Problem 29 - Basic Technical Mathematics with Calculus

Question

Use a calculator to graph all three parabolas on the same coordinate system. Describe (a) the shifts and (b) the stretching and shrinking. (a) \(y=x^{2}\) (b) \(y=3 x^{2}\) (c) \(y=\frac{1}{3} x^{2}\)

Step-by-Step Solution

Answer

(a) has no shift, (b) is narrower (stretched), (c) is wider (compressed).

Step 1: Graph the Basic Parabola

Graph the parabola for the equation (a) \( y = x^2 \). This graph opens upwards and has its vertex at the origin (0,0). The parabola is symmetric around the y-axis, and has the shape of a U.

Step 2: Graph the Stretched Parabola

For equation (b) \( y = 3x^2 \), the parabola is also centered at the origin, but it will be narrower compared to \( y = x^2 \). The factor 3 stretches the parabola vertically, meaning it grows faster as \( x \) moves away from zero.

Step 3: Graph the Compressed Parabola

For equation (c) \( y = \frac{1}{3}x^2 \), this parabola is wider than \( y = x^2 \). The factor \( \frac{1}{3} \) compresses the parabola, making it grow more slowly as \( x \) moves away from zero.

Step 4: Summary of Shifts and Transformations

None of the parabolas (a, b, or c) have been shifted horizontally or vertically; they all remain centered at the origin. In terms of stretching and shrinking, (b) stretches the parabola making it narrower, while (c) compresses it making it wider compared to (a).

Key Concepts

Graphing Transformations

Graphing transformations involve changing the position and shape of a graph. For parabolas, transformations typically include shifts and stretches or shrinks. When graphing a parabola like the one described in the exercise, you start with the basic parabola, which is rooted at the function \(y = x^2\). This basic shape looks like a U and has its vertex at the origin (0,0).

Transformations can alter this graph by changing its shape or position on the coordinate plane. For example, shifting the graph means moving the entire parabola without altering its shape. Shifts can be horizontal or vertical. However, in the given exercise, none of the parabolas have been shifted as per the solutions, meaning their vertices remain at the origin.

Understanding transformations allows you to predict and sketch how different functions will behave without plotting numerous points.

Quadratic Functions

Quadratic functions are mathematical equations of the form \(y = ax^2 + bx + c\). These functions graph as parabolas, which can open upwards or downwards, depending on the sign of the coefficient \(a\). A positive \(a\) leads to an upward-opening parabola, while a negative \(a\) means it opens downward.

For the exercise provided, the focus is on specific quadratic functions of the form \(y = ax^2\) without the \(b\) and \(c\) components. Parabolas that are formed from these quadratic functions have their vertices at the origin (0,0). The coefficient \(a\) determines the width and direction.

The beauty of quadratic functions lies in their symmetry and predictability, making them a fundamental concept in algebra and geometry. When you understand the structure and behavior of quadratic functions, graphing and manipulating them becomes much simpler.

Vertical Stretching and Shrinking

Vertical stretching and shrinking refer to altering the width of a parabola. This happens by changing the coefficient \(a\) in the quadratic equation \(y = ax^2\). For instance, the basic graph \(y = x^2\) has an \(a\) value of 1, representing a standard parabola.

Vertical stretching occurs when \(a\) is greater than 1, as seen in the function \(y = 3x^2\) from the exercise. Here, the graph is narrower and rises more steeply. This indicates that the parabola grows faster as \(x\) values move away from zero, giving the impression that the graph is being pulled vertically.

On the other hand, vertical shrinking happens when \(a\) is between 0 and 1. For example, the function \(y = \frac{1}{3}x^2\) results in a wider graph. This shows the parabola grows more slowly as \(x\) moves away from the origin, as though the graph is being compressed vertically.

When dealing with quadratic functions, understanding the effect of \(a\) is crucial for precise graphing and for knowing how adjustments in \(a\) lead to changes in the graph's appearance.