Inverse proportion and rational graph question with calculator-free methods

Expert intro: This type of question is really two skills in one: proportional reasoning and algebraic graph reading. The neat part is that both rely on spotting structure rather than doing long calculations.

Detailed walkthrough

(a) Inverse proportion

Since $V$ is inversely proportional to $t$,

$Vt=k$

Use $V=10$ when $t=3.6$:

$k=10\times 3.6=36$

So

$V=\frac{36}{t}$

(i)

As $t$ increases, $V$ decreases.

(ii)

When $V=3$:

$3t=36$

$t=12$

(b) Graph of $y=\frac{a}{x+2}$

A graph of this form always has:

• vertical asymptote at $x=-2$
• horizontal asymptote at $y=0$

To find $a$, use a point from the graph if one is given. Substitute the coordinates into

$y=\frac{a}{x+2}$

and solve for $a$.

Then draw the rest of the curve so that it approaches, but never crosses, the asymptotes.

💡 Pitfall guide

A frequent error is swapping the asymptotes: $x=-2$ is vertical, not horizontal. Another one is forgetting that the graph never touches the vertical asymptote.

When finding $a$, make sure you substitute both $x$ and $y$ carefully. A sign slip here changes the whole sketch.

🔄 Real-world variant

If the given value had been $V=5$ instead of $3$, you would still use $Vt=36$ and solve

$5t=36$

so $t=\frac{36}{5}=7.2$.

If the graph were $y=\frac{a}{x+2}$ with a point like $(-1,4)$, then $4=\frac{a}{1}$, giving $a=4$. The rest of the sketch would then follow the same asymptotes.

🔍 Related terms

inverse variation, hyperbola, vertical asymptote