Question

Sketch the graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$4 x^{2}-y^{2}=16$$

Short answer

Answer: The vertices of the hyperbola are (±2, 0), the foci are (±2√5, 0), and the equations of the asymptotes are y = ±2x.

Step-by-step solution

Rewrite the equation in the standard form of a hyperbola

To rewrite the equation in the standard form, we need to perform the following steps: 1. Divide the given equation by the constant term on the right side of the equation. 2. Identify the values of a^2, b^2, and the type of hyperbola (horizontal or vertical). 3. Write the equation in the standard form. Divide the given equation by 16: $$\frac{4x^2}{16} - \frac{y^2}{16} = 1$$ Now the equation is in this form: $$\frac{x^2}{4} - \frac{y^2}{16} = 1$$ We have a^2 = 4 and b^2 = 16. Since the x term comes first and has the positive sign, this is a horizontal hyperbola.

Find the coordinates of the vertices

The coordinates of the vertices for a horizontal hyperbola are (±a, 0), using the value of a that we found in step 1. Vertices: (±2, 0)

Find the coordinates of the foci

To find the coordinates of the foci, we first need to find the value of c using the equation: $$c = \sqrt{a^2 + b^2}$$ Substitute the values of a^2 (4) and b^2 (16): $$c = \sqrt{4 + 16} = \sqrt{20} = 2 \sqrt{5}$$ The coordinates of the foci for a horizontal hyperbola are (±c, 0), so the foci are (±2√5, 0).

Find the equations of the asymptotes

For a horizontal hyperbola, the equations of the asymptotes are: $$y = \pm \frac{b}{a}x$$ Using the values of a and b that we found in step 1 (a=2 and b=4), the asymptotes are: $$y = \pm \frac{4}{2}x$$ $$y = \pm 2x$$

Check the work with a graphing utility

Use a graphing utility to plot the given equation (4x² - y² = 16) and verify that our findings for the vertices, foci, and asymptotes are correct. The graph should show a horizontal hyperbola with the given vertices, foci, and asymptotes. Summary: Vertices: (±2, 0) Foci: (±2√5, 0) Asymptotes: y = ±2x

Key concepts

Vertices

In the world of hyperbolas, vertices are crucial points that help define the shape and orientation of the graph. For a hyperbola in standard form, the vertices are derived from the equation. In our example, the hyperbola equation is \( \frac{x^2}{4} - \frac{y^2}{16} = 1 \). Here, \( a^2 = 4 \), so \( a = 2 \). Since this is a horizontal hyperbola, the vertices lie along the x-axis at \( (\pm a, 0) \). This means the vertices are located at \( (2, 0) \) and \( (-2, 0) \).
These points mark the closest approach of the hyperbola to its center, playing a pivotal role in its symmetric shape. Fully understanding the position of vertices helps in sketching and analyzing hyperbolas effectively.

Foci

The foci of a hyperbola are points inside the curves that suggest how open or narrow the hyperbola is. They are not on the curve itself but provide depth to the graph's structure. To locate the foci, we use the formula \( c = \sqrt{a^2 + b^2} \). In our example, \( a^2 = 4 \) and \( b^2 = 16 \), so \( c = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \).
For a horizontal hyperbola, the foci are positioned at \( (\pm c, 0) \). This places them at approximately \( (2\sqrt{5}, 0) \) and \( (-2\sqrt{5}, 0) \). Understanding the foci is crucial because they demonstrate the unique property of hyperbolas: the difference in distances from any point on the curve to the two foci is constant.

Asymptotes

Asymptotes provide guidelines for the hyperbola's behavior at infinity, showing the direction in which each branch of the curve heads. For a horizontal hyperbola like \( \frac{x^2}{4} - \frac{y^2}{16} = 1 \), the equations of the asymptotes are derived from the relationship \( y = \pm \frac{b}{a}x \). Here, \( a = 2 \) and \( b = 4 \), leading to the asymptote equations \( y = \pm 2x \).
These asymptotes, represented by lines that intersect at the center of the hyperbola, act like invisible barriers that the hyperbola approaches but never touches. Visualizing them helps understand the extended behavior of the graph.