To determine which graph represents the hyperbola given by the equation [tex]\(\frac{x^2}{5^2} - \frac{y^2}{4^2} = 1\)[/tex], we need to understand the standard form and properties of a hyperbola in this context.
The general form of the hyperbola given is:
[tex]\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\][/tex]
where [tex]\(a\)[/tex] is the semi-major axis length along the [tex]\(x\)[/tex]-axis, and [tex]\(b\)[/tex] is the semi-minor axis length along the [tex]\(y\)[/tex]-axis. For our specific equation:
[tex]\[
\frac{x^2}{5^2} - \frac{y^2}{4^2} = 1
\][/tex]
we identify [tex]\(a = 5\)[/tex] and [tex]\(b = 4\)[/tex].
### Step-by-Step Solution:
1. Identify the type of hyperbola:
- From the equation [tex]\(\frac{x^2}{5^2} - \frac{y^2}{4^2} = 1\)[/tex], we see that the [tex]\(x^2\)[/tex] term is positive and the [tex]\(y^2\)[/tex] term is negative. This indicates a hyperbola that opens horizontally, which is of the general form [tex]\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)[/tex].
2. Determine the vertices:
- The vertices of the hyperbola are located at [tex]\(\pm a\)[/tex] along the [tex]\(x\)[/tex]-axis since it opens horizontally. Thus, the vertices are at [tex]\((\pm 5, 0)\)[/tex], which means [tex]\((5, 0)\)[/tex] and [tex]\((-5, 0)\)[/tex].
3. Determine the asymptotes:
- The equations of the asymptotes for a hyperbola of this form are given by [tex]\(y = \pm \frac{b}{a} x\)[/tex].
- Plug in [tex]\(a = 5\)[/tex] and [tex]\(b = 4\)[/tex] to get the slopes:
[tex]\[
y = \pm \frac{4}{5} x
\][/tex]
- Therefore, the equations of the asymptotes are [tex]\(y = \frac{4}{5} x\)[/tex] and [tex]\(y = -\frac{4}{5} x\)[/tex].
### Conclusion:
A graph representing this hyperbola will have:
- Vertices at [tex]\((5, 0)\)[/tex] and [tex]\((-5, 0)\)[/tex].
- Asymptotes with equations [tex]\(y = \frac{4}{5} x\)[/tex] and [tex]\(y = -\frac{4}{5} x\)[/tex].
- The hyperbola will open horizontally towards [tex]\(x\)[/tex]-axis.
Make sure the graph you choose has these characteristics: two branches opening to the left and right along the [tex]\(x\)[/tex]-axis, through the points [tex]\((5,0)\)[/tex] and [tex]\((-5,0)\)[/tex], and approaching the lines [tex]\(y = \frac{4}{5} x\)[/tex] and [tex]\(y = -\frac{4}{5} x\)[/tex] as [tex]\(x\)[/tex] moves further away from the origin.
Thus, any graph matching this description is the correct representation of the hyperbola [tex]\(\frac{x^2}{5^2} - \frac{y^2}{4^2} = 1\)[/tex].
