Question

USING STRUCTURE Which function represe the parabola with the widest graph? Explain your reasoning. (A) \(y=2(x+3)^2\) (B) \(y=x^2-5\) (C) \(y=0.5(x-1)^2+1\) (D) \(y=-x^2+6\)

Short answer

Among function (A), (B), (C), and (D), function (C) represents the parabola with the widest graph

Step-by-step solution

Identify the Functions

First, identify the functions at hand. The given functions are: (A) \(y=2(x+3)^2\), (B) \(y=x^2-5\), (C) \(y=0.5(x-1)^2+1\), and (D) \(y=-x^2+6\).

Isolate Coefficients

Second, isolate the coefficients of each function. The coefficients 'a' are: For function (A) 'a' is 2, for function (B) 'a' is 1 (since no coefficient is visible, it has a default value of 1), for function (C) 'a' is 0.5, and for function (D) 'a' is -1.

Determine the Widest Parabola

Next, determine the widest parabola. The function with the neighborhood of 'a' that is smallest, regardless of the sign, will give the widest parabola. Thus, the function with 'a' = 0.5 (function C) has the widest parabola.

Key concepts

Parabolas

Parabolas are symmetrical, U-shaped graphs that represent quadratic functions. They play a crucial role in various fields, such as physics and engineering, due to their unique properties related to paths and shapes. A parabola can either open upwards or downwards depending on the signs of its coefficients.
The vertex of a parabola is the highest or lowest point, depending on the direction it opens, and it's a critical aspect of its shape.
Parabolas also have an axis of symmetry, which is a vertical line running through the vertex, dividing the parabola into two mirror-image halves. Understanding the geometric shape and properties of parabolas helps in visualizing and solving problems within quadratic equations.

Coefficients

Coefficients in quadratic functions can be thought of as the numbers in front of variables that determine the parabolic curve's width and direction. In the general quadratic equation form, \[y = ax^2 + bx + c\],

• The 'a' coefficient affects the openness of the parabola.
• Larger values of a (whether positive or negative) produce narrower parabolas.
• Smaller absolute values of a produce wider parabolas.
• If a is positive, the parabola opens upward, and if negative, it opens downward.

Coefficients heavily influence how steep or flat a parabola appears. Thus, correctly identifying 'a' helps determine the parabola's orientation and size, as seen in the exercise solution.

Graphing Quadratic Equations

Graphing quadratic equations allows us to visually interpret and analyze the behavior of quadratic functions. By translating the equation's algebraic form to a geometric shape on a coordinate plane, critical features such as intercepts, vertex, and axis of symmetry become clearer.
The process typically involves:

• Identifying the direction in which the parabola opens (upward or downward).
• Locating the vertex, which provides a central point for drawing.
• Using the axis of symmetry for plotting further points on the graph.

The ability to graph aids in both understanding and predicting the function's behavior, ensuring holistic problem-solving skills when dealing with quadratic equations.

Function Transformation

Function transformation involves altering the graph of a function in various ways including scaling, translating, and reflecting. In terms of quadratic functions, transformation determines the position and shape of a parabola on the graph.
Some common transformations are:

• Translation, shifting the graph vertically or horizontally without altering its shape.
• Dilation, adjusting the 'a' coefficient to widen or narrow the parabola.
• Reflection, changing the sign of 'a' to flip the parabola about the x-axis.

Among these, adjusting the value of 'a' directly influences the parabola’s width, as shown in the exercise where function C with the smallest |a| had the widest parabola. Mastering function transformation enables better handling of diverse mathematical models and graphs.