Question

the graph of a hyperbola is given. match the graph to its equation below.
choose the correct answer below.
$\frac{x^{2}}{25}-y^{2}=1$
$x^{2}-\frac{y^{2}}{25}=1$
$\frac{y^{2}}{25}-x^{2}=1$
$y^{2}-\frac{x^{2}}{25}=1$

Explanation:

Step1: Recall the standard form of hyperbola

The standard form of a hyperbola opening horizontally is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\), and opening vertically is \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\).

Step2: Analyze the graph

The given hyperbola opens horizontally. So its equation should be of the form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\).

Step3: Determine \(a\) value

From the graph, the vertices are at \(x=\pm5\). For \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\), \(a\) is the distance from the center to the vertex. So \(a = 5\), and \(a^{2}=25\).

Answer:

\(\frac{x^{2}}{25}-y^{2}=1\)