Problem 87 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

The statement does not make sense as the axis of symmetry of a parabola cannot be the y-axis. The axis of symmetry of a parabolic function is always a vertical line, which only coincides with the y-axis if the parabola opens to either right or left and the vertex of the parabola is at the origin. However, in this case, this parabola wouldn't be a function.

Step 1: Analyze the Statement

Let's start by analyzing the statement: 'I must have made an error when graphing this parabola because its axis of symmetry is the y-axis.' Here, it's implied that the axis of symmetry of the parabola is the y-axis.

Step 2: Recall the Properties of a Parabola

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 for a parabola is the line that divides the parabola into two equal halves. It is a vertical line for a standard upright or downside parabola and runs through the vertex of it.

Step 3: Evaluate The Statement Based on The Properties

Based on the properties of a parabola, the axis of symmetry cannot be the y-axis. The axis of symmetry only coincides with the y-axis if the vertex of the parabola is at the origin and the parabola opens to the right or to the left. But in this case, the parabola would not be a function which contradicts the initial assumption of graphing a function. Thus, the statement that implying the axis of symmetry of a parabolic function is the y-axis does not make sense.