Sketching Gradient Functions

A gradient function, denoted as $f'(x)$ or $\frac{dy}{dx}$, represents the rate of change of the original function $f(x)$ at any given point $x$.

The $y$-coordinate on the graph of $y = f'(x)$ is exactly equal to the slope of the tangent to the graph of $y = f(x)$ at that same $x$-value.

If $f(x)$ is a smooth, continuous curve, its derivative $f'(x)$ will also typically be a smooth curve, reflecting the continuous change in steepness along the original path.

Stationary points (local maxima, local minima, or points of inflection) on the graph of $f(x)$ occur where the gradient is zero.

Consequently, every stationary point on $y = f(x)$ corresponds to an x-intercept (root) on the graph of $y = f'(x)$.

When $f(x)$ has a local maximum, $f'(x)$ crosses the $x$-axis from positive to negative; for a local minimum, it crosses from negative to positive.

On intervals where $f(x)$ is increasing, the gradient is positive ($f'(x) > 0$), meaning the graph of $f'(x)$ must be located above the x-axis.

On intervals where $f(x)$ is decreasing, the gradient is negative ($f'(x) < 0$), meaning the graph of $f'(x)$ must be located below the x-axis.

The steeper the original curve, the further the gradient function is from the $x$-axis (higher for positive slopes, lower for negative slopes).

A point of inflection on $f(x)$ is where the curve changes concavity and the gradient reaches a local maximum or minimum value.

This implies that a point of inflection on $y = f(x)$ corresponds to a turning point (peak or valley) on the graph of $y = f'(x)$.

If the curve $f(x)$ is getting steeper, $f'(x)$ is moving away from the $x$-axis; if it is flattening out, $f'(x)$ is moving toward the $x$-axis.

Vertical Asymptotes: If $f(x)$ has a vertical asymptote at $x = k$, the gradient function $f'(x)$ will almost always have a vertical asymptote at the same $x$-coordinate.

Horizontal Asymptotes: If $f(x)$ approaches a horizontal asymptote as $x \rightarrow \infty$, its gradient is approaching zero. Therefore, $f'(x)$ will have a horizontal asymptote at y = 0.

It is vital to check the sign of the gradient near asymptotes to determine if $f'(x)$ approaches $+\infty$ or $-\infty$.

Consistency Check: Always verify that the regions where $f'(x)$ is above/below the axis match the increasing/decreasing behavior of $f(x)$.

Avoid Intercept Confusion: A common mistake is thinking the $x$-intercepts of $f(x)$ matter for $f'(x)$; they do not. Only the turning points of $f(x)$ determine the intercepts of $f'(x)$.

Degree Reduction: For polynomial-like shapes, remember that the derivative is one degree lower (e.g., a cubic $f(x)$ results in a quadratic $f'(x)$).

Labeling: In exams, clearly mark the $x$-coordinates of turning points from the original graph as the roots on your new gradient sketch.