Problem 17 - Beginning and Intermediate Algebra

Question

Given the following pairs of functions, explain how the graph of \(g(x)\) can be obtained from the graph of \(f(x)\) using the transformation techniques. $$f(x)=x^{2}, g(x)=-x^{2}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = -x^2\) can be obtained from the graph of \(f(x) = x^2\) by reflecting the graph of \(f(x)\) over the x-axis. The transformation that relates the two graphs is given by the equation \(g(x) = -f(x)\).

Step 1: Introduction

We are given the functions \(f(x) = x^2\) and \(g(x) = -x^2\). Our task is to find the transformation that turns the graph of \(f(x)\) into the graph of \(g(x)\).

Step 1: Identify the Transformation

Here, we can see that the difference between the given functions is in the sign of the term \(x^2\). The function \(g(x)\) has a negative sign in front of the \(x^2\) term, whereas the function \(f(x)\) has a positive sign. This means that the \(f(x)\) graph has been reflected over the x-axis, and this transformation can be written as follows: \[g(x) = -1\cdot f(x)\]

Step 2: Describe the Reflection

The graph of \(g(x) = -x^2\) is obtained by reflecting the graph of \(f(x) = x^2\) over the x-axis. To reflect the graph over the x-axis, we simply take the opposite values of the function's output. If a point on the graph of \(f(x)\) has a coordinate pair \((x,y)\), then its reflected point on the graph of \(g(x)\) has the coordinate pair \((x,-y)\).

Step 3: Apply the Transformation

Apply the reflection transformation as defined in step 2 to transform the graph of \(f(x)\) into the graph of \(g(x)\). Doing so, we can derive the relation:\ \[g(x) = -f(x)\] This relation now clearly shows the transformation that relates the two given functions.

Step 5: Conclusion

The graph of \(g(x) = -x^2\) can be obtained from the graph of \(f(x) = x^2\) by reflecting the graph of \(f(x)\) over the x-axis. The transformation that relates the two graphs is given by the equation \(g(x) = -f(x)\).

Key Concepts

Reflection Over X-Axis

Reflecting a graph over the x-axis is a straightforward transformation. Imagine flipping the graph upside down along the x-axis. For any function, to carry out a reflection over the x-axis, simply multiply the output or the function itself by \-1. Let’s consider the quadratic function \( f(x) = x^2 \). When we want to find the function \( g(x) \) that reflects \( f(x) \) over the x-axis, this operation results in \( g(x) = -x^2 \).

To understand this better, look at any single point, \( (x, y) \) on the graph of \( f(x) = x^2 \). After the reflection, its new position will be \( (x, -y) \). This simple change affects the entire graph, turning every point’s vertical distance from the x-axis into its opposite, effectively flipping the whole graph.

• Effect: The top of the parabola opens downwards after reflection.
• Equation: Reflect by changing from \( y \) to \( -y \).

Notably, the x-values remain unchanged, but their corresponding y-values are inverted. This concept lies at the heart of reflecting any function about the x-axis.

Quadratic Functions

Quadratic functions have a specific and recognizable shape known as a parabola. The general form of a quadratic function is \( f(x) = ax^2 + bx + c \). In its most basic form, without the extra terms, the function \( f(x) = x^2 \) plots a symmetric parabola that opens upwards.

These functions are notable for their:

• Vertex: The turning point of the parabola, for \( f(x) = x^2 \), it's at the origin \( (0, 0) \).
• Axis of Symmetry: A vertical line through the vertex, given as \( x = 0 \).
• Direction: Determined by the sign of the leading coefficient \( a \). Positive \( a \) means the parabola opens upwards, while negative \( a \) implies it opens downwards.

Understanding quadratics is essential because they appear in various contexts from physics, representing projectiles’ motions, to financial calculations, like calculating trajectories in profit graphs. This universality makes grasping the nature and transformations of quadratic functions invaluable.

Function Transformations

Function transformations are systematic changes applied to basic functions; they affect graphs in predictable ways. Transformations include translations, reflections, scalings, and rotations. They form the toolkit for modifying functions and their graphs to achieve a desired shape or position.

Key transformation types of interest include:

• Translations: Moving the graph up, down, left, or right without altering its shape. For example, \( f(x) + k \) translates the graph upwards by \( k \) units.
• Reflections: Flipping over an axis. As we saw, multiplying a function by \-1 reflects it over the x-axis.
• Scalings: Stretching or compressing graphs. Multiplying the input by a factor scales the function horizontally, while multiplying the output scales it vertically.

Combining these transformations allows for versatile graph manipulations, modeling real-life situations more accurately. By mastering these concepts, you can predict how any function will look after undergoing a series of transformations.