Problem 35 - Algebra 2

Question

Sketch the graph of each rational function. $$ y=\frac{2 x+3}{x-5} $$

Step-by-Step Solution

Answer

The plotted graph of the function \( y = \frac{2x + 3}{x - 5} \) shows a vertical asymptote at \( x = 5 \) and the function intersects the x-axis at \( x = -3/2 \). Points on either side of the asymptote are used to depict the behavior of the function near its asymptote, with the function getting infinitely close to the asymptote as \( x \) approaches 5 from either side.

Step 1: Identify the Vertical Asymptote

The vertical asymptote of the function is determined by setting the denominator equal to zero and solving for \( x \). Therefore, \( x - 5 = 0 \) implies \( x = 5 \). This makes our vertical asymptote \( x = 5 \)

Step 2: Identify the Zero of the Function

The zero of the function is determined by setting the function equal to zero. In this case, the function equals zero when the numerator equals zero. Therefore, solving \( 2x + 3 = 0 \), we get \( x = -3/2 \). Hence, the graph of the function intersects the x-axis at \( -3/2 \)

Step 3: Select Points and Sketch the Graph

Select some random \( x \) values and use them to calculate corresponding \( y \) values using the formula \( y = \frac{2x+3}{x-5} \). Plot these points along with the vertical asymptote and zero of the function. Now, sketch the graph of the function which should approach the vertical asymptote without crossing it, and pass through the zero of the function.

Key Concepts

Vertical Asymptote

A vertical asymptote in a rational function is a vertical line that the graph of the function approaches but never actually crosses. It represents a value of \( x \) that makes the function undefined. In our given function, \( y = \frac{2x+3}{x-5} \), we find the vertical asymptote by setting the denominator equal to zero. This is because division by zero is undefined, indicating where our asymptote will lie.

For this particular function, setting \( x-5 = 0 \) leads to \( x = 5 \). This tells us that as \( x \) approaches 5, the function's values will increase or decrease without bound, creating a vertical asymptote at \( x = 5 \). When sketching the graph, this line will be a boundary that the curve will approach but never touch.

Zero of the Function

The zero of a function is another important feature in graphing rational functions. It refers to the value of \( x \) where the output or \( y \)-value of the function is zero. For rational functions, zeros occur where the numerator equals zero, as this results in a quotient of zero.

In our case with \( y = \frac{2x + 3}{x - 5} \), we set the numerator \( 2x + 3 \) equal to zero and solve for \( x \). By doing so, we find \( 2x + 3 = 0 \) which simplifies to \( x = -\frac{3}{2} \). This point, \( (-\frac{3}{2}, 0) \), indicates where the graph of the function will cross the x-axis. Such a point is crucial because it offers an exact spot through which the graph must pass when sketching.

Graph Sketching

Graph sketching of a rational function involves combining the insights from both the vertical asymptotes and the zeros of the function, as well as calculating additional points to ensure a smooth and accurate curve. Selecting various \( x \) values in the domain can help generate corresponding \( y \) values that offer more context for the function's behavior.

When you plug in selected \( x \)-values into the function, calculate the \( y \)-values and plot the points on a graph. For \( y = \frac{2x + 3}{x - 5} \), make sure to calculate points on either side of the vertical asymptote at \( x = 5 \). Drawing the curve includes having the graph approach the asymptote without ever touching it, while ensuring the graph passes through the zero at \( x = -\frac{3}{2} \).

Finally, observing the behavior of the function as \( x \) moves towards the asymptote, and towards positive or negative infinity, will help confirm the general shape. This comprehensive approach ensures you're able to successfully sketch the rational function and understand its behavior fully.