Question

In the following exercises, graph.

$\frac{{\left(y-4\right)}^2}{9}-\frac{{\left(x-2\right)}^2}{25}=1$

Short answer

The graph is as below

Step-by-step solution

Step 1. Given information

The equation is $\frac{{\left(y-4\right)}^2}{9}-\frac{{\left(x-2\right)}^2}{25}=1$

Draw the graph.

Step 2. Calculation

The y² term is positive, the transverse axis is vertical.

The hyperbola opens up and down.

Find the centre

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

So, Centre $\left(h,k\right)=\left(2,4\right)$

$a^2=9\Rightarrow a=\pm 3$and $b^2=25\Rightarrow b=\pm 5$

Sketch the rectangle that goes through the points 3 units above/below the centre

5 units left/right of centre

Step 3. Draw the graph

Sketch the graph using asymptotes.

Draw the graph is