Problem 43 - College Algebra

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. $$ f(x)=\frac{3 x^{2}-14 x-5}{3 x^{2}+8 x-16} $$

Step-by-Step Solution

Answer

x-intercepts: roots of \(3x^2 - 14x - 5 = 0\), y-intercept: \(\frac{5}{16}\), vertical asymptotes: roots of \(3x^2 + 8x - 16 = 0\), horizontal asymptote: \(y=1\).

Step 1: Finding Horizontal Intercepts

To find the horizontal intercepts (or x-intercepts) of the function \( f(x) = \frac{3x^2 - 14x - 5}{3x^2 + 8x - 16} \), set the numerator equal to zero: \( 3x^2 - 14x - 5 = 0 \). Solve for \( x \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = -14 \), \( c = -5 \). Substitute and solve to find the x-values.

Step 2: Finding Vertical Intercept

The vertical intercept (or y-intercept) is found by evaluating the function at \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \) and simplify: \( f(0) = \frac{3(0)^2 - 14(0) - 5}{3(0)^2 + 8(0) - 16} = \frac{-5}{-16} = \frac{5}{16} \). Therefore, the vertical intercept is \( \left(0, \frac{5}{16}\right) \).

Step 3: Finding Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero). Set the denominator equal to zero: \( 3x^2 + 8x - 16 = 0 \). Solve this quadratic equation using the quadratic formula to find the x-values of the vertical asymptotes.

Step 4: Determining Horizontal Asymptotes

Since the degrees of the numerator and denominator are equal (both are 2), the horizontal asymptote can be found by dividing the leading coefficients of the numerator and the denominator. The horizontal asymptote is \( y = \frac{3}{3} = 1 \).

Step 5: Sketching the Graph

Using the intercepts and asymptotes, sketch the graph. Plot the vertical intercept at \( \left(0, \frac{5}{16}\right) \) and horizontal intercepts calculated in Step 1. Draw vertical asymptotes at x-values found in Step 3. Add a horizontal line at \( y = 1 \) for the horizontal asymptote. Ensure the curve approaches these asymptotes appropriately.

Key Concepts

Horizontal Intercepts

The horizontal intercepts of a function, also known as x-intercepts, are the values of \(x\) where the function crosses the x-axis. This happens when the output of the function \(f(x)\) is zero. For rational functions such as \(f(x) = \frac{3x^2 - 14x - 5}{3x^2 + 8x - 16}\), you find the horizontal intercepts by setting the numerator equal to zero and solving for \(x\).

• Set \(3x^2 - 14x - 5 = 0\).
• This forms a quadratic equation where finding \(x\) solutions involves using the quadratic formula.

Finding the horizontal intercepts provides crucial points that help in graphing the function. These points allow us to understand where the graph touches or crosses the x-axis.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. For rational functions such as \(f(x) = \frac{3x^2 - 14x - 5}{3x^2 + 8x - 16}\), vertical asymptotes occur at the x-values where the denominator becomes zero, but the numerator is not zero. To find vertical asymptotes:

• Set the denominator equal to zero: \(3x^2 + 8x - 16 = 0\).
• Solve for \(x\) using the quadratic formula or factorization, if possible.

The graph will approach these x-values, but never actually meet them. Understandably, these asymptotes influence the shape of the graph significantly by acting as boundaries.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The solutions to the equation are given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to calculate the exact values of \(x\) that make the quadratic equation zero. In terms of rational functions, the quadratic formula is utilized to find both horizontal intercepts (when applied to the numerator) and vertical asymptotes (when applied to the denominator). Understanding this formula is key:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic expression.
• The expression \(b^2 - 4ac\) inside the square root is known as the discriminant.
• The discriminant helps determine the number and type of solutions.

Rational Functions

Rational functions are functions expressed as the quotient of two polynomials. A rational function like \(f(x) = \frac{3x^2 - 14x - 5}{3x^2 + 8x - 16}\) is defined for all x-values, except where the denominator equals zero. Key components of such functions include:

• Horizontal and vertical intercepts: Points where the graph crosses the axes.
• Asymptotes: Lines where the graph approaches but never touches.

Understanding how to find and interpret intercepts and asymptotes is essential in analyzing and graphing rational functions. These features provide insights into the behavior of the function and help in accurately plotting its graph.