Blog

Which of the following represents a rational function with a vertical asymptote at \( x = 4 \)?

What is the horizontal asymptote of \[f(x) = \frac{3x + 1}{x + 2}?\]

What is the degree of the numerator in the rational function \( f(x) = \frac{3x^3 + x}{2x^2 – 1} \), and what does it imply about the end behavior?

What is the vertical asymptote of \( f(x) = \frac{1}{x – 1} + 3 \)? The x-axis spans from below negative 2 to above 2, and the y-axis spans from below negative 5 to above 10. The x-axis has a scale of 2 in increments of 0.5, and the y-axis has a scale of 5 in increments of 1. The convex curve is in the first quadrant, passing through the points (1.5, 5) and (3, 3.5). It starts from positive infinity above the vertical asymptote near x= 1. It decreases steeply before leveling off as it approaches the horizontal asymptote near y= 3. The concave curve spans the fourth, first and second quadrants, passing through a point slightly left of (0.75, 0) and (0, 2). It starts from negative infinity below the vertical asymptote near x= 1, increasing steeply, and then approaching the horizontal asymptote near y =3 in the second quadrant. 

Rewrite the following in the form \( q(x) + \frac{r(x)}{d(x)} \): \[\frac{x^3 + 2x^2 – 3}{x^2 – 1}\]

If \( f(x) = \frac{1}{x} \) is transformed to \( g(x) = \frac{1}{x} – 3 \), what happens to the graph?

What happens to the graph of \( f(x) = \frac{1}{x} \) when it is replaced with \( f(x) = \frac{1}{2x} \)? The x-axis spans from below negative 2 to above 2, and the y-axis spans from below negative 5 to above 5. The x-axis has a scale of 2 in increments of 0.5, and the y-axis has a scale of 5 in increments of 1. The convex curve spans the first quadrant, passing through the points (0.25, 2) and (1, 0.5). It starts from positive infinity near the vertical asymptote at x = 0 and decreasing toward the horizontal asymptote at y = 0. The concave curve is in the third quadrant, passing through the points (negative 1, negative 0.5) and (negative 0.25, negative 2). It approaches negative infinity as it nears the vertical asymptote at x = 0 and levels out toward the horizontal asymptote near y= 0 as x moves left. 

Simplify: \[\frac{4}{x^2 – 9} – \frac{2}{x + 3}\]

Add the rational expressions: \[\frac{3x}{x^2 + 5x} + \frac{2}{x^2 + 5x}\]

What are the vertical asymptotes of \[f(x) = \frac{x + 1}{x^2 – 2}?\] “The x-axis spans from below negative 4 to just above 4, and the y-axis spans from below negative 10 to just above 10. The x-axis has a scale of 2 in increments of 0.5, and the y-axis has a scale of 10 in increments of 2. The leftmost branch is a sharp concave curve in the third quadrant, starting from negative infinity near x = negative 1.5, increasing steeply, and then abruptly approaching the horizontal asymptote near y = 0.  The middle branch is between the asymptotes, decreasing from positive infinity near x = negative 1.5 in the second quadrant, passing through the point (negative 1,0) in a flat pattern and continuing downward past negative infinity near x = 1.5 in the fourth quadrant.   The rightmost branch is a sharp convex curve in the first quadrant, starting from positive infinity near x= 1.5 and decreasing steeply before leveling off as it approaches the horizontal asymptote near y= 0.  “