D. Characteristics of Functions · Rates of Change

Reading Rates of Change from Graphs

D1.5 D1.7 D1.9About 70 minutes

Learning goals

I can tell from a graph where the rate of change is positive, negative or zero, and where it is greatest or least.
I can estimate tangent slopes at several points and sketch the graph of the rate of change.
I can describe a function's graph from the graph of its rate of change.
I can explain that a zero rate of change does not always mean a maximum or minimum.

Words to know

turning point
A point where a graph changes from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). The rate of change there is 0.

local maximum / minimum
A point higher (or lower) than all nearby points on the graph.

point of inflection
A point where the graph changes from concave up to concave down, or the reverse. Often the steepest point nearby.

slope graph (rate-of-change graph)
A graph of the tangent slope of f against x. In MCV4U it is called the graph of the derivative.

A graph is full of rates

Look at any graph and you can read rates of change without a single calculation: rising means positive, falling means negative, flat means zero, steep means large. In this lesson you will make that reading precise and go one step further: you will graph the rate of change itself.

That new graph is a preview of the derivative, the main idea of Grade 12 Calculus and Vectors (MCV4U).

What to look for

Where the graph is increasing, the instantaneous rate of change is positive.
Where it is decreasing, the rate is negative.
At a local maximum or local minimum (a turning point) of a smooth graph, the tangent is horizontal, so the rate is 0.
The rate is greatest where the graph climbs most steeply and least (most negative) where it falls most steeply. These often happen at points of inflection, where the bend changes direction.

Remember from Polynomial Functions and Equations: a cubic like f(x) = x³ − 3x has up to two turning points. They are exactly the places where its rate of change is 0.

For f(x) = x³ − 3x, the rate of change is positive, then negative, then positive. It is 0 at the local maximum (−1, 2) and the local minimum (1, −2), and most negative at the origin.

Worked example

Example 1: Building a slope graph

For f(x) = x³ − 3x, estimate the tangent slopes at x = −2, −1, 0, 1 and 2 with centred intervals (h = 0.001). Plot the points (x, slope) and describe the graph of the rate of change.

x = −2: (f(−1.999) − f(−2.001)) ÷ 0.002 ≈ 9.
x = −1: ≈ 0. x = 0: ≈ −3. x = 1: ≈ 0. x = 2: ≈ 9.
Plot (−2, 9), (−1, 0), (0, −3), (1, 0), (2, 9). They form a U-shape: a parabola opening up with zeros at x = ±1 and vertex (0, −3).
Check: this parabola is y = 3x² − 3, and 3(2)² − 3 = 9 ✓.

The rate-of-change graph of the cubic f(x) = x³ − 3x is the parabola y = 3x² − 3. It is positive where f increases, negative where f decreases, and crosses 0 at f's turning points.

Top: f(x) = x³ − 3x with short tangent segments. Bottom: the slope at each of those points. Where f has a turning point (dotted lines), the slope graph crosses 0.

Reading backwards

You can also go from a slope graph to the function:

Slope graph above the x-axis → f is increasing there.
Slope graph below the axis → f is decreasing.
Slope graph crosses from + to − → f has a local maximum. From − to + → a local minimum.
Slope graph has a peak or a valley → f is steepest there.

Worked example

Example 2: From a slope graph to the story

The rate of change of the water level in a reservoir (in cm per day) is negative from day 0 to day 10, zero at day 10, and positive from day 10 to day 25, with its largest value at day 18. Describe the water level.

Days 0 to 10: rate negative, so the level is falling.
Day 10: the rate changes from − to +, so the level reaches its lowest point (local minimum).
Days 10 to 25: rate positive, so the level is rising.
Day 18: the rate is greatest, so the level is rising fastest then (perhaps after heavy rain or spring snowmelt).

The level falls until day 10, reaches a minimum, then rises, climbing fastest around day 18.

Worked example

Example 3: Zero rate without a turning point

Show that f(x) = x³ has a rate of change of 0 at x = 0, but no maximum or minimum there.

Centred interval at x = 0, h = 0.001: (0.001³ − (−0.001)³) ÷ 0.002 = 0.000001. As h → 0 this approaches 0. (The slope formula from The Difference Quotient lesson, 3a², also gives 0.)
But f(−0.1) = −0.001 < 0 = f(0) < 0.001 = f(0.1). The function is increasing on both sides.
So the tangent at the origin is horizontal (y = 0), but the graph keeps rising through it.

A rate of change of 0 does not guarantee a maximum or minimum. Check the sign of the rate on both sides: it must change sign for a turning point.

Using technology

Graphing tools make these ideas quick to explore. In Desmos or GeoGebra you can graph f, add a point that slides along it, and draw the tangent at that point (many tools have a tangent command, or you can use the slider method from Secants Approaching a Tangent). Have the tool trace the point (x, slope) as you slide: the trace is the slope graph. Use this to check your hand sketches, not to replace them; tests often ask you to sketch by hand.

Just for fun

Calculus teachers have a famous party trick: show them any smooth graph and they instantly sketch its slope graph. Nobody has ever been impressed by this at a party, but on a test it is a superpower.

Hands-on activity

Swap and sketch

You need: two sheets of graph paper, a pencil, a ruler and a partner (a family member is perfect).

Each person draws a smooth, wavy graph across the page with at least two peaks or valleys. Keep it a function (no loops, no vertical parts).
Swap papers. On a second set of axes directly below the curve (same x-scale), sketch the slope graph of your partner's curve:

Mark every peak and valley: the slope graph is 0 there.
Mark the steepest uphill and downhill points: the slope graph has its peaks and valleys there.
Decide the sign of the slope between them.

Use a ruler to draw tangents at three points and measure their slopes. Check them against your sketch.
Swap back and check each other's work.

What to look for: every peak and valley of the curve lines up with a zero of the slope graph, every steepest point lines up with a peak or valley of the slope graph, and the slope graph is above the axis exactly where the curve rises.

Your turn

For f(x) = x³ − 3x, on which intervals is the rate of change positive? Negative?

Where is the rate of change of f(x) = x³ − 3x equal to 0? What happens at those points?

What is the rate of change of f(x) = x³ − 3x at x = 0? Why is that significant on the interval −1 < x < 1?

A parabola opens up with its vertex at x = 2. Describe its slope graph.

A function's slope graph is positive for x < 3, zero at x = 3 and negative for x > 3. What happens to the function at x = 3?

Is it true that every point with rate of change 0 is a maximum or a minimum? Give an example.

A ball's height graph is a downward parabola with its top at t = 2 s. Describe the graph of its rate of change (its velocity).

More practice: warm-up

If a function is increasing on an interval, what is the sign of its rate of change there?

What is the rate of change at a local maximum of a smooth graph?

What is the slope graph of a constant function such as y = 7?

What is the slope graph of y = 2x + 1?

Where is a graph steepest, in terms of its slope graph?

More practice: on your own

A table gives x = 0, 1, 2, 3, 4, 5, 6 and f(x) = 0, 5, 8, 9, 8, 5, 0. Use centred intervals to estimate the rate of change at x = 1 to 5, and describe the slope graph.

A town's population grows slowly, then quickly, then slowly again as it approaches a limit. When is its rate of change greatest? Describe the slope graph.

A function's slope graph is always positive and increasing. Describe the function.

For f(x) = x⁴ − 2x², estimate the slopes at x = −2, −0.5, 0.5 and 2 (centred, h = 0.001), and find where the rate is 0.

Using technology (or careful reasoning), explain how you would find where the graph of y = x³ − 3x has slope 9.

Challenge

Use the difference quotient to prove that the tangent slope of f(x) = x³ − 3x at x = a is 3a² − 3.

Explain why, at a point where the slope graph has a local maximum, the original graph changes from concave up to concave down.

Sketch (describe) the slope graph of y = sin x, then the slope graph of that slope graph. What do you notice?

Check your understanding

Quiz

If a smooth function is decreasing on an interval, its instantaneous rate of change there is:

At a local maximum or minimum of a smooth function, the rate of change is:

What is the slope graph of y = x²?

A slope graph crosses the x-axis at x = 3, going from positive to negative. What does the function do at x = 3?

What is the rate of change of f(x) = x³ − 3x at x = 0?

f(x) = x³ has a rate of change of 0 at x = 0. Which is true at x = 0?

For parents and teachers

Big ideas (D1.5, D1.7, D1.9): a graph shows its own rates of change: positive while rising, negative while falling, zero at turning points, largest at the steepest points. Plotting tangent slopes against x gives the slope graph, which MCV4U calls the derivative.

How to help: Look at any graph together (a stock price, the temperature over a day, a hiking trail's elevation profile) and ask: "Where is it changing fastest? Where does it stop changing for a moment? Is that a peak, a valley, or neither?"

Common difficulties:

Confusing the value of a function with its rate. A high point on f does not mean a high point on the slope graph (it means a zero).
Placing zeros of the slope graph at the zeros of f instead of at its turning points.
Assuming every zero rate is a maximum or minimum (x³ at 0 is the classic counterexample).
Drawing the slope graph on a different x-scale, so features do not line up.