D1.7 Connect secant and tangent slopes with rates

Ontario Grade 12 Mathematics

Characteristics of Functions

Using graphs and numerical values to describe average and instantaneous change

A graph can show how one quantity changes as another quantity changes. For example, a graph of distance against time can show how a trip progresses. The slope between two points describes the average change over an interval. The slope of a tangent at one point describes the rate of change at that input. This lesson connects these two ideas using graphs and numerical values.

What you will learn

Explain how a secant slope represents an average rate of change.
Connect a tangent slope at a point to the rate of change at that input.
Estimate a tangent slope using nearby function values.

1. Prerequisite bridge: slope and rate

Slope measures how much the vertical value changes compared with the horizontal value. For two points on a graph, subtract their vertical coordinates and divide by the difference in their horizontal coordinates. A positive slope means the graph rises from left to right. A negative slope means it falls. A slope of zero means the two points have the same vertical value.

A rate of change uses the same calculation. It compares a change in one quantity with a change in another. If distance is measured in kilometres and time in hours, the slope has units of kilometres per hour. The units help explain what the rate means.

A function assigns an output to each allowed input. On its graph, the input is shown horizontally and the output vertically. When comparing outputs, use the same function at two different inputs.

m=\frac{y_2-y_1}{x_2-x_1}

Slope compares vertical change with horizontal change.
Rate units come from the output units divided by the input units.

2. Secant slopes: average change over an interval

A secant line is a straight line through two points on a graph. Its slope measures the average rate of change between the two corresponding inputs. The word average matters: the graph may rise or fall differently between those inputs, but the secant records the overall change from one endpoint to the other.

Imagine a distance-time graph with one point at the start of a trip and another point later. The secant slope between them gives the average speed over that time interval. It does not say that the speed was constant at every moment.

For a function, choose two inputs, such as

a

and

b

. Read the corresponding outputs,

f(a)

and

f(b)

. The average rate is the change in output divided by the change in input. Reversing the order of both differences gives the same slope, as long as the inputs are different.

\frac{f(b)-f(a)}{b-a}

A secant uses two distinct points on the graph.
Its slope describes average change across the interval.

3. Tangent slopes: change at one input

A tangent line at a point is a straight line that follows the graph’s direction at that point. Its slope describes the graph’s rate of change at that input. This is an instantaneous rate of change: a rate assigned to one input rather than spread across a wider interval.

A tangent slope can be estimated from a graph by drawing a tangent line and choosing two convenient points on that line. Those points do not have to lie on the curve. Calculate the slope between them. A careful graph and well-chosen points make the estimate more useful.

A numerical estimate can also come from secant slopes. Keep one point fixed and choose a second point closer and closer to it. The secant slopes then give information about the tangent slope. If the nearby slopes settle around one value, that value is a reasonable estimate for the tangent slope.

A graph with a sharp corner may not have one clear direction at that point, so a single tangent slope may not be available there. For a smooth part of a graph, the tangent line gives a useful local description. In this lesson, estimates are based on the graph or nearby numerical values.

\text{rate at }a\approx\frac{f(a+h)-f(a)}{h}

A tangent slope represents the rate at one input.
Nearby secant slopes can help estimate a tangent slope.
A rate at one input has the same type of units as the corresponding secant rates.

4. Connect the representations

The graph, the numbers, and the rate description tell the same story in different ways. On a graph, a secant is a line through two curve points. In a table, its slope comes from two input-output pairs. In context, that slope is the average rate over the interval.

For a tangent, the graph shows the direction at one point. Numerically, nearby secant slopes estimate that direction. In context, the tangent slope describes the rate at that input. For example, if the input is time and the output is distance, the tangent slope estimates the speed at that time.

Always name the interval or input when describing a rate. Saying that a rate is

4

is incomplete if the units and the relevant interval are unknown. A complete interpretation might say that the average output increases by about

4

units for each input unit from one specified input to another.

Secant: two points, an interval, and an average rate.
Tangent: one point and a rate at that input.
Interpret the sign, size, and units of a slope in context.

Nearby secant slopes for the worked example

Second input	Output at second input	Secant slope from $x=3$
$2.9$	$8.41$	$5.9$
$3$	$9$	Not a second point
$3.1$	$9.61$	$6.1$

Worked example

Estimating a rate at one input

For

f(x)=x^2

, estimate the rate of change at

x=3

using nearby values. Interpret the result as a tangent slope estimate.

Find the central output
The function gives the vertical value at each input. At the input $3$ , its output is $9$ . This point will stay fixed while we compare it with nearby points.
$f(3)=3^2=9$
Compare with a nearby input on the left
Use $2.9$ as the second input. The secant slope from $x=3$ to $x=2.9$ is the change in output divided by the change in input. Both changes are negative, so the slope is positive.
$\frac{f(2.9)-f(3)}{2.9-3}=\frac{8.41-9}{-0.1}=5.9$
Compare with a nearby input on the right
Now use $3.1$ . The output is $9.61$ , so this nearby secant slope is slightly larger. The two slopes are close to $6$ and lie on opposite sides of it.
$\frac{f(3.1)-f(3)}{3.1-3}=\frac{9.61-9}{0.1}=6.1$
Estimate the tangent slope
The nearby secant slopes suggest that the graph’s direction at $x=3$ has slope about $6$ . This is a numerical estimate of the rate at that input, not an average over a wide interval.
$m_{\text{tangent}}\approx 6$

Answer: The rate of change at

x=3

is estimated as about

6

output units per input unit.

Check: The left and right secant estimates,

5.9

and

6.1

, are close to one another. Their closeness supports an estimate near

6

.

Common mistakes and how to avoid them

Calling a secant slope the rate at one exact input.

Correction: A secant slope is an average over the interval between two inputs. A tangent slope describes the rate at one input.

Using the change in output but forgetting to divide by the change in input.

Correction: A rate compares two changes. Divide the output change by the input change.

Changing the order in only one difference.

Correction: Keep the order consistent. If the output change is second point minus first point, the input change must use the same point order.

Treating a numerical tangent estimate as exact.

Correction: Nearby secants provide an estimate. Report it as approximate unless the information given establishes an exact value.

Lesson summary

A secant line passes through two points on a graph.
A secant slope gives the average rate of change between two inputs.
A tangent line follows the graph’s direction at one point.
A tangent slope represents the rate of change at that input.
Nearby secant slopes can be used to estimate a tangent slope.

Check your understanding

Question 1

A function’s output rises by

15

while its input rises by

5

. What is the average rate of change over that interval?

$3$ output units per input unit
$10$ output units per input unit
$75$ output units per input unit
$\frac{1}{3}$ output units per input unit

Show answer and explanation

3

output units per input unit

Divide the output change by the input change:

15\div 5=3

. This is an average rate over the interval.

Question 2

Which statement best describes a tangent slope at an input?

It gives the average rate across any two inputs.
It describes the rate of change at that input.
It is always zero.
It is the vertical change without considering horizontal change.

Show answer and explanation

It describes the rate of change at that input.

A tangent slope describes the graph’s direction at one input, so it represents the rate at that input.

Question 3

Nearby secant slopes from the left and right of an input are about

4.8

and

5.2

. What is a reasonable tangent slope estimate?

About $0.4$
About $5$
About $10$
About $25$

Show answer and explanation

About

5

The two nearby slopes are close to

5

, so about

5

is a reasonable estimate.

Key terms

Slope: The vertical change divided by the horizontal change between two points.
Rate of change: A comparison of how much one quantity changes for a change in another quantity.
Secant line: A straight line that passes through two points on a graph.
Tangent line: A straight line that follows the direction of a graph at one point.
Average rate of change: The rate calculated across an interval between two inputs.
Instantaneous rate of change: The rate of change at one input, described by the tangent slope.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D1.7. It is a study resource, not an official curriculum publication.

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D1.7 · Connect secant and tangent slopes with rates

Ontario Grade 12 Mathematics

Using graphs and numerical values to describe average and instantaneous change

What you will learn

1. Prerequisite bridge: slope and rate

2. Secant slopes: average change over an interval

3. Tangent slopes: change at one input

4. Connect the representations

Nearby secant slopes for the worked example

Estimating a rate at one input

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson and its review