Question

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+3)2/25−y2/16=1

Accepted Answer

Identify the center of the hyperbola from the equation $$\frac{(x+3)^2$}{25} - \frac{y^2$}{16} = 1. $$The center is at $$(-3, 0)$$ because the equation is in the form $$\frac{(x-h)^2$}{a^2$} - \frac{(y-k)^2$}{b^2$} = 1$$ where $$(h, k) is $$the center. Determine the values of $$a^2$$ and $$b^2$$ from the denominators: $$a^2 = 25$ and b^2$ = 16. $$Then find $$a$$ and $$b by $$taking the square roots: $$a = 5$$ and $$b = 4. $$Locate the vertices of the hyperbola. Since the $$x-$$term is positive and comes first, the transverse axis is horizontal. The vertices are $$a$$ units left and right from the center along the $$x-$$axis, so the vertices are at $$(-3 - 5, 0)$$ and $$(-3 + 5, 0). $$Find the foci using the relationship $$c^2$$ = $$a^2$$ + $$b^2. $$Calculate $$c by $$taking the square root of $$a^2$$ + $$b^2. $$The foci are located $$c$$ units left and right from the center along the $$x-$$axis, so their coordinates are $$(-3 - c, 0)$$ and $$(-3 + c, 0). $$Write the equations of the asymptotes. For a hyperbola with a horizontal transverse axis, the asymptotes are given by $$y - k = \pm \frac{b}{a}(x - h). $$Substitute $$h = -3$$, $$k = 0$$, $$a = 5$$, and $$b = 4 to $$get the equations of the asymptotes.

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+3)²/25−y²/16=1

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Concetti chiave

Standard Form of a Hyperbola

Vertices and Foci of a Hyperbola

Equations of the Asymptotes