Wemazi

Topic 5.9

Identify which function a graph represents

Builds on matching a function with its derivative. 6 minutes.

Three graphs on one axis

Graphs A, B and C share one x-axis. Drag the orange point: on A the orange line is the tangent, and the readout below gives its slope. On B the orange bar shows the height at the same x.

At every \(x\), the slope of A equals the height of B. At \(x = 2\), both are 9.

Press "Show the slope of B". The same holds for B and C: at \(x = 2\), the slope of B and the height of C are both 12.

Graph B records the slope of A, and C records the slope of B. Each is the derivative of the graph above it.

Any three graphs

Each dashed line joins a flat point of one graph to a zero of another graph. At a flat point the slope is 0, so the graph that records the slope has height 0 there.

Press "Sort the graphs". Each graph slides above the graph that records its slope, and its letter becomes its name.

The top graph is \(f\), the middle graph is \(f'\) and the bottom graph is \(f''\).

Notation

  1. The derivative \(f'\) records the slope of \(f\): its height at \(x\) is the slope of \(f\) at \(x\).

    \[\cy{f'(x)} = \text{slope of } f \text{ at } x\]

  2. The second derivative \(f''\) records the slope of \(f'\) in the same way.

    \[\cy{f''(x)} = \text{slope of } f' \text{ at } x\]

  3. To name a graph, find a flat point. The slope there is 0, so the graph that records this slope crosses the x-axis at the same \(x\).

  4. If a problem says "the graph of \(f'\)", its height is the slope of \(f\), whatever the shape of the graph.

When it fails

Flip the middle graph upside down and call it \(g\). It is still zero wherever \(f\) is flat, but it is not the derivative of \(f\).

At the point shown \(f\) rises, so its slope is positive, but \(g\) is below the axis. Matching zeros are not enough, because the signs must match too.

Example 1: name three graphs

The graphs P, Q and R show a function and its first two derivatives, in some order. P is a W shape, Q is a cubic and R is a parabola. Name each graph.

  1. P is flat at \(x = -1\), 0 and 1, where Q is zero. Q is also positive where P rises, so Q records the slope of P.

  2. Q is flat at about \(x = \pm 0.6\), where R is zero. R is also positive where Q rises, so R records the slope of Q.

  3. P records the slope of neither Q nor R, so P is \(f\), Q is \(f'\) and R is \(f''\).

Example 2: where does f have a maximum?

The graph shows \(f'\), a parabola that crosses the x-axis at 0 and 2 and peaks at \(x = 1\), where it is 3 high. Where does \(f\) have a local maximum?

  1. The graph of \(f'\) peaks at \(x = 1\).

  2. A tempting wrong move: The graph peaks at \(x = 1\), so \(f\) has a local maximum at \(x = 1\).

    \[f \text{ has a maximum at } x = 1\]

    The graph shows the slope of \(f\), not its height. At \(x = 1\) the slope is 3, so \(f\) is still rising.

  3. The slope of \(f\) is 0 where \(f'\) meets the x-axis, at \(x = 0\) and \(x = 2\).

    \[f'(0) = 0,\quad f'(2) = 0\]

  4. Between 0 and 2, \(f'\) is above the axis, so \(f\) rises. Right of 2, \(f'\) is below the axis, so \(f\) falls and has a local maximum at \(x = 2\).

Practice

The graphs U, V and W are a function and its first two derivatives, in some order. U is a line, V is a parabola and W is a cubic. Which graph is \(f\)?

Move the point to \(x = 2\). Compare the slope of each graph with the heights of the other two.

Solution

  1. At \(x = 2\), the slope of W is 4 and the height of V is 4, so V records the slope of W.

  2. At \(x = 2\), the slope of V is also 4 and the height of U is 4, so U records the slope of V.

  3. W records the slope of neither U nor V, so W is \(f\).

Questions

Graphs A, B and C are \(f\), \(f'\) and \(f''\) of one function, in some order.

  1. Which graph is \(f\)?

    • Graph A
    • Graph B
    • Graph C
    • It cannot be told without formulas
    Answer

    Graph C

  2. What is the slope of graph C at \(x = 0\)?

    • 0
    • 1
    • −1
    • It cannot be read from the graphs
    Answer

    1

  3. The line \(y = 2x - 6\) is the graph of \(f'\). Where is the lowest point of \(f\)?

    • \(x = 3\)
    • \(x = -6\)
    • \(x = 0\)
    • \(f\) has no lowest point
    Answer

    \(x = 3\)

Summary

The height of \(f'\) is the slope of \(f\), and the height of \(f''\) is the slope of \(f'\).

To name a graph, compare its height with the slopes of the other graphs.

Use it when graphs come without names, or when a problem gives the graph of \(f'\) and asks about \(f\).

This is one of five skills in topic 5.9 of Which & Why: AP Calculus.