To determine the type of graph represented by the given equation [tex]\(\frac{x^2}{5} - y^2 = 1\)[/tex], let's analyze it step by step.
1. Identify the standard form:
The general form for a hyperbola is given by:
[tex]\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\][/tex]
2. Compare given equation to the standard form:
The given equation is:
[tex]\[
\frac{x^2}{5} - y^2 = 1
\][/tex]
When compared with the standard form of a hyperbola, we notice that:
- The term [tex]\(\frac{x^2}{a^2}\)[/tex] corresponds to [tex]\(\frac{x^2}{5}\)[/tex] indicating that [tex]\(a^2 = 5\)[/tex].
- The term [tex]\(\frac{y^2}{b^2}\)[/tex] corresponds to [tex]\(y^2\)[/tex]. Here, [tex]\(y^2\)[/tex] implies [tex]\(\frac{y^2}{1}\)[/tex], indicating that [tex]\(b^2 = 1\)[/tex].
3. Conclusion:
Given that the equation [tex]\(\frac{x^2}{5} - y^2 = 1\)[/tex] matches the form [tex]\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)[/tex] with positive constants [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex], it fits the standard form of a hyperbola.
Therefore, the graph of the equation [tex]\(\frac{x^2}{5} - y^2 = 1\)[/tex] will be a hyperbola.
