 at x = 2.

y = −x² + 2x + 3

1 Find the point

The line touches the curve at x = 2.

Its y is f(2): put 2 in.

We get 3.

f(2) = −2² + 2(2) + 3 = −4 + 4 + 3 = 3

So the point is (2, 3).

point = (2, 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) + 2 × 1 + 0

Now we have f′(x) = −2x + 2.

= −2x + 2

At x = 2: put 2 in.

The slope is m = −2.

f′(2) = −2(2) + 2 = −4 + 2 = −2

Careful!

The slope of a line is one number.

f′(x) changes with x, so we put x = 2 in.

Careful: not m = −2x + 2 ✗

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 (2, 3) to (x, y).

The rise is y − 3: the change in y.

The run is x − 2: the change in x.

rise = y − 3

run = x − 2

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

y − 3 = −2(x − 2)

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 × (−2)

Work out each multiplication.

Minus times minus is plus.

Now we have y − 3 = −2x + 4.

y − 3 = −2x + 4

We want y by itself.

The − 3 is in the way.

To remove − 3, we add 3.

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

y ~~− 3 + 3~~ = −2x + 4 + 3

− 3 and + 3 make zero.

Cross them out.

4 plus 3 is 7.

So y = −2x + 7.

y = −2x + 7

5 Check

Let's check on a picture.

The curve y = −x² + 2x + 3.

Our line touches it at (2, 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