Problem 58 - Introductory Algebra for College Students

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 it is symmetric with respect to the \(y\) -axis

Step-by-Step Solution

Answer

The statement 'I must have made an error when graphing this parabola because it is symmetric with respect to the y-axis' does not make sense because a parabola can be symmetric with respect to the y-axis if its general equation is of the form \(y = ax^2 + c\), where \(b = 0\).

Step 1: Identify the axis of symmetry of a general parabola

The axis of symmetry of a parabola defined by the equation \(y = ax^2 + bx + c\) is the line \(x = -\frac{b}{2a}\).

Step 2: Compare with the y-axis

The y-axis is the line \(x = 0\). For a parabola to be symmetric with respect to the y-axis, the axis of symmetry must coincide with the y-axis. That is, \(-\frac{b}{2a}\) must be equal to 0, which implies that \(b\) must be 0.

Step 3: Analyze the statement

The given statement is 'I must have made an error when graphing this parabola because it is symmetric with respect to the y-axis.' This implies that the speaker believes that a parabola cannot be symmetric about the y-axis, which is not necessarily true. It is possible for a parabola defined by \(y = ax^2 + c\) (where \(b = 0\)) to be symmetric with respect to the y-axis.

Key Concepts

Axis of Symmetry

The axis of symmetry is an important concept when dealing with parabolas, especially in quadratic functions. A parabola is perfectly symmetric around this axis, which acts like a mirror line dividing the parabola into two equal halves.

For a quadratic function written in the standard form, \(y = ax^2 + bx + c\), the axis of symmetry is given by the formula \(x = -\frac{b}{2a}\). This vertical line helps in finding the vertex, the highest or lowest point on the parabola, which is an essential feature while graphing.

In essence, if you were to fold the parabola along the axis of symmetry, both sides would match perfectly. Understanding the axis of symmetry helps not just in graphing but also in solving and analyzing quadratic equations.

Graphing Parabolas

Graphing parabolas involves creating a visual representation of a quadratic function. It provides a clear understanding of the function's behavior, such as where it opens upwards or downwards and how wide or narrow it is.

When graphing a parabola:

• First, determine the direction of the parabola by looking at the sign of \(a\). If \(a\) is positive, the parabola opens upwards, resembling a smile. If \(a\) is negative, it opens downwards, resembling a frown.
• Find the vertex using the formula for the axis of symmetry. Substitute \(x = -\frac{b}{2a}\) back into the equation to find the \(y\)-coordinate of the vertex.
• Plot the vertex and a few additional points on either side of the axis of symmetry to guide the curve.
• Draw a smooth curve through the points, ensuring it is symmetrical about the axis.

Mastering graphing allows one to easily analyze the real-world situations modelled by quadratic functions, like projectile motion and optics.

Quadratic Functions

Quadratic functions have a distinct characteristic that sets them apart from linear functions. The general form of a quadratic function is \(y = ax^2 + bx + c\). These are used extensively in algebra and calculus for modelling various phenomena.

Key properties of quadratic functions:

• They always form a parabola on a graph.
• The coefficient \(a\) determines the width and the direction of the parabola.
• Quadratic functions can have either one vertex (a single peak or trough) located at \((x, y)\), where \(x = -\frac{b}{2a}\).

Quadratic equations often pop up in problems involving areas, physics, and when calculating maximum and minimum values. By understanding these functions well, you can easily tackle real-life problems that involve quadratic relationships. The simplicity and power of quadratic functions make them a fundamental mathematical tool.