Problem 22 - Algebra 2

Question

Graph each rational function. $$ f(x)=\frac{5 x}{x+1} $$

Step-by-Step Solution

Answer

Graph with asymptote at \(x = -1\) and \(y = 5\), intercept at (0,0).

Step 1: Identify the Asymptotes

The vertical asymptote occurs where the denominator is zero, so set \(x+1 = 0\). Solving this gives the vertical asymptote at \(x = -1\). This function has a horizontal asymptote at \(y = 5\), because as \(x\rightarrow \infty\), \(f(x)\rightarrow \frac{5}{1} = 5\).

Step 2: Find the Y-intercept

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into \(f(x)\):\[f(0) = \frac{5\cdot 0}{0+1} = 0\]. Hence, the y-intercept is \((0, 0)\).

Step 3: Find the X-intercept

The x-intercept occurs where \(f(x) = 0\). This is when the numerator is zero:\[5x = 0 \]Solving gives \(x = 0\), so the x-intercept is \((0, 0)\).

Step 4: Plot Key Points and Asymptotes

Plot the x-intercept \((0, 0)\) and the asymptotes along \(x = -1\) and \(y = 5\). Choose additional points such as \(x = -2, 1, 2\), and calculate their corresponding \(y\) values for accuracy.

Step 5: Sketch the Graph

Using the asymptotes and points, sketch the curve approaching the asymptotes. Ensure that the graph gets closer to \(x = -1\) and \(y = 5\) without touching them.

Key Concepts

Understanding Asymptotes

Asymptotes are lines that a graph of a function approaches but never actually touches. They are essential in understanding the behavior of rational functions. There are two types of asymptotes relevant to rational functions: vertical and horizontal.
The vertical asymptote occurs where the denominator becomes zero. For the function \(f(x) = \frac{5x}{x+1}\), it can be found by setting the denominator equal to zero: \(x+1 = 0\). Solving this gives \(x = -1\). This means that the graph will approach the line \(x = -1\) but never cross it. This creates a division in the graph as it cannot take a value at \(x = -1\).
Horizontal asymptotes analyze the behavior of a function as \(x\) approaches infinity. For this function, the horizontal asymptote can be determined by the ratio of the leading coefficients of the polynomial in the numerator and the polynomial in the denominator. Here, both are of degree one, so the horizontal asymptote is at \(y = 5\), aligning with \(\frac{5}{1}\). This tells us that as \(x\) grows large positively or negatively, the function will stabilize towards the line \(y = 5\) without actually reaching it.

Finding Intercepts

Intercepts are where the graph crosses the axes, offering crucial insights into the function's performance at these intersections. There are two primary types of intercepts for rational functions: x-intercepts and y-intercepts.
The y-intercept can be easily found by setting \(x = 0\) and calculating \(f(0)\). For our function, \(f(0) = \frac{5\times 0}{0 + 1} = 0\), revealing that the y-intercept is at \((0,0)\). This is where the graph crosses the y-axis.
The x-intercept, on the other hand, occurs where the entire function \(f(x) = 0\). This requires the numerator to be zero while the denominator is not zero. In \(5x = 0\), solving gives \(x = 0\), so the x-intercept is also at the origin \((0,0)\). This point marks where the graph crosses the x-axis, which in this case, it coincides exactly with the y-intercept.

Graph Sketching Essentials

Graph sketching is all about plotting the function's behavior across its domain accurately. It involves combining the information about asymptotes, intercepts, and additional key points.
Begin by plotting the known intercepts and asymptotes. For \(f(x) = \frac{5x}{x+1}\), mark the asymptotes at \(x = -1\) (vertical) and \(y = 5\) (horizontal). Locate the intercept at \((0,0)\) where both intercepts coincide.
Next, select a few additional points on either side of the vertical asymptote to determine how the graph behaves as it approaches \(x = -1\). Calculate values of \(y\) for convenient \(x\) values like \(x = -2,\ 1,\ 2\). These points will help to outline the curve within different regions distinguished by the vertical asymptote.
Finally, sketch the graph, ensuring it smoothly approaches but never crosses the asymptotes. Watch as the graph moves steeply upward or downward near \(x = -1\) and smoothly trends to \(y = 5\) at larger positive or negative \(x\) values, capturing the full behavior of this rational function.