Step by step
The question
Question: find the tangent line to y = x² − 4x at x = 2.
y = x2 − 4x
1 Find the point
The line touches the curve at x = 2.
Its y is f(2): put 2 in.
We get −4.
f(2) = 22 − 4(2) = 4 − 8 = −4
So the point is (2, −4).
point = (2, −4)
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 = 2: put 2 in.
The slope is m = 0.
f′(2) = 2(2) − 4 = 4 − 4 = 0
Careful!
The slope of a line is one number.
f′(x) changes with x, so we put x = 2 in.
Careful: not m = 2x − 4 ✗
3 Write the line
A slope of 0 means a flat line.
It goes through (2, −4), so every y on it is −4: y = −4.
y = −4
4 Check
Let's check on a picture.
The curve y = x² − 4x.
Our line touches it at (2, −4), with the same steepness.
Right!
Let's look back.
First: find the point.
Then: find the slope.
Then: write the line.
Last: check.
Answer
y = −4