Question

In Exercises 25–32, graph the function. State the domain and range. $$ y=\frac{x+6}{4 x-8} $$

Short answer

The vertical asymptote is \( x = 2 \). The domain is \( x \neq 2 \), and the range is all real numbers. The graph of the function is a hyperbola with vertical asymptote at \( x=2 \).

Step-by-step solution

Find the Vertical Asymptote

Solve \( 4x-8=0 \) to find the vertical asymptote, which is \( x=2 \). This is because rational functions have vertical asymptote where the denominator is zero.

Determine the Domain

The domain is all real number except the x value of the vertical asymptote. So, the domain is \( x \neq 2 \). The x value 2 is excluded because it leads to division by zero which is undefined in mathematics.

Determine the Range

Looking at our function it may be seen that the outputs could be any real number, so the range is all real numbers.

Graph the Function

To graph this function, plot the vertical asymptote at x=2. Then plot a few points on either side of the asymptote. As x approaches 2 from the left or right, the y-values go to negative or positive infinity. This graph is a hyperbola.

Key concepts

Vertical Asymptote

A vertical asymptote is a vertical line on a graph where the values of a function approach infinity as the input gets closer to a certain value. In rational functions like \( y=\frac{x+6}{4x-8} \), vertical asymptotes occur where the denominator of the fraction equals zero. These points are crucial as they help us understand the behavior of the graph near certain x-values.
To find a vertical asymptote, solve for the x-value that makes the denominator zero. In this function, setting \( 4x-8=0 \) gives \( x=2 \). This means as x approaches 2, the function's output goes towards positive or negative infinity. Recognizing this point on the graph involves:

• Setting the denominator equal to zero
• Solving for x
• Drawing a dashed line on the graph at this x-value

These steps ensure that you comprehend where and why vertical asymptotes appear in rational functions.

Domain and Range

The domain and range are fundamental concepts in understanding the scope and scale of functions.

The **domain** of a function refers to all the possible input values (x-values) that can be plugged into the function without causing any errors such as division by zero. For the function \( y=\frac{x+6}{4x-8} \), the domain is all real numbers except \( x=2 \), expressed as \( x eq 2 \). This exclusion is because having \( x=2 \) makes the denominator zero, leading to an undefined scenario.

The **range** is all possible output values (y-values) a function can produce. In our rational function example, although calculating range can sometimes be complex, in this context, observe that there are no restrictions on y-values based directly on the function's structure. Thus, the range is all real numbers, meaning any real number could potentially be a result of y.

Graphing Hyperbolas

Graphing hyperbolas, particularly when they arise from rational functions, involves understanding how these curves behave near asymptotes.
For the function \( y=\frac{x+6}{4x-8} \), graphing starts by marking the vertical asymptote at \( x=2 \). Next, select several x-values on both sides of this asymptote to compute corresponding y-values. Plot these points on the coordinate plane.
Observe that the graph forms two distinct curves that mirror each other with the vertical asymptote standing between them. As x moves closer to 2 from either side, the y-values will shoot up to infinity or dive down to negative infinity, indicating the function’s rapid increase or decrease.
When graphing:

• Note the approach of curves towards asymptotes without touching them
• Ensure the shape resembling hyperbola branches
• Assess symmetry if any, about vertical or horizontal axes

This visual skill helps in deeper explorations of mathematical behaviors and curve trends described by rational functions.