Problem 17 - Precalculus : Building Concepts and Connections

Question

This set of exercises will reinforce the skills illustrated in this section. Graph each pair of functions on the same set of coordinate axes, and find the domain and range of each function. $$f(x)=(x+2)^{2}, \quad g(x)=(x-2)^{2}$$

Step-by-Step Solution

Answer

The vertex of \(f(x)=(x+2)^2\) is at (-2, 0), and of \(g(x)=(x-2)^2\) is at (2, 0). The domain for both functions is all real numbers as any real number can be a valid input. The range for both is all real numbers greater than or equal to 0, because the quadratic function's lowest point is at the vertex, which is 0 in both cases.

Step 1: Understanding the Functions

The first step is to understand what type of functions these are. Both \(f(x)=(x+2)^2\) and \(g(x)=(x-2)^2\) are quadratic functions, which have a typical U-shaped curve. In a quadratic function of the form \(f(x)=(x-a)^2\), the graph will have its vertex at the point \((a, 0)\). Therefore, \(f(x)\) will have its vertex at (-2, 0) and \(g(x)\) at (2, 0).

Step 2: Plotting the Graphs

Next, graph each function using points around the vertex. For example, for \(f(x)\), choose points like (-3, 1), (-1, 1), and (0, 4). For the function \(g(x)\), choose points like (3, 1), (1, 1), and (0, 4). Plot these points and draw the curves.

Step 3: Identifying the Domain and Range

The domain of a function is all the possible input values (x-values), and the range is all the possible output values (y-values). For these quadratic functions, the domain is all real numbers, because we can plug in any real number for x. The range for each function is all real numbers greater than or equal to 0, because the smallest value for both \(f(x)\) and \(g(x)\) is 0, which happens when x is -2 for \(f(x)\) and 2 for \(g(x)\).

Step 4: Finalization

Now, with both graphs plotted and the domain and range identified, the task for the given functions is completed. Remember to always check your graphs for any inconsistencies or errors before finalizing your answer.

Key Concepts

Quadratic Functions

The concept of quadratic functions is a fundamental topic in algebra and gives rise to the classic U-shaped curve known as a parabola. A quadratic function is typically expressed in the form of \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) is not equal to zero.

The beauty of these functions lies in their symmetry, which is centered around the vertex, the peak or valley of the parabola. The vertex represents the highest or lowest point of the graph. In the given problem, you're dealing with quadratic functions that are in vertex form, \(f(x) = (x - h)^2 + k\), where \(h\) and \(k\) tell you the vertex's coordinates (\(h, k\)).

The exercise provided two specific quadratic functions, \(f(x) = (x+2)^2\) and \(g(x) = (x-2)^2\), which have their respective vertices at \((-2, 0)\) and \((2, 0)\), because they are expressed in the form where \(h\) is added or subtracted from \(x\). It is crucial to remember that when graphing these functions, the value of \(a\) affects the width and direction of the parabola – if \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.

Domain and Range of a Function

In mathematics, every function has a domain and a range that define the set of possible input and output values, respectively.

The domain of a function refers to all the values that \(x\) can take where the function is defined. For quadratic functions, such as those given in the exercise, the domain is usually all real numbers, denoted as \(-\text{infinity to infinity}\), because you can insert any real number for \(x\) and you'll get a valid output.

The range, on the other hand, refers to all possible values that the function can output, depending on the domain. For the given quadratic functions \(f(x)\) and \(g(x)\), since they both open upwards and have vertices at the points \((-2, 0)\) and \((2, 0)\) respectively, the range would be from 0 to infinity. This represents that the output of these functions can never be less than 0, which corresponds to the vertex, the lowest point on the graph of the function. It's essential to carefully evaluate the vertex and the direction of the parabola to accurately determine the range.

Coordinate Axes Plotting

Plotting functions on the coordinate axes is a visual process that helps one understand the behavior of mathematical functions. The coordinate plane consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis), which intersect at the origin (0,0). When graphing quadratic functions such as \(f(x) = (x+2)^2\) and \(g(x) = (x-2)^2\), it's helpful to start by marking the vertices on the graph.

For example, plot the vertex of \(f(x)\) at \((-2, 0)\) and then also plot additional points on either side of the vertex. By selecting appropriate \(x\) values and calculating the corresponding \(y\) values (outputs), you will see that the points lie along a U-shaped curve. After plotting several points, you can draw the curve through them, ensuring that it's symmetrical around the vertex.

Remember to not only focus on points close to the vertex but also on points further away to fully capture the parabola's shape. Consistency in the spacing of the plotted points can enhance the accuracy of your graph. Coordinate axes plotting is an excellent tool for visualizing the characteristics of a function, such as domain and range, intercepts, and overall shape.