Problem 504 - Precalculus

Question

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$z(x)=\frac{(x+2)^{2}(x-5)}{(x-3)(x+1)(x+4)}$$

Step-by-Step Solution

Answer

Horizontal intercepts: \(x = -2, 5\). Vertical intercept: \((0, \frac{5}{3})\). Vertical asymptotes: \(x = 3, -1, -4\). Horizontal asymptote: \(y = 1\).

Step 1: Find Horizontal Intercepts

To find horizontal intercepts, set the numerator of the function equal to zero because these are the points where the function crosses the x-axis. For \(z(x) = \frac{(x+2)^{2}(x-5)}{(x-3)(x+1)(x+4)}\), set \((x+2)^2(x-5) = 0\). This gives horizontal intercepts at \(x = -2\) (multiplicity 2) and \(x = 5\).

Step 2: Find the Vertical Intercept

To find the vertical intercept, evaluate the function at \(x = 0\). Substitute \(x = 0\) into \(z(x)\): \(z(0) = \frac{(0+2)^2(0-5)}{(0-3)(0+1)(0+4)}\). Simplifying, we get \(z(0) = \frac{4(-5)}{-12} = \frac{20}{12} = \frac{5}{3}\). So the vertical intercept is at \((0, \frac{5}{3})\).

Step 3: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function equals zero and the numerator is non-zero. Set the denominator \((x-3)(x+1)(x+4) = 0\), leading to vertical asymptotes at \(x = 3, x = -1, x = -4\).

Step 4: Determine Horizontal Asymptote or Slant Asymptote

Since the degrees of the numerator and denominator are both 3, to find the horizontal asymptote, divide the leading coefficients. Both leading coefficients are 1 (from \(x^3\)), thus the horizontal asymptote is \(y = 1\).

Step 5: Sketch the Graph

Use the identified features to sketch the graph. Mark horizontal intercepts at \(x = -2\) (with a bounce) and \(x = 5\), the vertical intercept at \(y = \frac{5}{3}\), the vertical asymptotes at \(x = 3, -1,\) and \(-4\), and the horizontal asymptote at \(y = 1\). The curve will approach the asymptotes, bounce at \(x = -2\), and cross at \(x = 5\).

Key Concepts

Horizontal Intercepts

Horizontal intercepts, also known as x-intercepts, are the points where the graph of a function crosses the x-axis. This crossing occurs when the output value of the function is zero. For rational functions like \(z(x) = \frac{(x+2)^{2}(x-5)}{(x-3)(x+1)(x+4)}\), finding the horizontal intercepts involves setting the **numerator** equal to zero and solving for \(x\).

• For the numerator \((x+2)^2(x-5)\), the points where the expression equals zero are \(x = -2\) and \(x = 5\).
• The solution \(x = -2\) has a multiplicity of 2, meaning the graph will "bounce" off the x-axis at this point, rather than crossing it.
• At \(x = 5\), the graph crosses the x-axis because the factor appears only once.

Clearly stating and understanding these intercepts is crucial since they guide how the graph interacts with the x-axis, showing real-world outcomes like profit-loss trends or population growth predictions.

Vertical Asymptote

Vertical asymptotes are vertical lines that the graph of a function approaches but never touches or crosses. For a rational function, these occur where the **denominator is zero** and the numerator is not zero.

• From the denominator \((x-3)(x+1)(x+4)\), solve \((x-3)(x+1)(x+4) = 0\) to find \(x = 3\), \(x = -1\), and \(x = -4\).
• Each of these values indicates a vertical asymptote; the graph will draw nearer to these x-values but not intersect or touch them.

Vertical asymptotes signify undefined behaviors or limits within the rational function, useful for various applications such as determining the limits of certain reactions in chemistry or physics.

Horizontal Asymptote

Horizontal asymptotes show the behavior of a function as it heads towards infinity or nears the horizontal axis. For rational functions, the degree of the numerator and denominator polynomials decides their placement.

• Here, the function \(z(x) = \frac{(x+2)^{2}(x-5)}{(x-3)(x+1)(x+4)}\) has both the numerator and denominator of degree 3 (highest power of \(x\)).
• This equal degree setup results in a horizontal asymptote found by the ratio of the leading coefficients, which is \(\frac{1}{1} = 1\).
• Thus, the horizontal asymptote is \(y = 1\), meaning the graph will approach the line \(y = 1\) as \(x\) goes to positive or negative infinity.

In real-life scenarios, horizontal asymptotes can help predict steady states or growth limitations in biological and economic models.

Vertical Intercept

A vertical intercept, also known as the y-intercept, pinpoints where the graph of a function crosses the y-axis. It is found by evaluating the function at \(x = 0\).

• For \(z(x) = \frac{(x+2)^{2}(x-5)}{(x-3)(x+1)(x+4)}\), substitute \(x = 0\) and simplify: \