Sketching Gradient Functions

A gradient function $f'(x)$ (or $\frac{dy}{dx}$) is a mathematical representation of the slope of the original function $f(x)$ at every point along its domain.

The $y$-coordinate of any point on the gradient function graph corresponds exactly to the numerical value of the gradient of the tangent to the original curve at that same $x$-coordinate.

If the original function $f(x)$ is a polynomial of degree $n$, its gradient function $f'(x)$ will be a polynomial of degree $n-1$, meaning a cubic curve differentiates into a quadratic, and a quadratic differentiates into a linear function.

Sign Correspondence: When $f(x)$ is increasing, the gradient is positive, so the graph of $f'(x)$ must lie above the x-axis. Conversely, when $f(x)$ is decreasing, $f'(x)$ must lie below the x-axis.

Stationary Points: Any point where the original curve has a gradient of zero (local maxima, local minima, or stationary points of inflection) becomes an x-intercept on the gradient function graph.

Curvature and Inflection: A non-stationary point of inflection on $f(x)$ represents the point where the gradient is at its steepest (maximum) or shallowest (minimum), which corresponds to a turning point on the $f'(x)$ graph.

• Step 1: Identify Stationary Points: Locate all points on $f(x)$ where the tangent is horizontal. Mark these $x$-values as intercepts on your new $f'(x)$ axes.

• Step 2: Determine the Sign: Look at the intervals between stationary points. If the curve goes 'up' from left to right, sketch $f'(x)$ in the positive $y$ region. If it goes 'down', sketch it in the negative $y$ region.

• Step 3: Analyze Steepness: Identify where the original curve is steepest. This is where the $f'(x)$ graph will reach its maximum or minimum distance from the $x$-axis.

• Step 4: Handle Asymptotes: If $f(x)$ has a vertical asymptote, $f'(x)$ will typically share that vertical asymptote. If $f(x)$ approaches a horizontal asymptote, the gradient is flattening out, so $f'(x)$ will approach zero ($y=0$).

It is vital to distinguish between the value of a function and the gradient of a function during the sketching process.

• Check the Degree: If the original graph looks like a parabola ($x^2$), ensure your gradient sketch looks like a straight line ($x$). If the original is a cubic, the derivative must be a parabola.

• Vertical Alignment: Always sketch the gradient function directly below the original function. Use dashed vertical lines to align stationary points with $x$-intercepts to ensure accuracy.

• Ignore y-intercepts: The $y$-intercept of the original function $f(x)$ has no direct impact on the shape or position of $f'(x)$; only the slope at that point matters.

• Verify Signs: A common check is to pick a point, estimate the slope (e.g., 'steep and negative'), and ensure the $f'(x)$ graph has a corresponding $y$-value (e.g., 'far below the $x$-axis').

• Confusing Intercepts: Students often mistakenly think that where $f(x)$ crosses the $x$-axis, $f'(x)$ must also cross. In reality, $f'(x)$ only crosses the axis where $f(x)$ has a turning point.

• Incorrect Inflection Mapping: Many fail to realize that a point of inflection on $f(x)$ results in a turning point on $f'(x)$. If the gradient stops increasing and starts decreasing, the gradient function has reached a peak.

• Smoothness: Unless the original function has a sharp 'corner' (discontinuity in gradient), the gradient function should be drawn as a smooth, continuous curve.