Hyperbola and Exponential Transformations

Hyperbola and Exponential Transformations

Hyperbolas and exponentials are recognised by their asymptotes and growth behaviour. For $y=a/(x-p)+q$, the vertical asymptote is $x=p$ and horizontal asymptote is $y=q$. For $y=ab^x+q$, $y=q$ is the horizontal asymptote; $b>1$ gives growth and $0<b<1$ gives decay. Domain and range follow the asymptotes.

Worked reasoning

For $y=2/(x-1)-3$, asymptotes are $x=1$ and $y=-3$. The graph never crosses either line, so $x\ne1$ and $y\ne-3$.

Exam method

1. Identify the parent family. 2. Read shifts from brackets/outside constants. 3. Draw asymptotes first, then transform key behaviour and state restrictions.

Restrictions are part of the answer

A graph sketch is incomplete without its domain and range restrictions. For a shifted hyperbola, values making the denominator zero are excluded from domain, and the horizontal-asymptote value is excluded from range. For an exponential, outputs may approach an asymptote forever without reaching it. Write these restrictions alongside dashed asymptote lines. This links formula, graph and set language, and avoids the common error of drawing a curve crossing an impossible line.

One-minute retrieval: Hyperbola and Exponential Transformations

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.