Problem 39 - College Algebra Essentials

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 graphs, describe how the graph of g is related to the graph of \(f\). $$ f(x)=x, g(x)=x+3 $$

Step-by-Step Solution

Answer

The function g(x) is a vertical translation of the function f(x), specifically a shift three units upward.

Step 1: Set up the x-values

Use the x-values provided in the exercise, i.e., -2,-1, 0, 1, 2.

Step 2: Substitute x-values into f(x)

Substitute each x-value into the function f(x) = x, yielding the points: (-2,-2), (-1,-1), (0,0), (1,1), (2,2). Use these to graph the function f.

Step 3: Substitute x-values into g(x)

Substitute each x-value into the function g(x) = x + 3, yielding the points: (-2,1), (-1,2), (0,3), (1,4), (2,5). Use these to graph the function g.

Step 4: Graphic Comparison

Upon drafting the graphs of f and g on the same chart, it's noticeable that g(x) is simply the graph of f(x) shifted 3 units up.

Step 5: Analytical Comparison

From the graphs, we can see that for each x-value, the corresponding y-value on the g-graph is always 3 greater than on the f-graph. This implies that g(x) = f(x) + 3.

Key Concepts

Linear Functions

Linear functions are one of the most straightforward types of functions you'll encounter in algebra and graphing. A linear function is a function of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These functions graph as straight lines on a coordinate plane.

In the given exercise, the function \( f(x) = x \) is a classic example of a linear function. Here, the slope \( m \) is 1, and there is no y-intercept since \( b = 0 \). This means the line passes through the origin (0,0), rising one unit up for every unit it moves to the right. This linearity is what makes linear functions both predictable and easy to graph using simple points.

Understanding linear functions is crucial as they provide a foundation for more complex function types, and their properties make analyzing and predicting the behaviour of graphical data much simpler.

Function Shifts

Function shifts describe how the graph of a function can be moved around on the coordinate plane without altering its shape. There are generally two types of shifts: vertical and horizontal, and they are represented in the function's equation.

In the case of the function \( g(x) = x + 3 \) from the exercise, it's an example of a vertical shift. Unlike \( f(x) = x \), the function \( g(x) \) adds 3 to the output of \( f(x) \), effectively shifting the entire graph of \( f(x) \) upwards by 3 units. You will see that every point on \( g \) is 3 units higher than its corresponding point on \( f \).

Vertical shifts do not affect the slope of the line; they only change the y-intercept. Here, while \( g(x) \) and \( f(x) \) are parallel, their y-intercepts differ because of this shift. Understanding how the algebraic form of a function relates to its graph helps in visualizing changes without drawing them.

Coordinate Systems

Coordinate systems, specifically the rectangular or Cartesian coordinate system, are essential for graphing functions. On this system, every point on a plane is defined by an \( (x, y) \) pair. The x-value indicates the position on the horizontal axis, while the y-value indicates the position on the vertical axis.

In our exercise, graphs of \( f(x) = x \) and \( g(x) = x + 3 \) are both plotted using this two-dimensional coordinate plane. Using integers as x-values simplifies plotting because you can easily predict where each point will land. Whether negative or positive, these values effectively sketch a visual representation of the function's behavior on the graph.

Understanding the coordinate system is fundamental when learning to graph any function, as it allows you to translate mathematical equations into visual forms, making comprehension of relationships between variables clearer.