Question

In this exercise, you will investigate the graphical effect of completing the square. a. Graph each pair of functions in the same coordinate plane. $$ \begin{array}{ll} y=x^2+2 x & y=x^2-6 x \\ y=(x+1)^2 & y=(x-3)^2 \end{array} $$ b. Compare the graphs of \(y=x^2+b x\) and \(y=\left(x+\frac{b}{2}\right)^2\). Describe what happens to the graph of \(y=x^2+b x\) when you complete the square.

Short answer

The process of completing the square for a quadratic function \(y=x^2+bx\) results in a horizontal shift of the graph. The shift is to the left if b is positive and to the right if b is negative. The magnitude of shift is equivalent to \( \frac{b}{2}\).

Step-by-step solution

Graph the functions

The functions need to be plotted on the same coordinate system. Each pair would occupy a different graph for better comparison. These are the pairs: 1) \(y = x^2+2x\) and \(y = (x+1)^2\) 2) \(y = x^2-6x\) and \(y = (x-3)^2\). You need to use a graphing tool or perform manual plotting by selecting an array of x-values, calculating corresponding y-values, and marking these points on the graph and drawing smooth curves.

Compare the graphs

After plotting, compare the two graphs in each pair to observe changes in position and shape. By comparing, it can be noted that a shift to the left or right is occurring when moving from \(y = x^2+bx\) to \(y = (x + \frac{b}{2})^2\). Coordination of both graphs helps you understand better how the transformation occurred.

Compare \(y=x^2+bx\) and \(y=(x+ \frac{b}{2})^2\)

This comparison is a generalisation of step 2. It is required to describe the behaviour of a graph when a square is completed. It can be deduced that completing the square for a quadratic function effectively results in a horizontal shift of the graph. The shift is determined by the \( \frac{b}{2}\) value.

Key concepts

Graph Transformation

In mathematics, graph transformations involve changing the appearance or position of a graph on a coordinate plane. In the context of quadratic functions, one common transformation is completing the square. This technique allows us to transform the equation of a quadratic function so that its graph can be easily analyzed.

When you apply graph transformations, you can predict how a graph will shift or change based on alterations to its equation. By completing the square, a quadratic equation in the form of \(y=x^2+bx\) becomes \(y=\left(x+\frac{b}{2}\right)^2\), resulting in a modified graph shape. Comparing these transformed equations helps us see how the graph's vertex position changes, which is crucial for understanding overall graph behavior.

Quadratic Functions

Quadratic functions are a type of polynomial function characterized by the equation \(y=ax^2+bx+c\). These functions create parabolas when graphed, and their standard form allows us to easily manipulate and understand the behavior of their graphs.

In the exercise, you deal with specific quadratic forms such as \(y=x^2+2x\) and \(y=x^2-6x\). These equations can be rewritten in vertex form by completing the square, revealing how the graphs shift. Quadratic functions are inherently symmetric, and understanding how to transform them aids in predicting and visualizing their behavior on a coordinate plane. This simplification through completing the square enhances graph interpretation.

Horizontal Shift

A horizontal shift in a graph occurs when the entire graph moves left or right on the coordinate plane. This shift does not alter the graph's shape, only its location.

To visualize this, consider the transformation from \(y=x^2+bx\) to \(y=(x+\frac{b}{2})^2\). Completing the square results in a graph that shifts horizontally by \(\frac{b}{2}\) units. For instance, \(y=x^2+2x\) changes to \((x+1)^2\), representing a shift to the left by 1 unit. Meanwhile, \(y=x^2-6x\) turns into \((x-3)^2\), meaning the graph moves to the right by 3 units. Mastering this concept lets you immediately determine how changes in the equation impact graph positioning.

Coordinate Plane

The coordinate plane serves as the backdrop for all graph transformations. It is a two-dimensional grid defined by a horizontal axis, \(X\), and a vertical axis, \(Y\). Each point on this plane is identified by a pair of numerical coordinates \((x, y)\).

In the given exercise, you plot graphs on the coordinate plane to observe their transformations. As you graph each quadratic function pair, you see firsthand how completing the square dictates their movement and position. By understanding how points shift on this plane, you gain insights into the effects of different operations on graph behavior. Observing these alterations on the coordinate plane highlights the visual impact of algebraic transformations.

Functional Pairs

Functional pairs are sets of functions expressed in relation to each other, offering a direct comparison in behavior and properties. In graphing exercises, they become invaluable tools to understand transformations like those seen in quadratic functions.

For example, in the task you compare \(y=x^2+2x\) with \(y=(x+1)^2\) as one functional pair. This pairing reveals the effects of completing the square directly through their graphs. By evaluating the position and shape of the parabolas next to each other, functional pairs illustrate how slight changes in equations lead to predictable graph transformations. This approach makes understanding complex algebraic concepts accessible and intuitive.