Problem 60 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{2}+2 x+1}{x+3}$$

Step-by-Step Solution

Answer

The x-intercepts are -1 and -1 (double root at -1), the y-intercept is \(\frac{1}{3}\), and the slant asymptote is \(x - 1\). And the graph should be drawn according to these.

Step 1: Determining the x-intercepts

The x-intercepts are the values for x when y or h(x) equals zero. By setting \( h(x) = 0 \) we get: \( \frac{{x^2 + 2x + 1}}{{x + 3}} = 0 \). By multiplying both sides by \(x + 3\) all that remains is \(x^2 + 2x + 1 = 0\). The x-intercepts can now be found by solving this quadratic equation.

Step 2: Determining the y-intercept

The y-intercept is found by substituting x = 0 into the rational function. So, the y-intercept is \(h(0) = \frac{{0^2 + 2*0 + 1}}{{0 + 3}} = \frac{1}{3}\).

Step 3: Determining the Slant Asymptote

The slant asymptotes are by performing polynomial (or long) division of \(x^2 + 2x + 1\) by \(x + 3\). The result of the division is the slant asymptote.

Step 4: Draw the Graph

With the x-intercepts, y-intercept, and the slant asymptote, the graph of the rational function \( h(x) = \frac{{x^2 + 2x + 1}}{{x + 3}} \) can now be drawn. The curve approaches but never crosses the slant asymptote.

Key Concepts

X-Intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. In the case of a rational function, such as \( h(x) = \frac{{x^2 + 2x + 1}}{{x + 3}} \), we find the x-intercepts by setting the numerator of the function equal to zero and solving for x. This is because the output of the function (h(x)) must be zero for a point to lie on the x-axis.

By solving the quadratic equation \( x^2 + 2x + 1 = 0 \), which is in the numerator, we find the x-intercept(s). For \( h(x) \), the equation factorizes to \( (x+1)^2 = 0 \), indicating that \( -1 \) is the only x-intercept, and it is a double root. Remember that if the quadratic equation does not factor easily, you may use the quadratic formula to find the intercepts.

Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. To find this point for a rational function, we set x to zero and evaluate the function. So for our function \( h(x) \), the calculation would be \( h(0) = \frac{{0^2 + 2*0 + 1}}{{0 + 3}} \).

In simpler terms, the y-intercept of our graph is at \( (0, \frac{1}{3}) \). This single point helps to give an initial shape to the graph when you begin plotting, providing a reference point where the curve will pass through.

Slant Asymptotes

A slant asymptote, also known as an oblique asymptote, occurs when the degree of the numerator is exactly one more than the degree of the denominator in a rational function. To find a slant asymptote, we perform polynomial division between the numerator and denominator. In the case of our function \( h(x) \), we divide \( x^2 + 2x + 1 \) by \( x + 3 \).

The quotient from this division gives the equation of the slant asymptote. For our function, the division does not leave a remainder, indicating that the slant asymptote and the rational function coincide for large values of x. Slant asymptotes are important because they provide a clear description of the behavior of the graph at the extremes of the x-axis.

Polynomial Division

The process of polynomial division is similar to long division but involves polynomials instead of numbers. The technique is especially useful for simplifying rational functions and finding slant asymptotes as part of graphing these functions. When we have a rational function where the degree of the numerator is higher than the degree of the denominator, polynomial division can be used to rewrite the function into a more manageable form.

For the function \( h(x) \) provided, we would divide \( x^2 + 2x + 1 \) by \( x + 3 \). This operation can reveal not only the slant asymptote but also simplify the function, allowing us to understand its long-term behavior as \( x \) approaches infinity or negative infinity. This simplification is crucial when graphing the function as it gives insight into the end behavior of the curve.