Question

In the following exercises, graph.

$\frac{{\left(x-2\right)}^2}{4}-\frac{{\left(y-3\right)}^2}{16}=1$

Short answer

The graph is as below

Step-by-step solution

Step 1. Given information

The equation is $\frac{{\left(x-2\right)}^2}{4}-\frac{{\left(y-3\right)}^2}{16}=1$

Draw the graph.

Step 2. Calculation

The x² term is positive, the transverse axis is horizontal.

The hyperbola opens left and right.

Find the centre

Compare the equation with standard form$\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$

So, $\left(h,k\right)=\left(2,3\right)$

$a^2=4\Rightarrow a=\pm 2$and $b^2=16\Rightarrow b=\pm 4$

Sketch the rectangle that goes through the points 2 units left/right of centre 4 units above/ below the centre .

Step 3. Draw the graph

Sketch the asymptotes and draw the graph using asymptotes .