Question

given the following equation, identify the correct graph for the hyperbola. (type your letter response below: a,b,c or d) ((y - 4)^2 - \frac{(x - 1)^2}{4} = 1)

Explanation:

Step1: Identify hyperbola type

The equation is \((y - 4)^2-\frac{(x - 1)^2}{4}=1\), which is in the form \(\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1\) (vertical transverse axis hyperbola), center \((h,k)=(1,4)\), \(a = 1\), \(b = 2\).

Step2: Analyze graph features

• Vertical transverse axis means branches open up/down.
• Center at \((1,4)\) (close to \(x = 1,y = 4\) on grid).
• Graphs B (horizontal transverse) and C (vertical but center lower) and D (horizontal/center wrong) are eliminated. Graph A has vertical branches, center near \((1,4)\) (adjusted for grid), matching the equation.

Answer:

A