Problem 2 - A Graphical Approach to College Algebra

Question

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. (Column II) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote. (Column I) $$f(x)=\frac{x+10}{x+2}$$

Step-by-Step Solution

Answer

The y-intercept is (0, 5).

Step 1: Understanding the Graph of Rational Functions

Rational functions are of the form \( f(x) = \frac{N(x)}{D(x)} \), where \(N(x)\) and \(D(x)\) are polynomials. The characteristics of these functions, such as intercepts and asymptotes, depend on these polynomials.

Step 2: Determine the x-intercept

The \(x\)-intercept of a function is where \(f(x) = 0\), meaning the numerator \(N(x) = 0\). For \(f(x)=\frac{x+10}{x+2}\), set \(x+10=0\), giving \(x = -10\). Hence, the \(x\)-intercept is \((-10, 0)\).

Step 3: Determine the y-intercept

To find the \(y\)-intercept, set \(x = 0\) and solve for \(f(0)\). For \(f(x)=\frac{x+10}{x+2}\), \(f(0) = \frac{0+10}{0+2} = 5\). Thus, the \(y\)-intercept is \((0, 5)\).

Step 4: Determine the vertical asymptote

Vertical asymptotes occur where the denominator \(D(x) = 0\) and the numerator is non-zero. For \(f(x)=\frac{x+10}{x+2}\), set \(x+2=0\), giving \(x = -2\). Thus, there is a vertical asymptote at \(x = -2\).

Step 5: Determine the horizontal asymptote

For a rational function, if the degrees of the numerator and denominator are equal, the horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of the numerator and denominator, respectively. Here, it is \(y = \frac{1}{1} = 1\).

Step 6: Checking for holes and oblique asymptotes

A hole occurs if \(N(x)\) and \(D(x)\) share a common factor which is not the case here. An oblique asymptote occurs if the degree of \(N(x)\) is one more than \(D(x)\), which is not satisfied here.

Key Concepts

x-intercept

In the graph of a rational function, the \( x \)-intercept represents the point where the graph crosses the \( x \)-axis. To find this point, set the function \( f(x) \) to zero and solve for \( x \). This essentially means we solve for the values of \( x \) that make the numerator zero, while ensuring the denominator is not zero at these points.

For the function \( f(x) = \frac{x+10}{x+2} \), set the numerator equal to zero, which means solving \( x+10=0 \). This gives \( x = -10 \). Therefore, the \( x \)-intercept of this function is \((-10, 0)\). This means that the graph will cross or touch the \( x \)-axis at this coordinate point.

y-intercept

The \( y \)-intercept is the point where the graph intersects the \( y \)-axis. This happens when \( x = 0 \). To find it, substitute \( x=0 \) into the rational function and solve for \( f(x) \). This will give you the \( y \)-coordinate of the intercept.

For the given function \( f(x) = \frac{x+10}{x+2} \), when \( x = 0 \), the calculation is \( f(0) = \frac{0+10}{0+2} \). Simplifying this, we find \( f(0) = 5 \).

Hence, the \( y \)-intercept is \((0, 5)\), indicating that the graph crosses the \( y \)-axis at this point.

vertical asymptote

Vertical asymptotes of a rational function are lines where the graph proceeds to infinity, causing the function value to increase or decrease without bound. They occur wherever the denominator equals zero while the numerator does not simultaneously equal zero.

To find the vertical asymptote for \( f(x) = \frac{x+10}{x+2} \), determine where the denominator \( x+2=0 \). Solving gives \( x = -2 \). This means there is a vertical asymptote at \( x = -2 \).

On the graph, this line represents a location where the function cannot take a finite value, leading to a steep structure approaching this line.

horizontal asymptote

Horizontal asymptotes represent the value that a function approaches as \( x \) tends towards positive or negative infinity. In rational functions, these depend on the degrees of the numerator and denominator polynomials.

For the function \( f(x) = \frac{x+10}{x+2} \), both the numerator and denominator are of degree one. In such cases, the horizontal asymptote is found by dividing the leading coefficients. Here, it's \( y = \frac{1}{1} = 1 \).

This line is where the function 'flattens out' at very large or very small values of \( x \). The graph will get closer and closer to \( y = 1 \), but never actually touch it at extreme values of \( x \).

graph characteristics

To gain a complete understanding of rational functions, it is essential to look at various graph characteristics which include intercepts, asymptotes, and any peculiarities like holes.

• **Intercepts:** These are the points where the graph crosses the axes; namely, the \( x \)-intercept where \( y = 0 \) and the \( y \)-intercept where \( x = 0 \).
• **Vertical Asymptotes:** Vertical lines where functions go to infinity due to division by zero in the denominator alone.
• **Horizontal Asymptotes:** Lines reflecting the end behavior of the function as \( x \) goes to infinity.
• **Holes:** Points where both the numerator and the denominator have a common factor, leading to indeterminate points. Not relevant to this problem, but useful in others.

For \( f(x) = \frac{x+10}{x+2} \), the lack of shared factors avoids holes, and the degrees ensure there are no oblique asymptotes, focusing the graph attributes on the determined intercepts and asymptotes.