Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g\). (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=x+1, \quad g(x)=(x-1)^{2}\) (a) \(-2\) (b) 2 (c) 10 (d) 4 (e) \(\underline{8}\)

Step-by-Step Solution

Answer

The value that best approximates the area of the region is (d) 4, judged by visual inspection of the drawn graphs.

Step 1: Sketch the Graphs of f(x) and g(x)

Begin by graphing the two functions \(f(x) = x + 1\) and \(g(x) = (x-1)^{2}\) on the same Cartesian plane. The first function \(f(x) = x+1\) is a linear function with a slope of 1 and a y-intercept at 1. The second function \(g(x) = (x-1)^{2}\) is a quadratic function with its vertex at (1, 0) opening upwards.

Step 2: Identify the Bounded Region

Observe where the two graphs intersect and the area they bound together. They intersect at points where \(f(x) = g(x)\) i.e. \(x+1 = (x-1)^{2}\), which gives two solutions, these will be points of intersection. This bounded region is the area that needs to be approximated.

Step 3: Estimate the Area

Examine the bounded area and make a rough estimation of its size. Without performing any calculations, this becomes a visual task, taking into consideration the shape of the area (which may likely be a kind of irregular polygon) and comparing it to known area sizes or shapes.

Key Concepts

Functions intersection

To determine the area between two curves, you first need to identify where the curves intersect. This intersection is crucial as it tells us the boundaries of the region we are interested in. The points of intersection are where the two functions have the same y-values, i.e., when their expressions are equal. Here, for the functions given as

• \( f(x) = x + 1 \)
• \( g(x) = (x-1)^2 \)

The points of intersection can be found by setting these two functions equal: \[ x + 1 = (x-1)^2 \]This equation can be solved for \( x \) to find the x-coordinates where both functions meet. Whether graphically or algebraically, these intersection points will form the limits for which we estimate the area between the curves below and above the x-axis.

Linear and quadratic functions

Understanding the difference between linear and quadratic functions is important when sketching and analyzing graphs. A linear function like

• \( f(x) = x + 1 \)

represented graphically as a straight line with a constant slope, which is 1 in this case. It means it rises one unit vertically for every unit it moves horizontally.

On the other hand, a quadratic function like

• \( g(x) = (x-1)^2 \)

appears as a parabola on the graph. This specific quadratic has its vertex at (1,0) and opens upwards, indicating that as \( x \) moves away from the vertex in either direction, the value of \( g(x) \) gets larger. This difference is critical for understanding how the two functions interact and eventually how the area of their bounded region can be approximated, as areas further from the vertex of the quadratic contribute differently than those closer.

Graph sketching

Sketching the graphs of functions is a fundamental skill for visualizing mathematical concepts. Begin by plotting any known points, intercepts, and shapes based on the function types and characteristics, such as slopes or turning points. For the function

• \( f(x) = x + 1 \),

start at the y-intercept (0,1) and move by the slope of 1 to draw the line.

For

• \( g(x) = (x-1)^2 \),

identify the vertex at (1,0) and sketch the parabola opening upwards. Ensure both functions are clearly drawn over the range that includes their intersection points. Observing the visual intersection helps predict which function is greater over specific intervals, aiding in assessing the shape and size of the area they encompass.

By using these techniques, one can make an educated guess about the bounded area's size without needing to delve into complex calculations, especially when asked to choose the best approximate value from a list of options.