12.10PureStretch

Sketching Gradient Functions

Given the graph of $y = f (x)$ , you can sketch the graph of its gradient function $y = f^{'} (x)$ without any algebra — just by reading the slope of the original curve at each point.

25 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.10 Sketching Gradient FunctionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Read the gradient of a curve qualitatively
Turn stationary points into roots of the gradient graph
Sketch f′(x) from the graph of f(x)
Relate features of the two graphs

Stationary points become roots

Where the original graph is flat (a stationary point), its gradient is zero — so the gradient graph $f^{'} (x)$ $crosses the x-axis$ there. Each turning point of $f$ becomes a root of $f^{'}$ .

f^{'} (x) = 0 where f (x) has a stationary point

Flat on

f

⟶ a root on

f^{'}

Positive and negative regions

Where $f$ slopes $up$ , $f^{'}$ is $above$ the axis (positive); where $f$ slopes $down$ , $f^{'}$ is $below$ it. The steeper the original, the further $f^{'}$ is from the axis.

Tip — Scan left to right: is f going up or down? That sets whether f′ is above or below the x-axis.

Degree drops by one

Differentiating lowers the degree: a cubic $f$ gives a quadratic $f^{'}$ (a parabola), a quadratic gives a line. So expect the gradient graph to be one “shape simpler” than the original.

1Two stationary points ⟶ two roots on

f^{'}

2Degree drops to 2 ⟶ a parabola.

Answera quadratic (parabola) crossing the axis twice

Formula recap

f^{'} (x) = 0 \leftrightarrow stationary point of f

Turning points ⟶ roots.

f ↗\Rightarrow f^{'} > 0, f ↘\Rightarrow f^{'} < 0

Slope ⟶ sign.

de g (f^{'}) = de g (f) - 1

Degree drops by one.

Common mistakes to avoid

Drawing f′ with the same number of turning points as f.

f′ has roots where f has turning points; its degree is one lower.

Putting f′ above the axis where f is decreasing.

f decreasing ⟶ f′ below the axis (negative).

Key takeaways

Stationary points of f become roots (x-axis crossings) of f′.
f sloping up ⟶ f′ positive; sloping down ⟶ f′ negative.
The gradient function is one degree lower than f.

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