Exponentials, Hyperbolas and Circles
Recognise the connection between algebraic and graphical representations of exponential relations and solve related exponential equations.
Three more curve families. Each has one giveaway feature that identifies it instantly.
Method
Exponential
Passes through $(0,1)$. Has a horizontal asymptote at $y = 0$.
Hyperbola
Two branches. Asymptotes on both axes. Never touches either.
Circle centred at the origin
Not a function — it fails the vertical line test.
An asymptote is approached, never reached
This is what makes these curves different from a parabola.
Exponential growth beats any polynomial eventually
Worked example
e.g.
- The first has in the index, so it is exponential.
- Find its key point and asymptote.
- The second has in the denominator, so it is a hyperbola.
- State its asymptotes.
- The third has both variables squared and summed — a circle.
- Read the radius.
Practice
Look at where sits. That identifies the family.
1
Answer
2
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3
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4
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5
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6
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