Problem 39 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$g(x)=\frac{3-x}{x+4}$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = \frac{3-x}{x+4}\) has a vertical asymptote at \(x = -4\), a horizontal asymptote at \(y = -1\), a y-intercept at (0,3/4), and an x-intercept at (3,0).

Step 1: Identifying the vertical asymptote

The vertical asymptote of a rational function can be found by setting the denominator equal to zero. In this case, solve the equation \(x + 4 = 0\) for x. This gives \(x = -4\) as the vertical asymptote.

Step 2: Identifying the horizontal asymptote

The horizontal asymptote of a rational function depends on the degrees of the polynomial in the numerator and the denominator. Since the degree of the numerator and the denominator are both one and the leading coefficients are -1 and 1 respectively, the horizontal asymptote is given by \(-1/1\) or \(-1\).

Step 3: Identifying the x-intercept

The x-intercept(s) can be found by setting the function equal to zero and solving for x. For this function, solve \(3-x = 0\), resulting in \(x = 3\) as the x-intercept.

Step 4: Identifying the y-intercept

The y-intercept is found by evaluating the function at x=0. Substituting 0 into \(g(x)\) yields \(g(0) = 3/4\) as the y-intercept.

Step 5: Sketching the function

With the vertical asymptote at \(x=-4\), the horizontal asymptote at \(y=-1\), the y-intercept at (0,3/4), and the x-intercept at (3,0), we can now sketch the graph.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are like invisible walls in a graph where the rational function approaches but never actually reaches. They occur when the denominator of the rational function is equal to zero, as division by zero is undefined. For the function \(g(x) = \frac{3-x}{x+4}\), we find this by setting the denominator \(x+4\) equal to zero. Solving \(x+4 = 0\) yields \(x = -4\) as the vertical asymptote.

• Think of vertical asymptotes as boundaries. As the graph gets closer to \(x = -4\), the graph shoots off towards infinity or negative infinity.
• This means the function never touches or crosses this line.
• Vertical asymptotes often indicate points where the function changes direction or behavior dramatically.

Understanding vertical asymptotes helps us see how the function behaves near certain critical values.

Horizontal Asymptotes

Horizontal asymptotes provide insight into the long-term behavior of a function as \(x\) approaches infinity or negative infinity. For rational functions, the horizontal asymptote is determined by comparing the degrees of the polynomials in the numerator and the denominator. For our function \(g(x) = \frac{3-x}{x+4}\), both the numerator and the denominator are degree 1.

• When degrees are the same, divide the leading coefficients: -1 from the numerator and 1 from the denominator.
• This results in a horizontal asymptote at \(y = \frac{-1}{1} = -1\).
• This means no matter how large \(x\) gets, the value of \(y\) will approach -1 but never actually reach it.

Knowing the horizontal asymptote is crucial for understanding the end behavior of the graph.

x-intercepts

The x-intercept of a function is the point where the graph crosses the x-axis, which means the value of \(g(x)\) is zero. To find the x-intercept, set the numerator equal to zero because the numerator determines when the overall equation equals zero. In our example, set \(3-x = 0\).

• Solving this gives \(x = 3\).
• The coordinates of the x-intercept are \( (3, 0) \).
• This tells us that at \(x = 3\), the graph will cross the x-axis.

Finding x-intercepts helps in constructing a graph as it indicates where the function outputs a value of zero.

y-intercepts

Y-intercepts indicate where the graph crosses the y-axis, which is where \(x\) equals zero. To find the y-intercept, simply substitute \(x=0\) into the function. For \(g(x) = \frac{3-x}{x+4}\), substituting 0 gives:

• \(g(0) = \frac{3 - 0}{0 + 4} = \frac{3}{4}\).
• This provides the coordinates \((0, \frac{3}{4})\).
• At this point, the graph crosses the y-axis at \(y = \frac{3}{4}\).

The y-intercept gives a starting point for graphing the function and understanding its initial value.