Problem 41 - 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 \(f\) $$ f(x)--2 x, g(x)--2 x-1 $$

Step-by-Step Solution

Answer

The graph of \(g(x) = 2x - 1\) is the graph of \(f(x) = 2x\) shifted one unit down.

Step 1: Graph the first function

Graph the function \(f(x) = 2x\) on a coordinate plane. For \(x= -2, -1, 0, 1, 2\), compute the corresponding \(y\)-values, which will be \(-4, -2, 0, 2, 4\) respectively. Plot these points and draw the line passing through them.

Step 2: Graph the second function

Graph the function \(g(x) = 2x - 1\) on the same coordinate plane. For \(x= -2, -1, 0, 1, 2\), compute the corresponding \(y\)-values which will be \(-5, -3, -1, 1, 3\) respectively. Plot these points and draw the line passing through them.

Step 3: Analyze the relationship

Look at both graphs to determine their relationship. Note that both graphs have the same slope (rate of change), but \(g(x)\) is one unit down from \(f(x)\), indicating a vertical shift.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These axes divide the plane into four quadrants and provide a framework for plotting points, lines, and curves based on ordered pairs Every point on this plane is represented by an ordered pair of numbers, such as where the first number indicates the horizontal distance from the origin (the point where the two axes intersect) and the second number represents the vertical distance. Coordinates are used to graph mathematical functions like linear functions, which in our exercise include then plotting points selected from integer values of , ranging from to .

Function Transformation

Function transformation involves altering the appearance of a graphed function by applying certain operations or manipulations. These changes can be of various types such as shifting (vertical/horizontal), stretching, compressing, or reflecting the function's graph. In the context of our exercise, when we move from the function to , we observe a vertical shift which is a specific type of function transformation. The graph of has been shifted downward by one unit, as indicated by the '-1' in the function's equation. Such transformations can help us understand how alterations in a function's equation are visually represented on the graph.

Slope of a Line

The slope of a line measures the steepness or tilt of the line on the coordinate plane. It's calculated by the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. To find the slope, use the formula where and are the coordinates of two different points on the line. In the case of our linear functions and , both have a slope of '2', indicating they rise 2 units for every 1 unit they run horizontally. This constant rate of change results in straight lines graphed on the rectangular coordinate system.

Vertical Shift

A vertical shift in a linear function occurs when each point of the graph moves up or down the same amount. This is represented by adding or subtracting a constant from the function's equation. In the solution provided, moving from to showcases a vertical shift of one unit downward, as evidenced by the subtraction of '1'. The vertical shift doesn't change the slope of the line, but it does change the line's y-intercept, which is the point where the line crosses the y-axis. It's important for students to understand that a vertical shift translates the graph parallel to the y-axis, without affecting the line's original direction or its x-intercept (the point where the line crosses the x-axis).