Question

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 4y2−x2=1

Accepted Answer

Rewrite the given equation in the standard form of a hyperbola. The equation is $$4y^2$$ - $$x^2 = 1$. Divide both sides by 1 to isolate the right side, then express it as $$\frac{$$y^2$$}{\frac{1}{4}} - \frac{$$x^2$$}{1} = 1$$. Identify the values of a^2$ and b^2$ from the standard form $$\frac{$$y^2$$}{$$a^2$$} - \frac{$$x^2$$}{$$b^2$$} = 1$$. Here, a^2$ = \frac{1}{4}$$ and $$b^2 = 1$. Determine the orientation of the hyperbola. Since the y^2$ term is positive and comes first, the hyperbola opens vertically (up and down). The center is at the origin (0$$,0)$$ because there are no x$ or y$ shifts. Find the coordinates of the vertices using a$. The vertices are located at (0$$, \pm a)$$, so calculate a$$ = \sqrt{\frac{1}{4}}$$ and write the vertices as (0$$, \pm a)$$. Find the foci using c$, where c^2$ = a^2$ + b^2$. Calculate c$$ = \sqrt{$$a^2$$ + $$b^2$$}$$, then write the foci coordinates as (0$$, \pm c)$$. Next, find the equations of the asymptotes, which for a vertical hyperbola are y$$ = \pm \frac{a}{b} x$$. Substitute the values of a$ and b$ to write the asymptote equations.

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 4y²−x²=1

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Key Concepts

Standard Form of a Hyperbola

Vertices and Foci of a Hyperbola

Equations of Asymptotes