What is the vertical asymptote of \( y = \frac{6x + 1}{x^2 + x} \)? “The x-axis spans from below 0 to just above 5, and the y-axis spans from below negative 30 to above 10. The x-axis has a scale of 5 in increments of 1, and the y-axis has a scale of 10 in increments of 2. The graph represents a rational function with three branches.   The left most branch is a concave curve in the third quadrant, starting from negative infinity near x= negative 1, increasing steeply, and then leveling off towards the horizontal asymptote near the negative x-axis.  The middle branch extends between the two vertical asymptotes, decreasing from positive infinity near x= negative 1 in the second quadrant. It crosses the x axis at slightly left of the origin (0,0) and continues downward past negative infinity near x= 0.  The rightmost branch is a convex curve in the first quadrant, starting from positive infinity near x= 0 and decreasing steeply before leveling off toward the horizontal asymptote close to the positive x-axis. “

For \[f(x) = \frac{1}{x – 3},\] which value of \( x \) is excluded from the domain?

What are the vertical and horizontal asymptotes of \[f(x) = \frac{5}{x + 1}?\] The x-axis spans from below zero to just above 5, and the y-axis spans from below negative 10 to just above 10. The x-axis has a scale of 5 in increments of 1, and the y-axis has a scale of 10 in increments of 2. The convex curve spans the first and second quadrants, passing through the points (4, 1) and (negative 0.5, 10). It starts from positive infinity above the vertical asymptote near x = negative 1 in the second quadrant. It decreases steeply before leveling off as it approaches the horizontal asymptote near y= 0 in the first quadrant. The concave curve is in the third quadrant, passing through the points (negative 3.5, negative 2) and (negative 1.5, negative 10). It starts from negative infinity below the vertical asymptote near x = negative 1, increasing steeply, and then approaching the horizontal asymptote near y = 0. 

Which characteristic of the function \[f(x) = \frac{3}{x – 1}\] does not change if the numerator becomes \( 5 \)?

Which is an alternate form of  \( (x + 3)(x – 3) \), creating a polynomial identity?

What are the zeros of the polynomial represented in the graph? The x-axis spans from below negative 5 to 5, and the y-axis spans from below negative 20 to above zero. The x-axis has a scale of 5 in increments of 1 and the y-axis has a scale of 10 in increments of 2. The red curve represents a polynomial function with four turning points. It begins from the bottom-middle of the third quadrant, reaches a local maximum at (negative 3, 0), then drops steeply to a local minimum near (negative 1.5, negative 21). It rises again to another local maximum near (0.5, 0.8), falls to a local minimum around (1, 0), and then steeply extends upward out of view. 

For the function \( f(x) = \frac{1}{x} – 2 \), the horizontal asymptote is at:

What is the horizontal asymptote of \( f(x) = \frac{5x}{x + 2} \)?

What is the minimum value of the polynomial function? The x-axis spans from below negative 5 to above 5, and the y-axis spans from below negative 20 to 10. The x-axis has a scale of 5 in increments of 1 and the y-axis has a scale of 10 in increments of 2. The blue parabola opens upward, with its vertex at approximately (negative 0.5, negative 20). The curve is symmetric around the vertical line passing through the vertex. It intersects the y-axis at (0, negative 20). It crosses the x-axis at (negative 5, 0) and (4, 0), extending out of view at both ends. 

If \( y = \frac{12}{x} \), which of the following is true about the graph?