Problem 48 - Algebra 1

Question

Graph the function. Describe the domain. $$y=\frac{-2 x+11}{x-5}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\frac{-2x+11}{x-5}\) is all real numbers except 5, or in interval notation, (-∞, 5) U (5, ∞). The function is graphed as a curve slanting down to the right, with a vertical asymptote at x = 5 and a y-intercept at -2.2.

Step 1: Determine the Value That Makes the Denominator Zero

First, solve 'x - 5 = 0' to find out the value for x that makes the denominator zero. So, x = 5 is the value that makes the denominator zero.

Step 2: Describe the Domain

The domain of the function is all real numbers except the value that makes the denominator zero. The value from the previous step is x = 5, so the domain of the function is all real numbers except 5. In interval notation, this can be written as (-∞, 5) U (5, ∞).

Step 3: Graph the Function

To graph the function, first plot a point on the y-axis where x = 0. This will be the y-intercept. For the function \(y=\frac{-2x+11}{x-5}\), when x = 0, y = -2.2. Then plot a vertical asymptote at x = 5 and draw the curve of the function respecting the asymptote and going through the y-intercept, slanting down to the right.

Key Concepts

Domain of a Function

When working with rational functions, an important concept to understand is the domain of the function. The domain refers to all possible values of the variable for which the function is defined. In simpler terms, it's all the values of 'x' you can use without making the math "break". For rational functions, like \( y=\frac{-2x+11}{x-5} \), the main thing to watch out for is not allowing the denominator to become zero, since division by zero is undefined.

To find the domain here, look at the denominator \( x - 5 \). Set it equal to zero and solve for 'x' to find the "problem spot":

• \( x - 5 = 0 \)
• \( x = 5 \)

This means that \( x = 5 \) will make the denominator zero, and thus cannot be in the domain.

So, the domain of this function is all real numbers except 5. You can write this in interval notation as \((-\infty, 5) \cup (5, \infty)\). This tells us we can use any number less than 5 or more than 5, but not exactly 5.

Vertical Asymptotes

Another key concept when graphing rational functions is understanding vertical asymptotes. A vertical asymptote is a vertical line that a graph approaches but never touches or crosses. For the function \( y=\frac{-2x+11}{x-5} \), vertical asymptotes occur at the values of 'x' that make the denominator zero, similar to the problem spots in the domain.

In this exercise, we've already found that setting the denominator equal to zero gives:

• \( x - 5 = 0 \)
• \( x = 5 \)

This tells us there is a vertical asymptote at \( x = 5 \). When you graph this function, you should draw a dashed line along \( x = 5 \). The dashed line isn't part of the function but is a guide showing where the graph will approach but never touch. This asymptote creates a boundary on the graph, influencing how the function's curve behaves on either side.

Y-Intercept

Finding the y-intercept of a rational function is another important step when graphing. The y-intercept is the point where the graph crosses the y-axis, meaning where \( x = 0 \). To find it, substitute \( x = 0 \) into the function and solve for 'y'.

For the function \( y=\frac{-2x+11}{x-5} \):

• Substitute \( x = 0 \) into the equation: \( y = \frac{-2(0) + 11}{0 - 5} = \frac{11}{-5} \)
• Simplify to get \( y = -2.2 \)

Therefore, the y-intercept of this function is \( -2.2 \), which corresponds to the point (0, -2.2) on the graph.

Knowing the y-intercept is useful because it helps you start plotting the function and gives you a reference point on the vertical axis.