Problem 76 - Algebra and Trigonometry

Question

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=c x^{2}$$ (a) \(c=1, \frac{1}{2}, 2,4 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=1,-1,-\frac{1}{2},-2 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Step-by-Step Solution

Answer

The sign of \(c\) affects the opening direction of the parabola, and the absolute value \(|c|\) affects its width.

Step 1: Analyze Function Family Characteristics

The family of functions given is \(f(x) = c x^2\). This represents a set of parabolas opening upwards if \(c > 0\), and downwards if \(c < 0\). The parameter \(c\) changes the width and direction (up or down) of the parabola.

Step 2: Graph Functions in Part (a)

In Part (a), we graph \(f(x) = c x^{2}\) for \(c = 1, \frac{1}{2}, 2, 4\) within the viewing rectangle \([-5,5]\) by \([-10,10]\). - For \(c=1\), the graph is a standard parabola.- For \(c=\frac{1}{2}\), the parabola is wider than when \(c=1\).- For \(c=2\) and \(c=4\), the parabolas are narrower than when \(c=1\), with \(c=4\) being the narrowest among those values.

Step 3: Graph Functions in Part (b)

In Part (b), we graph \(f(x) = c x^{2}\) for \(c = 1, -1, -\frac{1}{2}, -2\) within the same viewing rectangle.- \(c=1\) repeats the standard upward parabola.- For \(c=-1\), the parabola opens downwards and has the same width as for \(c=1\). - For \(c=-\frac{1}{2}\), the downward-opening parabola is wider than with \(c=-1\).- For \(c=-2\), the parabola opens downwards and is narrower than for \(c=-1\).

Step 4: Analyze How the Constant 'c' Affects the Graph in Part (c)

The value of \(c\) affects both the direction and the width of the parabola:- A positive \(c\) results in parabolas opening upwards.- A negative \(c\) results in parabolas opening downwards.- The absolute value \(|c|\) determines the width: smaller \(|c|\) values mean wider parabolas, while larger \(|c|\) values mean narrower parabolas.

Key Concepts

Quadratic Functions

Quadratic functions form a fundamental concept in algebra and are often represented by the equation \(f(x) = ax^2 + bx + c\). The most basic form is the one centered around the origin, \(f(x) = ax^2\). This particular equation is pivotal as it describes parabolic graphs.

A parabola is a U-shaped curve that can open either upwards or downwards. The direction in which the parabola opens is determined by the sign of the coefficient \(a\) (which corresponds to \(c\) in our specific problem \(f(x) = cx^2\)).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

Parabolas have a unique center point called the vertex, and a line of symmetry that typically runs vertically through this vertex point. Understanding quadratic functions is crucial because they form the basis for exploring more complex relationships in mathematics.

Graph Transformations

Graph transformations help us understand how changing certain parameters in an equation affects the shape and position of the graph. In the equation \(f(x) = c x^2\), the parameter \(c\) controls the width and the direction of the parabola. Adjusting \(c\) allows us to visualize these changes effectively.

**Effect of \(c\) on Graph Width and Direction** When \(c\) changes, the parabola becomes either wider or narrower:

• When \(|c| < 1\), the graph widens compared to the standard parabola.
• When \(|c| > 1\), the graph narrows.
• For \(c > 0\), the graph opens upwards, while for \(c < 0\), it opens downwards.

These transformations are crucial in understanding how a seemingly simple change in a parameter can lead to entirely different graph behaviors, providing deeper insights into function characteristics.

Function Families

Function families consist of a group of functions that share specific common characteristics, making it easier to study them as a unit. This allows us to understand variations within a set of functions using a single variable. In our problem, we explored a family of quadratic functions expressed as \(f(x) = c x^2\).

**Characteristics of the Quadratic Function Family**

• Each member of the family differs by the value of \(c\), determining the exact shape of its graph.
• This family of functions shares the same general form, all parabolas, but vary in width and direction based on \(c\).'
• Investigating these differences helps us make predictions about the behavior of similar functions and apply these in various real-world scenarios.

By analyzing function families, students can develop a keen ability to predict and understand how changing parameters affect a function's graph. This understanding is integral not only in mathematics but also in fields like physics, engineering, and economics where these concepts are applied.