Tangent line to y = x^2 - 5 at x = -2 Step by Step

Step by step

The question

Question: find the tangent line to y = x² − 5 at x = −2.

y = x² − 5

1 Find the point

The line touches the curve at x = −2.

Its y is f(−2): put −2 in.

We get −1.

f(−2) = (−2)² − 5 = 4 − 5 = −1

So the point is (−2, −1).

point = (−2, −1)

2 Find the slope

The slope of the curve is the derivative.

Use the power rule on each term.

f′(x) = 2x^(2 − 1) + 0

Now we have f′(x) = 2x.

= 2x

At x = −2: put −2 in.

The slope is m = −4.

f′(−2) = 2(−2) = −4

Careful!

The slope of a line is one number.

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

Careful: not m = 2x ✗

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, −1) to (x, y).

The rise is y − (−1): the change in y.

The run is x − (−2): the change in x.

rise = y − (−1)

run = x − (−2)

Put them in: rise = m × run becomes y − (−1) = −4(x − (−2)).

y − (−1) = −4(x − (−2))

Taking away a negative is the same as adding.

− (−) = +

So it becomes y + 1 = −4(x + 2).

y + 1 = −4(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 −4 in front of the bracket multiplies every part inside.

So we write −4 times each part.

y + 1 = −4 × x − 4 × 2

Work out each multiplication.

Now we have y + 1 = −4x − 8.

y + 1 = −4x − 8

We want y by itself.

The + 1 is in the way.

To remove + 1, we take away 1.

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

y ~~+ 1 − 1~~ = −4x − 8 − 1

+ 1 and − 1 make zero.

Cross them out.

−8 minus 1 is −9.

So y = −4x − 9.

y = −4x − 9

5 Check

Let's check on a picture.

The curve y = x² − 5.

Our line touches it at (−2, −1), 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