Coaching Centre
Circles, exponentials and hyperbolas
Equations of circles (x − h)² + (y − k)² = r², exponential graphs and their asymptotes, and hyperbolas y = k/x with translations.
Circles
Find the centre and radius of $(x - 2)^2 + (y + 1)^2 = 9$.
Hyperbolas
The two asymptotes shift with h and v. A negative k puts the branches in the other pair of quadrants.
Centre of $(x - 2)^2 + (y + 1)^2 = 9$?
Signs flip inside the brackets.
Radius of that circle?
$\sqrt{9}$.
Radius of $x^2 + y^2 = 25$?
$\sqrt{25}$.
Does $(3, 4)$ lie on $x^2 + y^2 = 25$?
$9 + 16 = 25$.
Horizontal asymptote of $y = \dfrac{2}{x} + 1$: $y = $ ?
The $+1$ shifts it up.
Vertical asymptote of $y = \dfrac{1}{x - 3}$: $x = $ ?
The denominator is zero at $x = 3$.
$y = \dfrac{4}{x}$. Find $y$ when $x = 2$.
$4 \div 2$.
The graph of $y = 3^x$ has:
It approaches the $x$-axis as $x \to -\infty$.
New numbers every time. Keep going until you get five in a row.