Problem 42 - 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 \(f\). $$ f(x)=-2 x, g(x)=-2 x+3 $$

Step-by-Step Solution

Answer

The graph of \(g(x) = -2x + 3\) is an upward shift of the graph \(f(x) = -2x\) by 3 units along the y-axis.

Step 1: Determine the Coordinate Points

Start by substituting the value of \(x\) into each equation to find the corresponding \(y\) values. For \(f(x)=-2x\), substitute \(x\) with \(-2\), \(-1\), \(0\), \(1\), and \(2\). Repeat this process for \(g(x)=-2x+3\) to get the coordinates for \(g\).

Step 2: Plot the Coordinate Points

Once you have the coordinate points for each function, plot these points on the rectangular coordinate system. Make sure to label each point correctly with the corresponding function \(f\) or \(g\).

Step 3: Draw the Graph

Now, connect the corresponding points plotted for each function to get two separate lines. The lines representing the functions \(f(x)=-2x\) and \(g(x)=-2x+3\) should be clearly distinguishable.

Step 4: Describe the Relationship Between the Graphs

With both graphs plotted, analyze how they are related. Notice if one graph is a translation, dilation, reflection, or rotation of the other.

Key Concepts

Coordinate Points

Understanding coordinate points is essential when working with linear functions and their graphs. A coordinate point is a pair of numbers \( (x, y) \) that define a specific location on the cartesian plane, also known as a rectangular coordinate system. In our exercise, we determine the coordinate points by substituting chosen values of \( x \) into the function equations \( f(x) \) and \( g(x) \) to find the corresponding \( y \) values.

For instance, compute \( f(-2)=-2(-2)=4 \) to find the point \( (-2, 4) \) for the function \( f \) on our graph. Repeating this for values from \( -2 \) to \( 2 \) and applying it to both \( f(x) \) and \( g(x) \) will give us a set of coordinates that we can then plot to visualize the functions.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane with a horizontal axis (the x-axis) and a vertical axis (the y-axis). The intersection of these axes at the point \( (0, 0) \) is known as the origin. To graph a linear function on this system, we plot the coordinate points calculated from the function's equation and then draw a line through them.

Each point \( (x, y) \) on a graph corresponds to an \( x \) value from the horizontal axis and a \( y \) value from the vertical axis. By plotting points and connecting them, we can illustrate the relationship of the function's input values \( x \) to their corresponding output values \( y \) on the graph. This visualization is crucial for understanding the behaviour of linear equations.

Linear Equations

Linear equations like \( f(x)=-2x \) and \( g(x)=-2x+3 \) represent straight lines when graphed on the rectangular coordinate system. These equations are in the slope-intercept form \( y=mx+b \) where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) determines the steepness and direction of the line, while the y-intercept \( b \) indicates the point where the line crosses the y-axis.

In our exercise, \( f(x)=-2x \) has a slope of \( -2 \) and no y-intercept term, meaning it passes through the origin. The function \( g(x) \) has the same slope but with a y-intercept of \( 3 \) indicating it also will be a straight line but will intersect the y-axis at \( (0, 3) \) instead of the origin.

Graph Translation

Graph translation refers to shifting a graph horizontally and/or vertically without changing its shape or orientation. When comparing \( f(x)=-2x \) and \( g(x)=-2x+3 \) from our exercise, we see that the graph of \( g \) can be obtained by translating the graph of \( f \) vertically upward by three units. The \( +3 \) in \( g(x) \) signifies this upward shift.

Understanding graph translation is valuable especially when the equations share the same slope, as in our case. A shared slope means both lines will be parallel, and an additional constant term \( b \) in the slope-intercept equation simply lifts or lowers the line, revealing how one graph is related to the other through translation along the y-axis.