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Chapter 6: Problem 69

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 \(x^{2}-6 x-2 y+7=0\) represents a parabola. More specifically, it is a vertical parabola since the coefficient of the \(x^{2}\) term is positive.

Step by step solution

01

Identify the Form of the Equation

The equation given is \(x^{2}-6 x-2 y+7=0\). This equation contains only \(x^{2}\) term. There is no \(y^{2}\) term in the equation. This single squared term indicates that the equation represents a parabola.
02

Analyzing the Coefficients

As the coefficient of \(x^{2}\) is positive, this means we have a vertical parabola. A parabola can open upwards or downwards depending on if the leading term is positive or negative.
03

Rewriting the Equation

We can rewrite the equation in vertex form for better clarity: \(x^{2}-6 x=-2 y+7 \to x^{2} - 6x = -2(y - \frac{7}{2})\). This is the vertex form of a parabola \((x-h)^{2}=4p(y-k)\) where \((h, k)\) is the vertex.

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Most popular questions from this chapter

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