Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph \(f\) for \(c=-2,0\), and 2 in the same viewing window. (a) \(f(x)=\frac{1}{2} x+c\) (b) \(f(x)=\frac{1}{2}(x-c)\) (c) \(f(x)=\frac{1}{2}(c x)\) In each case, compare the graph with the graph of \(y=\frac{1}{2} x\).

Step-by-Step Solution

Answer

For equation (a), the function will shift vertically based on \(c\). Negative \(c\) values shift the graph down, and positive \(c\) values shift it up. For equation (b), the function will shift horizontally based on \(c\). Negative values shift the graph to the right, and positive values to the left. For equation (c), the value of \(c\) will rescale the slope of the function.

Step 1: Graph Base Function

Begin by graphing the base function \(y=\frac{1}{2} x\). The function is a straight line with a slope of 0.5 and passes through the origin (0,0).

Step 2: Graph (a)

To graph \(f(x)=\frac{1}{2} x+c\), start with the base function and shift it up or down according to the value of \(c\). For \(c=-2\), the line will shift down by 2 units. For \(c=0\), there will be no shift which means it will be the same as the base function. For \(c=2\), the graph will shift up by 2 units.

Step 3: Graph (b)

For \(f(x)=\frac{1}{2}(x-c)\), the graph will shift horizontally based on the value of \(c\). Here, when \(c=-2\) the graph will move 2 units to the right, for \(c=0\), it will not move, and for \(c=2\), it will move 2 units to the left.

Step 4: Graph (c)

For \(f(x)=\frac{1}{2}(c x)\), this will scale the slope of the graph. So when \(c=-2\), the slope will be -1, for \(c=0\), the function will flatline to \(y=0\), and when \(c=2\), the function will have a slope of 1.

Step 5: Conclusion

After drawing all the graphs, compare them with the original function \(y=\frac{1}{2} x\). Note the shifts, and changes in slope due to the different values of \(c\) and their positions in \(f(x)\).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and are represented by the general form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These equations create straight lines when graphed on a coordinate plane. The straight line nature comes from the fact that for any constant rate of change (slope), the outputs (y-values) change linearly with the inputs (x-values). In our base function \(y = \frac{1}{2}x\), the slope \(\frac{1}{2}\) indicates that for every unit increase in \(x\), \(y\) increases by half a unit. Understanding linear equations is crucial because many real-world phenomena can be modeled in this simple but powerful form. They help in making predictions and understanding relationships between variables.

Function Transformations

Function transformations involve altering a function's position or shape on a graph without changing its nature. Common transformations include:

• Vertical shifts: Moving a graph up or down. For instance, in \(f(x)=\frac{1}{2} x+c\), the value of \(c\) will shift the graph vertically.
• Horizontal shifts: Sliding a graph left or right. In \(f(x)=\frac{1}{2}(x-c)\), the value of \(c\) determines this shift.
• Scaling: Changes the steepness or slope. For \(f(x)=\frac{1}{2}(c x)\), \(c\) alters the slope.

Transformations help in understanding how changes in a function's formula affect its graph. They are like mathematical tools allowing adjustments to a base function to fit specific situations. For example, think of shifting a line to align with a data trend or scaling it to match proportions.

Slope and Intercept

The concept of slope and intercept is central to understanding linear functions. The slope, represented by \(m\), tells you how steep a line is. It's a ratio of the vertical change (rise) to horizontal change (run) between any two points on the line. In equations like \(y = \frac{1}{2}x\), the slope \(\frac{1}{2}\) means a rise of 0.5 units for every 1 unit of run. This uniform rate of change is why linear equations produce straight lines.

Intercepts are where the line crosses the axes. The y-intercept \(b\) is the point where the line crosses the y-axis (when \(x = 0\)), while x-intercept occurs where the line crosses the x-axis (\(y = 0\)). In the basic function \(y = \frac{1}{2}x\), the y-intercept is 0 because it passes through the origin (0,0). Understanding slope and intercept enables you to quickly sketch a graph or predict future points along the line, making linear equations an invaluable tool in mathematics and various applications.