Question

Graph the function. Describe the domain. $$y=\frac{6}{x+9}-7$$

Short answer

The domain of the function is (-∞, -9) ⋃ (-9, ∞). X-intercept is at \(\frac{-69}{7}\), y-intercept is at -1/3. The graph has a vertical asymptote at \(x = -9\) and horizontal asymptote at \(y = -7\) and passes through the x and y intercepts and asymptotically approaches the asymptote lines.

Step-by-step solution

Identify Domain

A function's domain is the set of all possible x-values that the function can have. Since this function is a fraction, and there is an x in the denominator, the denominator cannot equal zero as it will make the function undefined. So, set the denominator equal to zero and solve for x: \(x + 9 = 0\), this gives \(x = -9\). So, the domain is all real numbers except -9, which we can write as: (-∞, -9) U (-9, ∞).

Find the x and y Intercepts

For the x-intercept, set y = 0 and solve the equation: \(0=\frac{6}{x+9}-7\). This simplifies to \(x+9 = \frac{-6}{7}\), hence \(x = \frac{-6}{7} - 9 = \frac{-69}{7}\). So, the x-intercept is \(\frac{-69}{7}\). For the y-intercept, set x = 0: \(y = \frac{6}{0+9}-7\), that simplifies to \(y=-\frac{1}{3}\). So, y-intercept is -1/3.

Identify the Asymptotes

The vertical asymptote is at \(x=-9\) which we got from the domain. The horizontal asymptote is at \(y=-7\). The horizontal asymptote can be found by examining the degrees of the numerator and denominator. When the degree in the denominator is equal to the numerator (as it is in our case), the line \(y = \frac{(coefficient of the highest degree term in the numerator)}{(coefficient of the highest degree term in the denominator)}\) is a horizontal asymptote for the function.

Sketch the Graph

Now we can use the points and asymptotes to sketch the graph. Draw axes and locate the x and y intercepts. The plot also vertical asymptote at \(x = -9\) and a horizontal asymptote at \(y = -7\). Now sketch the graph approaching asymptotes as \(x→±∞\) and as \(x → -9\).

Key concepts

Domain of a Function

Understanding the domain of a function is crucial for graphing it accurately. In simple terms, the domain represents all the possible x-values that can be input into the function without causing issues, like dividing by zero. In our example, the function given is \(y=\frac{6}{x+9}-7\).

To find the domain, we set the condition that the denominator \(x+9\) cannot be zero because division by zero is undefined in mathematics. Solving the equation \(x+9=0\), we find \(x=-9\), which means \(x\) can be any real number as long as it is not \(x=-9\). Thus the domain is represented as the union of the intervals \( (-\infty, -9) \cup (-9, \infty) \) which excludes the single point where the denominator would be zero.

When graphing the function, remember that at \(x=-9\), the graph will not be defined and there will be a break. This is important for overall understanding and sketching the accurate behavior of the function across its entire domain.

X and Y Intercepts

The x and y intercepts of a graph are the points at which the graph crosses the x-axis and y-axis respectively. These points are helpful anchors in sketching the rough shape of a graph. In the function \(y=\frac{6}{x+9}-7\), to find the x-intercept, we set \(y=0\) and solve the equation for \(x\).

In our example, solving \(0=\frac{6}{x+9}-7\) leads to \(x=\frac{-69}{7}\), indicating the x-intercept. To find the y-intercept, we set \(x=0\) and solve for \(y\), yielding \(y=-\frac{1}{3}\), indicating the y-intercept.

These intercepts allow us to place exact points on the graph where the function crosses the axes. For graphing rational functions, the presence of intercepts can significantly influence the plot shape especially near the axes.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are critical to understanding the behavior of rational functions and are divided into two types: vertical and horizontal.

The vertical asymptote of a graph corresponds to the values of \(x\) that make the function undefined (where the denominator is zero). In our function, the vertical asymptote is at \(x=-9\). It dictates that as the function's graph gets close to this line on either side, the y-values will increase or decrease without bound.

On the other hand, a horizontal asymptote is a y-value that the function approaches as \(x\) heads towards positive or negative infinity. It's determined by comparing the degrees of the numerator and the denominator in the function. In this case, the horizontal asymptote is at \(y=-7\), based on the constant terms as the \(x\)-values become very large. When graphing, the function will approach, but not cross, this horizontal line. Knowing the asymptotes helps in visualizing the long-term behavior of the function and is an essential step before sketching the graph.