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Chapter 3: Problem 87

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I must have made an error when graphing this parabola because its axis of symmetry is the \(y\) -axis.

Short Answer

Expert verified
The statement makes sense if the parabola opens upwards or downwards and its vertex is at the origin. It does not make sense if the parabola opens sideways.

Step by step solution

01

Understanding the Concept of Parabola and Axis of Symmetry

A parabola is a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). The axis of symmetry is the line that divides the parabola into two equal parts. For a parabola, the axis of symmetry is a vertical line if the parabola opens upwards or downwards, and it's a horizontal line if the parabola opens sideways.
02

Assessing the Statement

The statement says 'I must have made an error when graphing this parabola because its axis of symmetry is the y-axis.' This statement could make sense because the axis of symmetry for a normally positioned parabola (i.e., one that opens upwards or downwards) is a vertical line, which could coincide with the y-axis if the vertex of the parabola is at the origin (0,0). However, if the parabola opens sideways, the axis of symmetry would be a horizontal line, which cannot be the y-axis.
03

Conclusion

So, whether the statement makes sense or not depends on the orientation of the parabola. If the parabola opens upwards or downwards, then the statement makes sense. If it opens sideways, the statement does not make sense

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Use a graphing utility to graph \(y-\frac{1}{x^{2}}, y-\frac{1}{x^{4}},\) and \(y-\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y-\frac{1}{x^{n}} ?\)

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