Lesson 5 of 5
Connect a positive repeated factor to intercept, range and asymptote.
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Worked example · 1 of 8
Worked example
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For , find the outputs at zero and one. Check: they are 2 and 6, so the ratio for one step is 3.
Big ideas
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For with , and , the y-intercept is . The base b is the output ratio for an input increase of one.
With , makes the function increasing, while makes it decreasing. Compare outputs from left to right rather than judging their heights alone.
For this unshifted positive-coefficient form, the domain is all real numbers and range is . The graph never reaches at a finite real input.
The horizontal asymptote is . A growth graph approaches it as x decreases without bound; a decay graph approaches it as x increases without bound.
Worked example
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For , plot , and . The graph decreases, has all real inputs and positive outputs, and approaches on the right without crossing it.
Step-through
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Worked example: sketch growth anchors
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The graph gets close to zero to the left but never reaches it.
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Moving left makes the output larger, not closer to zero.
Do this: Explain why the tempting shortcut fails.
Wrong idea: the horizontal asymptote is the y-intercept. For , the intercept is , while describes the far-left approach. One is an attained point; the other is a limiting line.
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Flashcards
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Card 1 of 8
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At , the power factor is 1.
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What is the y-intercept of ?
(0,7).
For , which bases make increase?
.
For , which bases make decrease?
.
What is the real domain of ?
All real numbers.
What is the range of ?
.
What is the horizontal asymptote of ?
.
What happens to as x decreases without bound?
It grows without bound.
Does an exponential graph require an x-intercept?
No. Positive never equals zero.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Write it
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Connect the ideas
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