Turn a curve sketch into its gradient graph

Original graph f(x)Source-faithful qualitative diagram; distances are schematic unless an exact function is stated.xyOA (−4a,0)B (−3a,2)C (−2a,0)D (−a,−2)ax=ay=0
Question diagram 1
Original graph f(x)Source-faithful qualitative diagram; distances are schematic unless an exact function is stated.xyOA (−4a,0)B (−3a,2)C (−2a,0)D (−a,−2)ax=ay=0

Given graph of f with A, B, C, D, origin and its asymptotes.

The diagram shows the graph of y=f(x)y=f(x). The curve passes through the origin OO, the points A(−4a,0)A(-4a,0) and C(−2a,0)C(-2a,0). The points B(−3a,2)B(-3a,2) and D(−a,−2)D(-a,-2) are stationary points and the lines y=0y=0 and x=ax=a are asymptotes of the curve.

Sketch the graph of y=f′(x)y=f'(x), indicating clearly the points corresponding to AA, BB, CC and DD, and the asymptotes, where applicable. marks

[modified 2010/RVHS/Prelim/II/1]

Solution

Derivative sketch

Gradient graphs

The graph of y=f(x)y=f(x) passes through OO, A(−4a,0)A(-4a,0) and C(−2a,0)C(-2a,0). B(−3a,2)B(-3a,2) and D(−a,−2)D(-a,-2) are stationary points, and y=0y=0 and x=ax=a are asymptotes. Sketch y=f′(x)y=f'(x), marking the points corresponding to AA, BB, CC, DD and the asymptotes.

Explore this idea

Hint

  1. Trace the gradient from left to right: positive when the curve rises, negative when it falls.

  2. Stationary points become xx-intercepts of y=f′(x)y=f'(x).

  3. Carry the vertical asymptote across and turn the horizontal asymptote into y=0y=0 for the gradient.

Qualitative derivative reconstruction
Qualitative derivative reconstructionScroll across to see the full diagram.

Answer · Derivative sketch

Zeros (−3a,0),(−a,0)(-3a,0),(-a,0); asymptotes x=a,y=0x=a,y=0.

A′ has positive slope ordinate, C′ negative; their numerical ordinates are unspecified. Limits follow the particular source sketch.

Explore the idea

Read the gradient from the curve (a = 1)

Drag along the curve. Upper graph: the curve and its tangent. Lower graph: the gradient. Rising stretches are shaded.

Position along the curvex = -3.5
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)
1.2
Estimated gradient f′(x)
2.6
positive

The model rises from A towards B; the gradient is positive inside this stretch.

Selected values are within the original drawing windows.

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

Turning points of the curve become xx-intercepts of the gradient graph; rising stretches map to the part above the axis.

Ideas to try
  1. Find the two places where the gradient graph crosses the axis. What are they on the original?
  2. Where is the gradient most negative?