Problem 4 - Algebra 2

Question

Which function has the widest graph? $$ \begin{array}{lllll}{\text { A } y=-4 x^{2}} & {\text { B } y=-1.2 x^{2}} & {\text { C } y=3.1 x^{2}} & {\text { D } y=11 x^{2}}\end{array} $$

Step-by-Step Solution

Answer

The function with the widest graph is \( y = -1.2x^2 \).

Step 1: Identify the Type of Functions

Each function given is a quadratic function of the form \( y = ax^2 \), where \( a \) is a constant. These are parabolas that open upwards or downwards based on the sign of \( a \). Since all coefficients (\( a \)) are positive or zero, the parabolas open upward.

Step 2: Determine What 'Widest Graph' Means

The 'widest graph' refers to the parabola with the smallest absolute value of the constant \( a \). For parabolas, the narrower the graph, the larger the absolute value of \( a \); conversely, the wider the graph, the smaller the absolute value of \( a \).

Step 3: Compare Coefficients of Each Function

List the coefficients: \(-4, -1.2, 3.1, 11\). Consider the absolute values: \( |-4| = 4\), \( |-1.2| = 1.2\), \( |3.1| = 3.1\), \( |11| = 11\). The smallest absolute value corresponds to \( -1.2 \).

Step 4: Identify the Function with the Smallest Absolute Coefficient

Since \( -1.2 \) has the smallest absolute value, the function \( y = -1.2x^2 \) has the widest graph among the options.

Key Concepts

Parabolas

Parabolas are U-shaped graphs that represent quadratic functions, which take the form of \( y = ax^2 + bx + c \). In the simplest case, like the exercise here, it's just \( y = ax^2 \). When you plot these functions, they create a symmetrical curve with a distinct high or low point called the vertex. Depending on the coefficient \( a \), the parabola can open upward or downward. Parabolas are a key concept in understanding quadratic functions and appear frequently in many mathematical contexts.

Coefficients

The coefficient \( a \) in the quadratic function \( y = ax^2 \) plays a crucial role in determining the shape and direction of the parabola. In general, this coefficient affects:

• The direction the parabola opens: upward if \( a > 0 \) and downward if \( a < 0 \).
• The width of the parabola, although this is determined by the absolute value of \( a \).

Coefficients are central to solving many problems involving parabolas because they define the graph's principal characteristics.

Graph Width

The width of a parabola's graph is determined by the absolute value of the coefficient \( a \). Smaller absolute values of \( a \) result in a wider graph, while larger absolute values make the parabola narrower. For instance, in the equation \( y = -1.2x^2 \), the coefficient \(-1.2\) has the smallest absolute value among the options given, hence it produces the widest graph. It's essential to account for the absolute size of \( a \) rather than its sign when deciding the graph's width.

Upward and Downward Opening

The direction that a parabola opens - up or down - is dictated by the sign of the coefficient in front of \( x^2 \). In the equation \( y = ax^2 \), if \( a \) is positive, the parabola will open upward, resembling a U shape. If \( a \) is negative, it opens downward, creating an inverted U. Understanding whether a parabola opens up or down is fundamental because it aids in predicting the behavior of the function for large values of \( x \). For example, even if \( a = -4 \) (a large negative), the parabola flips downward due to the negative sign.

Absolute Value

Absolute value, often indicated by straight brackets like \( |x| \), denotes the non-negative value of a number irrespective of its sign. It's used in parabola equations to focus on the magnitude of the coefficient \( a \), not whether it's positive or negative. In comparing functions for graph width, we assess the absolute value of the coefficients: \( |-4| = 4 \), \( |-1.2| = 1.2 \), \( |3.1| = 3.1 \), and \( |11| = 11 \). The smallest absolute value, like \( |-1.2| \), signals the widest graph amongst a set of equations.