Problem 27 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=-4 x^{2}\) can be obtained from the graph of \(y=x^{2}\) by vertically stretching by applying a factor of ___________ and reflecting across the ___________ -axis.

Step-by-Step Solution

Answer

The factor is 4; reflected across the x-axis.

Step 1: Identify the Base Graph

The base graph given in the problem is \( y = x^2 \). This is a standard parabola that opens upwards, with its vertex at the origin \((0, 0)\).

Step 2: Analyze the Transformed Equation

We are given the transformed equation \( y = -4x^2 \). In this equation, the coefficient of \( x^2 \) is \(-4\).

Step 3: Determine the Vertical Stretching Factor

The absolute value of the coefficient \(-4\) indicates the factor by which the graph is stretched. The graph is vertically stretched by a factor of \(4\), because the multiplicative factor applied to the square term is \(4\).

Step 4: Determine the Reflection Axis

Since the coefficient of \( x^2 \) is negative \(-4\), the graph is reflected across the x-axis. This is because inverting the sign of the function causes a reflection over the x-axis.

Key Concepts

Graph Transformations

Graphs of functions can often be changed through transformations. Understanding these transformations can help us see how a simple base graph changes into a more complex one. In this exercise, we start with the base graph of the function \( y = x^2 \), which is a classic quadratic function shaped like a parabola opening upwards.

Transformations come in a few different types: translations, reflections, stretching, and compressions. Each of these changes the graph in different ways. In our case, we will focus on applying reflections and stretching to change the appearance of the graph. This method is valuable because by visualizing and understanding transformations, you can quickly sketch the graph and predict the shape of complex quadratic functions with ease.

Vertical Stretch

When a function is multiplied by a number, it affects how "tall" or "short" the graph appears. This is known as vertical stretching or compression, depending on the factor. If the factor is greater than 1, the graph stretches vertically and appears taller. If the factor is between 0 and 1, the graph compresses and becomes shorter.

In the equation \( y = -4x^2 \), the factor is \(-4\). The absolute value of this factor, which is \(4\), tells us that the graph is vertically stretched. Essentially, every point on the graph of \( y = x^2 \) is moved farther away from the x-axis by a factor of 4, making it appear taller.

Here’s the rundown:

• Multiply positive and negative outputs by 4
• Graph becomes 'taller' compared to the original
• The other transformations are affected independently

This stretching doesn’t change the graph's general shape but rather its height at corresponding x-values.

Reflection Across Axes

Reflections are transformations that flip graphs across specific lines called axes. For quadratic functions like \( y = x^2 \), reflections often occur across the x-axis or the y-axis.

In our equation \( y = -4x^2 \), the negative sign indicates a reflection. Because the negative is placed before the coefficient of \( x^2 \), it tells us the reflection is across the x-axis. Imagine folding the graph over the x-axis, causing it to flip its "arms" from upwards to downwards.

Key points about reflecting across the x-axis:

• Positive outputs become negative outputs
• The entire graph turns upside down
• It visually inverts the graph

These reflections change the direction the graph opens, helping create a parabola that opens downwards. However, it doesn’t affect the width or height of the graph, which is only influenced by factors modifying the function, like stretches.