Question

As \(|p|\) increases, how does the width of the graph of the equation \(y=\frac{1}{4 p} x^2\) change? Explain your reasoning.

Short answer

As the absolute value of \(p\), that is \(|p|\), increases, the width of the parabolic curve of the equation \(y=\frac{1}{4 p} x^2\) increases.

Step-by-step solution

Understand the shape of the graph

Remember that the graph of a quadratic equation \(y = ax^2 + bx + c\) is a parabola. In this case, we don’t have the terms with \(x\) and constant term, so the parabola will open upwards (if \(a>0\)) or downwards (if \(a<0\)), and the vertex will be at the origin.

Determine the effect of \(p\) on the graph

The coefficient of \(x^2\) is \(\frac{1}{4p}\). This coefficient, being a reciprocal function of \(p\), will become larger as \(p\) gets smaller and vice versa. Therefore, as \(|p|\) (the absolute value of \(p\)) increases, the coefficient \(\frac{1}{4p}\) gets smaller.

Explain the change in the width of the graph

The width of the parabola is inversely proportional to the absolute value of the coefficient of \(x^2\). This means if the coefficient is large, the graph is 'narrow' and if the coefficient is small, the graph is 'wide'. As established in the previous step, as \(|p|\) increases, the coefficient \(\frac{1}{4p}\) decreases. Therefore, as \(|p|\) increases, the width of the graph increases.

Key concepts

Parabola Shape and Direction

Understanding the shape and direction of a parabola is crucial when studying quadratic equations. A parabola is the graph that represents quadratic equations of the form
\( y = ax^2 + bx + c \). When the leading coefficient, \( a \), is positive, the parabola opens upwards like a smile. Conversely, if \( a \) is negative, the parabola opens downwards, resembling a frown. The vertex of the graph is the highest or lowest point on the parabola, and for the equation \( y = \frac{1}{4p} x^2 \), the vertex is at the origin. This simplified quadratic equation without the \( x \) and constant terms highlights how changes in the coefficient affect the graph's shape without the influence of linear or constant shifts.
By exploring the parabola's shape and direction, we gain insight into the behavior of quadratic functions and their graphs.

Effect of Coefficient on Parabola Width

The parabola's width is significantly affected by the coefficient of the \( x^2 \) term in a quadratic equation. To see how this plays out, consider the quadratic function \( y = \frac{1}{4p} x^2 \). In this equation, \( \frac{1}{4p} \) represents the coefficient of the \( x^2 \) term.
As the absolute value of \( p \) increases, \( |p| \) gets larger and thus \( \frac{1}{4p} \) becomes smaller. The effect is a wider parabola since the width is inversely proportional to the coefficient's absolute value. If we imagine \( p \) shrinking, the absolute value of the coefficient would increase, leading to a narrower parabola. This relationship is crucial for predicting and understanding the effects of various coefficients on the spread of a parabola, allowing students to visualize and interpret the changes to the graph's shape as the coefficient varies.

Absolute Value Influence in Quadratic Equations

In the context of quadratic equations, like \( y = \frac{1}{4p} x^2 \), the absolute value plays a pivotal role in determining the graph's characteristics. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, \( |p| \) represents the magnitude of \( p \) without considering its sign.
When we talk about how \( |p| \) influences the graph, we focus on its size. As \( |p| \) increases, regardless of whether \( p \) is positive or negative, the coefficient \( \frac{1}{4p} \) decreases in absolute value, which means the parabola expands outward and becomes wider. Since the absolute value ensures that we only consider the magnitude, the direction of \( p's \) effect on the graph remains consistent—it affects only width, not direction. This allows students to isolate the impact of the coefficient's magnitude on the graph's width from other graph features like orientation and vertex position.