Problem 72 - 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 that 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) \(c=1,-1,-\frac{1}{2},-2 ; \quad[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Step-by-Step Solution

Answer

The value of \(c\) affects the width and direction of the parabola; larger \(|c|\) results in narrower parabolas and positive \(c\) opens upwards, while negative \(c\) opens downwards.

Step 1: Understand What the Function Represents

The family of functions given is \(f(x) = c x^2\). This represents a set of parabolas. The coefficient \(c\) determines the shape and direction of each parabola. When \(c\) is positive, the parabola opens upwards, and when \(c\) is negative, it opens downwards.

Step 2: Graph Functions for Part (a)

For \(c=1, \frac{1}{2}, 2, 4\) on the viewing rectangle [-5,5] by [-10,10], draw the graphs of each function:- For \(c=1\), the parabola is \(y = x^2\).- For \(c=\frac{1}{2}\), the parabola is \(y = \frac{1}{2}x^2\), which is wider than \(y=x^2\).- For \(c=2\), the parabola is \(y = 2x^2\), which is narrower than \(y=x^2\).- For \(c=4\), the parabola is \(y = 4x^2\), even narrower than \(y=2x^2\). Plot these parabolas on the same set of axes to see how changing \(c\) affects the width of the parabola.

Step 3: Graph Functions for Part (b)

For \(c=1, -1, -\frac{1}{2}, -2\) on the viewing rectangle [-5,5] by [-10,10], draw the graphs:- For \(c=1\), we already have \(y = x^2\) from part (a).- For \(c=-1\), the parabola is \(y = -x^2\), which opens downwards and is the reflection over the x-axis of \(y=x^2\).- For \(c=-\frac{1}{2}\), the parabola is \(y = -\frac{1}{2}x^2\), which is wider and opens downwards.- For \(c=-2\), the parabola is \(y = -2x^2\), which is narrower and opens downwards. Plot these on the same set of axes as the previous graphs to compare.

Step 4: Analyze and Conclude (Part c)

From the graphs, we observe:- As \(c\) increases (positive values), the parabola becomes narrower.- As \(c\) decreases (negative values), the parabola becomes narrower but opens downwards.- Changing \(c\)'s value affects the "width" of the parabola; larger \(|c|\) makes parabolas narrower and direction depends on the sign of \(c\).

Key Concepts

Parabolas

In the world of quadratic functions, parabolas are the most common shapes you encounter. These curves have a characteristic U-shape and can open either up or down, depending on certain factors within their equations.

In these functions, the highest power of the variable is two, making them quadratic. This essentially sets the stage for the family of functions represented as \(f(x) = c x^2\). The beauty of parabolas lies in their symmetry and how they intersect with the x-axis.

• They are symmetrical about a vertical line called the axis of symmetry.
• The turning point of the parabola is known as the vertex.
• Depending on the quadratic equation, the vertex can be a maximum or a minimum.

Understanding these properties is crucial for mastering any task involving parabolas.

Coefficient Effect

The coefficient \(c\) in quadratic functions \(f(x) = c x^2\) plays a crucial role in determining the parabolas' characteristics and orientation. It may seem like a simple number at first, but it has a significant effect! Let's explore:

When \(c\) is positive, the parabola will open upwards, creating a U-shape. However, if \(c\) is negative, the parabola will open downwards, forming an upside-down U-shape.

• If \(|c|\) is greater than 1, the parabola appears narrow.
• If \(|c|\) is less than 1, the parabola becomes wider.

This all happens because the coefficient \(c\) stretches or compresses the graph vertically. Thus, by altering \(c\), you change the width and orientation of the parabola.

Graphing Functions

The process of graphing quadratic functions involves plotting the parabolas on a coordinate system to visually analyze their structure. For the given problem, we are working with \([-5,5]\) by \([-10,10]\) viewing rectangle.

As you graph the parabolas:

• Notice how different values of \(c\) create distinct parabolas.
• For positive values of \(c\), like 1, \(\frac{1}{2}\), 2, and 4, the parabolas open upwards and vary in width.
• For negative values of \(c\), such as -1, -\(\frac{1}{2}\), and -2, the parabolas open downwards.

By graphing these different scenarios, it becomes easier to understand how \(c\) affects the shape and direction of each graph.

Function Family

The concept of a function family in mathematics is akin to having a collection of closely related individuals, where each member subtly varies in a particular way. Here, we explore the function family defined by \(f(x) = c x^2\).

This family of functions represents a collection of parabolas that all share the same basic formula framework. Despite this similarity, they vary based on their coefficient \(c\), which alters their graphical characteristics.

• When considering a family of functions within a viewing window like \([-5,5]\) by \([-10,10]\), you can easily see the visible impact of different values for \(c\).
• The same equation framework allows these graphs to show the vast range of transformations possible simply through variations in \(c\).

Studying function families deepens your understanding of how small changes in an equation can lead to different outcomes in graphs, illustrating one of algebra's powerful insights.