Problem 34 - Intermediate Algebra

Question

How does the graph of \(y=4 x^{2}\) compare to the graph of \(y=2 x^{2}\) ? Explain your answer.

Step-by-Step Solution

Answer

The graph of \(y=4x^2\) is narrower than \(y=2x^2\) because 4 > 2, leading to a steeper curve.

Step 1: Understanding the Parabola Equation

Both given equations are in the form of a parabola, specifically vertically oriented parabolas because they are quadratic in terms of 'x'. The general form for such parabolas is \( y = ax^2 \), where \( a \) is a constant determining the parabola's width and direction.

Step 2: Analyze the Coefficient's Role

The coefficient \( a \) in \( y = ax^2 \) affects the parabola's width and "stretch." If \( |a| > 1 \), the parabola is narrower compared to \( y = x^2 \), and if \( |a| < 1 \), it is wider.

Step 3: Compare the Coefficients

For \( y = 4x^2 \), the coefficient \( a \) is 4, and for \( y = 2x^2 \), \( a \) is 2. Since 4 > 2, the graph of \( y = 4x^2 \) is narrower than \( y = 2x^2 \).

Step 4: Conclusion on Graph Comparison

Given that both parabolas open upwards (since both coefficients are positive), \( y = 4x^2 \) will appear narrower (steeper) than \( y = 2x^2 \), because the larger coefficient results in a steeper ascent.

Key Concepts

Understanding Quadratic Functions

Quadratic functions are an essential concept in mathematics, mostly represented by the equation \(y = ax^2 + bx + c\). In the context of parabolas, where the interest is in vertical changes, this simplifies to \(y = ax^2\). Here, 'a' is the key player. It determines how the parabola will look like in terms of width and direction. Parabolas are symmetrical graphs shaped like a "U" or an inverted "U" depending on the sign of 'a'. If \(a > 0\), the parabola opens upward, resembling a bowl, while if \(a < 0\), it opens downward.

• The vertex of the parabola is the central point where it either peaks or troughs.
• The y-intercept is where the parabola crosses the y-axis, and in \(y = ax^2\), this is always the origin when \(b\) and \(c\) are zero.

Understanding the role of the quadratic component, \(ax^2\), lays the groundwork for exploring how changes in its coefficient affect the parabola's graph.

Exploring the Coefficient Effect

The coefficient 'a' in a quadratic function, \(y = ax^2\), plays a critical role in defining the parabola's appearance. Specifically, it influences the parabola's width and steepness. Here's how it works:

• If \(|a| > 1\), the graph of the parabola becomes narrower than that of \(y = x^2\), indicating it is "stretched" vertically.
• If \(|a| < 1\), the parabola widens, as it appears more "flattened."
• When \(a = 0\), the quadratic function doesn't form a parabola either, since it lacks the squared term.

Given the equations \(y = 4x^2\) and \(y = 2x^2\), both coefficients are greater than 1. However, \(4 > 2\), leading to \(y = 4x^2\) being narrower compared to \(y = 2x^2\). This difference highlights how even slight changes in the coefficient can significantly alter a parabola's graph.

Investigating Graph Transformations

Graph transformations describe how we can modify a graph's appearance by changing parameters within its equation. With quadratic functions, the coefficient 'a' is central to these transformations. Depending on its value, we see the parabolas "shift" in shape:

• "Stretching" occurs when \(|a|\) increases, causing the parabola to become steeper and narrower.
• "Compressing" happens if \(|a|\) decreases, resulting in a wider, more flattened graph.

These transformations permit robust comparisons, such as those between \(y = 4x^2\) and \(y = 2x^2\). For these two equations, since both are upwards-opening, the only visible difference is in their width, where \(y = 4x^2\) is visibly narrower due to a larger coefficient. Understanding how these parameters affect the overall graph contributes to a deeper grasp of quadratic functions and real-life applications they may relate to, like projectile motion or optimizing product shapes.