Problem 68 - Algebra and Trigonometry

Question

Describe how to graph \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)

Step-by-Step Solution

Answer

The graph of the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) opens to the right and left with the center of the hyperbola at the origin (0,0), vertices at (3,0) and (-3,0), and the foci at \((\sqrt{10},0)\) and \((- \sqrt{10},0)\).

Step 1: Identify the Center of the Hyperbola

The centre of the hyperbola is at the origin (0,0) because there are no \(h\) and \(k\) values on the right side of the equation \(\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1\)

Step 2: Determine the Vertices

The vertices are determined by the value under \(x^{2}\) which is 9. The square root of 9 is 3, so the vertices are (3,0) and (-3,0) because \(a=3\).

Step 3: Determine the Foci

The foci are determined from the equation \(c=\sqrt{a^{2}+b^{2}}\). In this case, \(a=3\) and \(b=1\), so \(c=\sqrt{3^{2}+1^{2}}=\sqrt{10}\). So the foci are at \((\sqrt{10},0)\) and \((- \sqrt{10},0)\)

Step 4: Draw the Horizontal and Vertical Axes

Drawing the line \(x=0\) and \(y=0\) will give us the horizontal and vertical axes. The hyperbola is symmetric about these axes.

Step 5: Plot the Vertices

Plotting the vertices (3,0) and (-3,0) on the same axis will help to visualize the hyperbola.

Step 6: Draw the Hyperbola

From the origin (0,0), the hyperbola spreads out to pass through the vertices plotted in Step 5. Recall that the hyperbola opens left and right. Draw the hyperbola so it does not intersect the vertical axis. It should appear pinched in the middle, and spread out indefinitely to the left and right.