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Chapter 6: Problem 65

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$x^{2}-6 x-2 y+7=0$$

Short Answer

Expert verified
The graph of the equation \(x^{2}-6x-2y+7=0\) represents a parabola.

Step by step solution

01

Identify The Form

The given equation \(x^{2}-6x-2y+7=0\) is a quadratic equation. To determine the type of conic section this equation represents (circle, parabola, ellipse, or hyperbola), one should pay attention to the power of \(x\) and \(y\). Given that there's no \(y^{2}\) term here and we have an \(x^{2}\) term, it implies that it is a parabola.
02

Convert Equation to Standard Form

For further clarity, we can rewrite the given equation in the standard form for a parabolic equation. The standard form of a parabolic equation is \(y = ax^2 + bx + c\). Rearranging the given equation to be expressed in terms of \(y\), we have \(y = 0.5x^{2} - 3x + -3.5\).
03

Graph Analysis

The graph of this equation would look like a parabola because the equation is quadratic (one variable, \(x\) in this case, is squared). When \(a>0\) as in our equation, the parabola opens upwards, giving a U-shape.

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