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Equation of a tangent or normal line

Start a set of 8
Step 1 of 5

For the tangent to f at x = 2, which quantity gives the gradient?

\[f(x)=x^2+2x+1\]
Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

Tangent and normal lines

The point on a curve is found from its function value, and the tangent gradient is found from its derivative. A normal line has the negative reciprocal gradient when the tangent gradient is non-zero

\[y-y_0=m\left(x-x_0\right)\]
  1. Find the point At x = a, the point on the curve has coordinates (a, f(a)) \(\displaystyle P=\left(a,f(a)\right)\)
  2. Find the tangent gradient Differentiate the function and substitute the x-coordinate of the point \(\displaystyle m_{\mathrm{t}}=f'(a)\)
  3. Find the normal gradient A normal is perpendicular to a non-horizontal tangent, so its gradient is the negative reciprocal \(\displaystyle m_{\mathrm{n}}=-\frac{1}{m_{\mathrm{t}}}\)
  4. Use point-gradient form Substitute the point and the required gradient into the equation of a straight line \(\displaystyle y-y_0=m\left(x-x_0\right)\)

Worked example

Find the equation of the normal at the given x-coordinate

\[g(x)=x^3+2x^2+x+1,\quad x=1\]
  1. \(\displaystyle g(1)=1^3+2(1)^2+1+1=5\quad\Rightarrow\quad P=(1,5)\)Evaluate the function to find the point on the curve
  2. \(\displaystyle g'(x)=3x^2+4x+1,\quad g'(1)=3+4+1=8\)Differentiate g and evaluate the tangent gradient at x = 1.
  3. \(\displaystyle m_{\mathrm{n}}=-\frac{1}{8}\)The normal gradient is the negative reciprocal of the tangent gradient
  4. \(\displaystyle y-5=-\frac{1}{8}\left(x-1\right)\quad\Rightarrow\quad y=-\frac{1}{8}x+\frac{41}{8}\)Use point-gradient form with P = (1,5) and the normal gradient

Common mistakes

Using the function value as the gradient
✕\(\displaystyle f(2)=9\quad\Rightarrow\quad m_{\mathrm{t}}=9\)✓\(\displaystyle f'(2)=6\quad\Rightarrow\quad m_{\mathrm{t}}=6\)
The function value gives the y-coordinate of the point, while the derivative value gives the tangent gradient
Using x as the y-coordinate
✕\(\displaystyle x=2\quad\Rightarrow\quad P=(2,2)\)✓\(\displaystyle x=2,\quad f(2)=9\quad\Rightarrow\quad P=(2,9)\)
The x-coordinate is given, but the y-coordinate must be found by evaluating the function
Negating instead of taking a reciprocal
✕\(\displaystyle m_{\mathrm{t}}=6\quad\Rightarrow\quad m_{\mathrm{n}}=-6\)✓\(\displaystyle m_{\mathrm{t}}=6\quad\Rightarrow\quad m_{\mathrm{n}}=-\frac{1}{6}\)
Perpendicular non-zero gradients have a product of -1, so the normal gradient is the negative reciprocal

Difficulty levels

introFind a tangent line to a quadratic at an integer x-coordinate
standardFind a normal line to a cubic at a point with a fractional gradient or coordinate
challengeFind a tangent line whose gradient is parallel to a given reference line
examFind a normal perpendicular to a reference line using a fractional or non-integer gradient

Questions about this topic

How is the tangent gradient found?+

Differentiate the function and substitute the x-coordinate of the point into the derivative

How are tangent and normal gradients related?+

For a finite nonzero tangent gradient, the normal gradient is its negative reciprocal and their product is −1. A horizontal tangent at x = a has a vertical normal x = a, whose gradient is undefined

Why is point-gradient form useful here?+

It gives the line directly from one known point and its gradient without first finding the y-intercept

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 5: Differentiation

In this chapter

Evaluate limits and test continuityGradient of a tangent from first principlesDifferentiate power functions and evaluate a derivativeExpand and factor polynomial expressionsPascal rows and binomial coefficient structure5A Limits, first principles and the power rule: Simplify and combine algebraic fractionsEvaluate a polynomial or recover a coefficient parameterSubstitute values into a formula5A Limits, first principles and the power rule: Solve a linear equationTest or construct a smooth piecewise joinRead and construct a gradient function from a function graphAnalyse a cubic graph or determine its ruleEvaluate and determine function rulesEquation of a tangent or normal lineRecover a polynomial rule from derivative and tangent conditionsFind the gradient of a line through two pointsFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a pair of simultaneous linear equationsDetermine parameters in a polynomial identity5C Tangents, normals and derivative conditions: Solve a linear equationDifferentiate natural exponential functions and solve simple derivative conditionsExponential growth and decay rate models5D Exponential derivatives and rate models: Simplify and combine algebraic fractions5D Exponential derivatives and rate models: Round to decimal places and significant figuresDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsGeometric optimisation with a trigonometric modelSinusoidal models and their rates of changeSolve a practical right-angled-triangle problemPerimeter and area of standard shapes5E Trigonometric differentiation and rates: Round to decimal places and significant figuresApproximate a cubic root using Newton's method