Problem 43 - College Algebra

Question

graph the given functions, \(f\) and \(g,\) in the same rectangular coordinate system. Select integers for \(x\), starting with \(-2\) and ending with \(2 .\) Once you have obtained your graphts, describe how the graph of \(g\) is related to the graph of \(f\) $$ f(x)-x^{2}, g(x)-x^{2}+1 $$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is the same as the graph of \(f(x)\) but shifted one unit upwards. This shift corresponds to the '+1' in the equation of \(g(x)\).

Step 1: Setting Up the Coordinates

Since we must choose integer \(x\) values between -2 and 2, we'll set up a table of values for each of the functions.

Step 2: Calculating Function Values

Substitute each \(x\) value into the equations for both \(f(x) = x^2\) and \(g(x) = x^2 + 1\) to find the corresponding \(y\) values.

Step 3: Plotting The Functions

Use a graphing tool to plot the values from steps 1 and 2. Draw smooth curves through the plotted points to represent each function. Each pair of \(x, y\) values represents one point on the graph of the function.

Step 4: Comparing the graphs

Compare the graphs of \(f(x)\) and \(g(x)\) to analyze their relationship. Note that the \(g(x)\) function is simply the \(f(x)\) function shifted one unit upwards.

Key Concepts

Coordinate System

The foundation of graphing any function starts with understanding the coordinate system. This is a way to visually represent numerical relationships in a plane using two perpendicular lines, typically referred to as axes. The horizontal axis is known as the x-axis, and the vertical one is the y-axis. The point where these axes intersect is called the origin, denoted as (0,0).

After setting up the coordinate system, each point on the plane is defined by an ordered pair of numbers (x, y), where 'x' represents the position along the horizontal axis and 'y' along the vertical. In the exercise provided, integer x-values from -2 to 2 are chosen to create a set of points that will represent the given functions when plotted and connected. This visualization is crucial as it allows students to understand the behavior and the properties of the graphed functions at a glance.

Function Transformation

When we talk about function transformation, we are discussing changes to the original function that alter its graph in some way. These changes could be translations (shifts), reflections, stretches, or compressions. In the context of graphing the functions f(x) = x^2 and g(x) = x^2 + 1, the ‘+1’ represents a vertical translation.

When you add a constant to every y-value of f(x), the entire graph shifts up by that constant amount. In the exercise, this shift is one unit upward for the function g(x). It's important to understand that transformations don't change the shape of the graph, just its position. Recognizing these transformations allows students to quickly sketch more complex functions based on simpler parent functions they are already familiar with.

Quadratic Functions

Quadratic functions are a type of polynomial with a degree of two, which means the highest exponent of the variable is two. The standard form of a quadratic function is f(x) = ax^2 + bx + c. The graphs of these functions are parabolas that open upwards if a is positive and downwards if a is negative.

In the exercise, the given functions f(x) = x^2 and g(x) = x^2 + 1 are both quadratic, with their graphs both being parabolas that open upwards since the coefficient of x^2 is positive. Understanding the nature of quadratic functions is essential because it allows students to predict the shape and direction of their graphs. For instance, because the coefficient of x^2 in both f(x) and g(x) is 1, we know their parabolas will be congruent, with the vertex of g(x) being exactly 1 unit above that of f(x), demonstrating a vertical shift, one of the simplest forms of function transformation.