Problem 2 - Algebra for College Students

Question

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{x^{2}}{x^{2}+2 x-3} $$

Step-by-Step Solution

Answer

The function has vertical asymptotes at \(x = -3\) and \(x = 1\), and a horizontal asymptote at \(y = 1\). There is no symmetry.

Step 1: Identify the Rational Function

The given rational function is \(f(x) = \frac{x^2}{x^2 + 2x - 3}\). A rational function is expressed as \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. Here, \(P(x) = x^2\) and \(Q(x) = x^2 + 2x - 3\).

Step 2: Find Symmetry

A function is symmetric with respect to the y-axis if \(f(-x) = f(x)\) and symmetric with respect to the origin if \(f(-x) = -f(x)\). Compute \(f(-x) = \frac{(-x)^2}{(-x)^2 + 2(-x) - 3} = \frac{x^2}{x^2 - 2x - 3}\). Here, \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\). Thus, there is no symmetry.

Step 3: Factor the Denominator

Factorize the denominator \(x^2 + 2x - 3\) to identify potential vertical asymptotes and removable discontinuities. The factorization gives \((x + 3)(x - 1)\).

Step 4: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. Set each factor equal to zero: \(x + 3 = 0\) gives \(x = -3\) and \(x - 1 = 0\) gives \(x = 1\). Both values contribute to vertical asymptotes since the numerator is non-zero for these x-values.

Step 5: Determine Horizontal Asymptotes

For horizontal asymptotes of \(\frac{P(x)}{Q(x)}\), if the degrees of \(P(x)\) and \(Q(x)\) are equal, the asymptote is \(\frac{a}{b}\) where \(a\) and \(b\) are the leading coefficients. Here, both degrees are 2, leading coefficients are 1, so the horizontal asymptote is \(y = \frac{1}{1} = 1\).

Step 6: Sketch the Graph

To sketch the graph, plot the vertical asymptotes at \(x = -3\) and \(x = 1\), the horizontal asymptote at \(y = 1\), and evaluate \(f(x)\) for various \(x\)-values to understand the curve's behavior. Check \(f(0) = \frac{0^2}{0^2 + 2*0 - 3} = 0\) to plot as a point.

Key Concepts

Symmetry in Functions

In the exploration of rational functions, symmetry helps us understand how a graph might reflect across certain lines.
For functions symmetric about the y-axis, plugging in \(f(-x) = f(x)\) should hold true.
For symmetry about the origin, \(f(-x) = -f(x)\) should do the job. In our example function, \(f(x) = \frac{x^2}{x^2 + 2x - 3}\), evaluating the symmetry criteria showed us neither condition holds.
This lack of symmetry indicates the graph doesn't mirror about the y-axis or the origin. Symmetry is particularly useful because it can reduce the effort needed; if a function is symmetric, you only need to compute half the graph.
Once you spot symmetry, graphing becomes less cumbersome and offers insights into the function's nature.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches.
Understanding asymptotes is crucial in graphing rational functions, as they describe the behavior of the function at extremes of the domain. Vertical asymptotes occur at x-values that make the denominator zero, provided the numerator isn't zero there too.

• In our function \(f(x) = \frac{x^2}{x^2 + 2x - 3}\), the factorization of the denominator becomes \(x^2 + 2x - 3 = (x+3)(x-1)\) allowing us to identify vertical asymptotes at \(x = -3\) and \(x = 1\).
• These are points where the function heads toward infinity, indicating the graph will sharply rise or fall at these values.

A horizontal asymptote similarly guides us, showing end behavior.
Given equal degrees for numerator and denominator polynomials, such as in this example, the horizontal asymptote is given by dividing their leading coefficients \(\frac{1}{1} = 1\).
This indicates that as \(x\) approaches infinity, the graph levels out, hovering near \(y = 1\). Such analyses make sketching rational functions grounded and structured.

Graphing Functions

Graphing rational functions can be made easier by understanding their key features. Begin by identifying and plotting asymptotes.
For our function, vertical asymptotes are at \(x = -3\) and \(x = 1\), and a horizontal asymptote at \(y = 1\).
These lines inform where the graph doesn't exist and how it behaves in the extremes. After placing the asymptotes, check a few points to see how the graph sits relative to these lines.

• For example, calculate \(f(0)\) to determine a y-intercept; here it's \(f(0) = 0\), giving the point \( (0,0) \).

Through all this, remember rational functions can bend and twist around and across these asymptotes.
By using symmetry and intercepts along with asymptotes, you equip yourself with a map of what the graph looks like.
This visualization process not only reinforces algebraic concepts but enhances your spatial understanding of mathematical functions.