Problem 4 Which function has the widest gr... [FREE SOLUTION]

Chapter 5: Problem 4

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

Short Answer

Expert verified

The function with the widest graph is $ y = -1.2x^2 $.

Step by step solution

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.

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 $.

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 $.

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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

Short Answer

Step by step solution

Identify the Type of Functions

Determine What 'Widest Graph' Means

Compare Coefficients of Each Function

Identify the Function with the Smallest Absolute Coefficient

Key Concepts

Parabolas

Coefficients

Graph Width

Upward and Downward Opening

Absolute Value

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