Problem 61 - 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. $$f(x)=\frac{3 x^{2}+5 x-2}{x+1}$$

Step-by-Step Solution

Answer

The x-intercepts are \(x = -2\) and \(x = 1/3\), y-intercept is \(y = -2\), slant asymptote is \(y = 3x - 3\), and vertical asymptote is \(x = -1\).

Step 1: Find the x and y intercepts

The x-intercept can be calculated by setting the function equal to zero and solving for x, which gives the roots of the numerator. So, solve \(3 x^{2} +5 x -2=0\). Similarly, the y-intercept is found by setting x as zero in the equation.

Step 2: Calculate the Slant Asymptote

The slant asymptote can be determined by long division of the numerator by the denominator. A slant asymptote exists when the degree of the numerator is exactly one more than the degree of the denominator.

Step 3: Identify the Vertical Asymptote

The vertical asymptote is found when the denominator \(x + 1\) is equal to zero, thus, \(x = -1\).

Step 4: Sketch the Graph

Using the x and y intercepts, asymptotes as guide, sketch the graph of the function.

Key Concepts

Understanding X-Intercepts in Rational Functions

An x-intercept is a point where the graph of a function crosses the x-axis. For rational functions, finding the x-intercepts involves finding the values of x that make the numerator of the function equal to zero. To find these points, set the numerator of the rational function equal to zero and solve for x.
For the given function \( f(x) = \frac{3x^{2} + 5x - 2}{x+1} \), you calculate the x-intercepts by solving \( 3x^{2} + 5x - 2 = 0 \).
By solving this quadratic equation, you will get the values of x where the curve intersects the x-axis. These solutions, often found using the quadratic formula or factoring, are crucial for understanding where the function has zero values.

Exploring Y-Intercepts in Rational Functions

Y-intercepts are points where the graph intersects the y-axis. To find the y-intercept of a rational function, substitute x with zero in the function’s equation and solve for f(x). This calculation gives you the y-value of the point where the graph crosses the y-axis.
In our function, substituting x = 0 gives \( f(0) = \frac{3(0)^{2} + 5(0) - 2}{0+1} = \frac{-2}{1} = -2 \).
Thus, the y-intercept is at point (0, -2), showing us how the curve behaves when starting from the y-intercept.

Slant Asymptotes and How They Appear

Slant asymptotes, also called oblique asymptotes, occur in rational functions when the degree of the numerator is exactly one higher than the degree of the denominator. They portray a slanting line that the function approaches but never touches as it extends toward infinity. To identify a slant asymptote, you perform polynomial long division on the rational expression.
In the case of \( f(x) = \frac{3x^{2} + 5x - 2}{x+1} \), you divide the numerator \(3x^{2} + 5x - 2\) by the denominator \(x+1\).
Ignoring the remainder, the quotient of this division will give you the equation of the slant asymptote, which guides how the graph behaves at its extremes.

Understanding Vertical Asymptotes in Rational Functions

Vertical asymptotes occur where the denominator of a rational function is zero and the function is undefined. These asymptotes are vertical lines that represent points the function cannot attain, showing the function’s tendency to climb towards positive or negative infinity as it nears these lines.
For \(f(x) = \frac{3x^{2} + 5x - 2}{x+1}\), the denominator \(x+1\) is zero when \(x = -1\).
Thus, there is a vertical asymptote at \(x = -1\). This means that as x approaches -1 from either side, the function values increase or decrease without bound, which is a distinctive feature on the graph of the rational function.