Problem 18 - 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. $$f(x)=-(x-4)^{2}, g(x)=-(x+4)^{2}$$

Step-by-Step Solution

Answer

The graphs of \(f(x) = -(x-4)^{2}\) and \(g(x) = -(x+4)^{2}\) are both downwards facing parabolas with vertices at (4,0) and (-4,0) respectively. The domain of both functions is all real numbers while the range is (-\infty, 0].

Step 1: Understand the given functions

Firstly, it's crucial to determine what kind of function we're dealing with. Both \(f(x) = -(x-4)^{2}\) and \(g(x) = -(x+4)^{2}\) are quadratic functions. They both have a negative leading coefficient which means they are both facing downwards on a graph.

Step 2: Plot the Functions

Next, plot the two functions, \(f(x)\) and \(g(x)\), on the same graph. The vertex form of a quadratic function is \(f(x) = a(x-h)^2 + k\), where (h,k) is the vertex of the parabola. Here, for \(f(x)\), the vertex is (4,0) and for \(g(x)\), it's (-4,0). This means \(f(x)\) has its maximum point at (4,0) and \(g(x)\) has its maximum point at (-4,0).

Step 3: Determine the Domain and Range

The domain of a function is the possible set of input values (x-values), while the range is the set of output values (y-values). For quadratic functions such as ours, the domain is all real numbers, as any real number can be substituted for 'x'. The range, considering that both functions open downwards, will be (-\infty, 0] for both functions.

Key Concepts

Understanding Domain and Range

When working with any function, it's essential to identify the domain and range to understand what x-values you can input into the function and what y-values you can expect as output. Quadratic functions are typically defined for all real numbers, so their domain is usually all real numbers. In the case of the functions provided,

• The domain for both functions, \(f(x) = -(x-4)^{2}\) and \(g(x) = -(x+4)^{2}\), is all real numbers: \((-\infty, \infty)\).

The range, however, is determined by the direction the parabola opens and its maximum or minimum point.

• Both functions have a maximum point, as they open downwards due to their negative leading coefficient.
• Therefore, the highest point on the y-axis occurs at the vertex. Since both functions have a vertex at y=0, the range is \(( -\infty, 0 ]\) for both functions.

Remember, understanding domain and range helps in both plotting and analyzing functions appropriately.

The Vertex Form of Quadratic Functions

Quadratic functions can be elegantly expressed in vertex form, which is \(f(x) = a(x-h)^2 + k\). Here, \((h, k)\) represents the vertex of the parabola. Knowing the vertex form helps in quickly identifying significant characteristics of the quadratic function:

• **Vertex**: This point is crucial as it represents the peak or trough of the parabola, depending on its orientation. For \(f(x) = -(x-4)^{2}\), the vertex is at \((4,0)\); for \(g(x) = -(x+4)^{2}\), it is \((-4,0)\).
• **a-value (Leading Coefficient)**: This affects the direction of the graph. If it's positive, the parabola opens upwards; if negative, it opens downwards.
Using the vertex form simplifies graphing as you can directly pick off the vertex coordinates and understand the shaping of the parabola through its leading coefficient.

Negative Leading Coefficient

In quadratic functions, the leading coefficient \(a\) in the vertex form \(f(x) = a(x-h)^2 + k\) has a vital role in the graph's orientation. A negative leading coefficient, like in our functions \(-1\), indicates that both parabolas open downwards. This pivotal characteristic creates several effects:

• **Opening Direction**: Parabolas that open downwards produce a curved arch shape, similar to an upside-down 'U'.
• **Maximum Point**: Since the parabola curves downward, the vertex represents the highest point. Thus, the vertex becomes a maximum point for the function.
• **Impacts on Range**: Opposite to a parabola opening upwards (where vertices are minimum), these downward-opening functions have ranges described by \(( -\infty, k ]\). In our functions, where the vertex y-value \(k\) is 0, the range is \((-\infty, 0]\).

Recognizing a negative leading coefficient at a glance helps predict the general behavior of the quadratic function on a graph.