Read stationary points and slope limits

Original graph f with maximumSource-faithful qualitative diagram; distances are schematic unless an exact function is stated.xyOy=f(x)(3,5)−2−123y=2
Question diagram 1
Original graph f with maximumSource-faithful qualitative diagram; distances are schematic unless an exact function is stated.xyOy=f(x)(3,5)−2−123y=2

Given graph of f with intercepts(-1,0),(0,3), maximum(3,5) and asymptotes.

The diagram below shows the graph of y=f(x)y=f(x). The curve cuts the axes at (−1,0)(-1,0) and (0,3)(0,3). It has a maximum point at (3,5)(3,5), and asymptotes x=−2x=-2 and y=2y=2.

Sketch the graph of y=f′(x)y=f'(x), showing clearly asymptotes, intercepts and coordinates of turning points, where available. marks

[modified 2010/ACJC/Prelim/I/3]

Solution

Derivative sketch

Gradient graphs

The graph of y=f(x)y=f(x) cuts the axes at (−1,0)(-1,0) and (0,3)(0,3), has a maximum point at (3,5)(3,5) and asymptotes x=−2x=-2 and y=2y=2. Sketch y=f′(x)y=f'(x), showing asymptotes, intercepts and the coordinates of turning points, where available.

Explore this idea

Hint

  1. Trace the gradient of the curve as xx increases.

  2. The maximum (3,5)(3,5) gives the xx-intercept (3,0)(3,0).

  3. y=2y=2 becomes y=0y=0; x=−2x=-2 stays.

Qualitative derivative from a maximum
Qualitative derivative from a maximumScroll across to see the full diagram.

Answer · Derivative sketch

Zero (3,0)(3,0); asymptotes x=−2,y=0x=-2,y=0.

Positive for x<−2 and −2<x<3; negative for x>3. Turning-point coordinates beyond this are unspecified.

Explore the idea

Follow the gradient across the asymptote

Drag along an example curve with these properties. The gradient is positive up to x = 3, zero at the maximum and negative afterwards.

Position along the curvex = 0
Try an example
Test a critical position

Buttons use full-precision reviewed positions; displayed coordinates are rounded.

This curve is an illustrative reconstruction of the source sketch. Its numerical heights and gradients describe the model; only the reviewed signs and labelled features come from the source.

f(x)
3
Estimated gradient f′(x)
1.5
positive

The curve rises towards its maximum: the gradient is positive.

Selected values are within the original drawing windows.

  • An example of y = f(x)
  • Tangent
  • Y = f′(x), and where the curve rises
  • Join P to Q; the dashed legs show Δx and Δy
Key idea

A horizontal asymptote y=2y=2 becomes y=0y=0 for the gradient. A maximum becomes an xx-intercept where the gradient changes from positive to negative.

Ideas to try
  1. Before comparing graphs, trace ff from left to right and say: rising, undefined, rising, flat, then falling. Predict the sign of f′f' in each region.
  2. Pause at x=3x=3. Why is f′(3)=0f'(3)=0, and why must the derivative change from positive to negative there?
  3. As x→∞x\to\infty, the curve approaches y=2y=2. What happens to its gradient?