A tangent touches the curve at a point and has the same gradient as the curve there. To find its equation, find the point (x₁, y₁) on the curve, find the gradient m = dy/dx at x₁, then use y − y₁ = m(x − x₁).
For y = 2x² − 4x at x = 1: y₁ = −2, dy/dx = 4x − 4, so m = 0. The tangent is y = −2, a horizontal line, because x = 1 is the bottom of the curve. Give the answer in the form asked, such as y = mx + c.
Worked example
Find the tangent to y = x² − 2x − 5 at x = 3.
- y = 9 − 6 − 5 = −2, and dy/dx = 2x − 2 = 4.
- y + 2 = 4(x − 3), so y = 4x − 14.
Answer:y = 4x − 14
Key words
- tangent
- a line touching the curve with the same gradient
- point of contact
- where the tangent touches
- y − y₁ = m(x − x₁)
- the equation of a line through a point
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