Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$h(t)=\frac{3 t^{2}}{t^{2}-4}$$

Step-by-Step Solution

Answer

The function \(h(t) = \frac{3t^{2}}{t^{2} - 4}\) has x and y intercepts at t=0. The function is symmetric across the y-axis. The function also has vertical asymptotes at \(t = -2\) and \(t = 2\), and a horizontal asymptote at \( t = 3 \).

Step 1: Checking for intercepts

The x-intercepts of the graph of the function are the values of \(t\) for which \(h(t) = 0\). Solving the equation \(3 t^{2} = 0\), it is found that the x-intercept is at t=0. The y-intercept is found when \(t = 0\), giving a y-intercept of 0 as well.

Step 2: Checking for symmetry

If the function is even, \(h(t) = h(-t)\), the graph of the function has symmetry about the y-axis. If the function is odd, \(h(-t) = -h(t)\), the graph of the function has symmetry about the origin. Substituting \(-t\) into the original function gives \(\frac{3(-t)^{2}}{(-t)^{2} - 4} = \frac{3t^{2}}{t^{2} - 4}\), equal to \(h(t)\), so the function has y-axis symmetry.

Step 3: Checking for vertical asymptotes

The vertical asymptotes occur at the values of \(t\) for which \(h(t)\) becomes undefined. The only values of \(t\) for which \(h(t)\) is undefined are \(t = 2\) and \(t = -2\). Thus, \(t = 2\) and \(t = -2\) are vertical asymptotes.

Step 4: Checking for horizontal asymptotes

If the degree of the polynomial in the denominator is greater than or equal to the degree of the polynomial in the numerator, then the x-axis is a horizontal asymptote. If the degrees are equal, then there is a horizontal asymptote at the quotient of the leading coefficients. For this function, the degrees of the numerator and denominator are equal and the leading coefficients are 3 and 1, therefore there is a horizontal asymptote at \( t = \frac{3}{1} = 3 \).

Key Concepts

Rational Function Graph

When visualizing a rational function graph, you're looking at the shape created by plotting the function on a coordinate plane. A rational function is one where the numerator and denominator are both polynomials, like the one given: \( h(t)=\frac{3 t^{2}}{t^{2}-4} \).

Understanding the graph involves identifying intercepts, which are points where the graph crosses the axes, as well as symmetry, which speaks to the balance and reflection of the graph across an axis or a point. Additionally, recognizing vertical and horizontal asymptotes will help predict the behavior of the graph as it approaches certain values of \( t \).

Vertical Asymptotes

Vertical asymptotes are the lines where the function heads off to infinity. These occur at values of \( t \) which make the denominator of our function zero, as the function value becomes undefined.

For the given function \( h(t) \), the denominator \( t^{2}-4 \) becomes zero at \( t = 2 \) and \( t = -2 \), resulting in vertical asymptotes at these points. This means the graph will approach these lines without touching or crossing them, displaying a kind of boundary for the function.

Horizontal Asymptotes

Horizontal asymptotes, on the other hand, define the behavior of a graph as \( t \) approaches infinity or negative infinity. These are particularly important for rational functions where the degrees of the numerator and denominator are equal.

For our example, since the degrees are indeed equal and the leading coefficient of the numerator (3) is divided by the leading coefficient of the denominator (1), we find a horizontal asymptote at \( y = 3 \). The graph will approach this line but typically not cross it, representing the function's end behavior.

Symmetry in Functions

Symmetry in functions helps us understand the reflective properties of a graph. A function is symmetrical about the y-axis if replacing \( t \) with \( -t \) results in the original function, and about the origin if the same replacement yields the negation of the function.

In the case of \( h(t) \), replacing \( t \) with \( -t \) gives us the same function, indicating it is symmetrical about the y-axis. This axis of symmetry allows us to predict the shape of one side of the graph by simply mirroring the other side.

Function Intercepts

Function intercepts are the points at which the graph crosses the axes. For x-intercepts, we set \( h(t) \) to zero and solve for \( t \). For our function \( h(t)=\frac{3 t^{2}}{t^{2}-4} \), the numerator must be zero to yield \( h(t) = 0 \), offering a single x-intercept at (0,0).

The y-intercept occurs when \( t = 0 \), also yielding the coordinate (0,0). Intercepts are crucial for plotting the initial shape of the graph and serve as a starting reference point.