Lesson 4 of 6
Calculate average change and distinguish increasing from positive.
One idea at a time. Try the example, then keep going.
Big ideas · 1 of 12
Big ideas
Do this: Read one idea, then explain it in your own words.
The Average rate of changeHow much the output changes per unit of input over an interval.Exactly: For , the average rate of change of over is . Its units are output units per input unit.Mix-up alert: Divide the output change by the input change, in the same order.See this in Key words from a to b is , with . Its units are output units per input unit.
A function is increasing on an interval when greater inputs give greater outputs. Outputs can be negative while the function increases.
An output describes a height; a rate describes change in height. A large output does not automatically mean a large rate.
A local extreme compares nearby outputs. An absolute maximum or minimum compares all outputs in the stated domain, including allowed endpoints.
Step-through
Do this: Follow the labeled graph. Use the four steps to connect its points with the equation.
Worked example: a secant rate
Whole picture, 4 steps
4 steps. Tap a number or Show me how it works.
Do this: Cover the next step and try it yourself.
It is not the total output change 15.
Do this: Cover the next step and try it yourself.
Do this: Explain why the tempting shortcut fails.
Wrong idea: negative outputs mean decreasing. Increasing/decreasing compares successive outputs as x moves right. The rule increases everywhere even when its output is below zero.
Big ideas
Do this: Read one idea and locate it on the example graph.
A function is positive where its graph lies above the x-axis and negative where it lies below. A zero belongs to neither strict sign interval.
An x-intercept has output zero; a y-intercept has input zero. For a straight segment, use its slope or linear interpolation to find a crossing between labeled points.
For this example, report increasing and decreasing behavior on open intervals between vertices. Include a finite domain endpoint in a sign interval when its output has the required sign.
Interior turning points compare nearby values on both sides. Absolute maximum and minimum values compare the whole domain, including filled endpoints; endpoint conventions for local extrema are not needed here.
Step-through
Do this: Sketch the stated segments, then follow the four steps beside the drawing.
The graph joins the five listed filled vertices by straight segments, in order, with no other points.
Whole picture, 4 steps
4 steps. Tap a number or Show me how it works.
Do this: Sketch the vertices and compare the interior peak with both endpoints.
Both interior troughs, at and , attain the absolute minimum −2.
Worked example
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Worked example
Do this: Predict the next step on paper, then compare the worked reasoning.
The rate is .
Tip
Do this: Open your own class copy at the listed section.
Class handout reference · Unit 2, section 2.5, pages 31-32, and key features section 2.6, pages 33-36. Use your class copy.
Watch
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Optional packet practice
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All 12 numbered questions on printed pages 31 and 32: 16 answer parts. Graph questions use the labeled source coordinates; original curves stay in the PDF. Worked steps and hints are original instructional support.
Question 1 of 16
Find the average rate of change of the function over the interval [32, 60].
Interval:
Average rate of change is change in output divided by change in input. Subtract in the same order in the numerator and denominator.
Use when your teacher permits it. Keep working alongside it; closing the calculator keeps your calculations.
Flashcards
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Card 1 of 8
0 Know it0 Still learning
Change in output over change in input.
On the card: Enter or Space flips. Left and Right move. After revealing, 1 marks Still learning and 2 marks Know it. Tab reaches every control.
What is the average rate formula from a to b?
, for .
What are units of a distance-versus-time rate?
Distance units per time unit, such as meters per second.
Can a function increase below the x-axis?
Yes. For example, x−10 increases while .
What does a negative average rate mean?
The final output is smaller than the initial output when .
What is a secant line?
A line through two distinct points on a graph.
What is a local minimum?
An output no greater than nearby outputs in the domain.
What must an absolute extreme comparison include?
All allowed inputs, including any included endpoints.
For , what is every average rate?
3, because every secant slope is the same.
True or false
Do this: Choose true or false; explain what would make a false statement correct.
Self-test
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6 questions. One at a time.
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Self-test
Do this: Try each problem without a timer. Read the feedback before moving on.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Write it
Do this: Write two or three sentences, then compare with the model.
Start here if it helps: I would first identify...
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Self-test
Do this: Sketch each coordinate description before choosing; check sign, trend and endpoints separately.
4 questions. One at a time.
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Write it
Do this: Draw on paper, write your conclusions here, then compare with the model.
Start here if it helps: My graph joins the vertices without extending past them. Its domain is...
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Write it
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Start here if it helps: My argument starts with...
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Match
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Connect the ideas
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Homework · 3 points
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Homework
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https://gradebridge-beryl.vercel.app/learn/precalculus/functions/rates-and-graphs#homework
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