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What is the end behavior of \( f(x) = \frac{3x}{x^2 + 1} \) as \( x \to \infty \)?

Which of the following best describes the approximate intervals where the polynomial is decreasing? The x-axis spans from below negative 4 to 4, and the y-axis spans from negative 40 to above 40. The x-axis has a scale of 2 in increments of 0.5 and the y-axis has a scale of 20 in increments of 5. The blue polynomial function has multiple turning points: it descends steeply from positive infinity in the second quadrant, reaches a local minimum at around (negative 2, 0), then rises to a local maximum at a coordinate with x values roughly halfway between negative 0.5 and negative 1, and y values roughly halfway between 40 and 45, followed by a symmetrical descent towards the first quadrant, crossing the y-axis at (0, 20). It then dips to short flat portion at (1, 0), before decreasing sharply toward negative infinity in the fourth quadrant.

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

How can the polynomial \( x^4 + 4x^2 + 4 \) be factored to identify its roots?

Multiply \( (2 + 3i)(2 – 3i) \).

Which are the zeroes of the polynomial shown? The x-axis spans from below negative 5 to above 5, and the y-axis spans from below negative 50 to just above 50. The x-axis has a scale of 5 in increments of 1 and the y-axis has a scale of 50 in increments of 10. The red polynomial function has multiple turning points: it starts from negative infinity in the third quadrant and ascends to a local maximum approximately around (negative 2.5, 55) in the second quadrant. Then it drops to a local minimum slightly below (negative 0.5, negative 0.5) in the third quadrant and rises to a local maximum near (1, 25) in the first quadrant. It then falls steeply into a deep local minimum near (3.5, negative 80) in the fourth quadrant, before increasing sharply toward positive infinity in the first quadrant.

What are the zeros of the polynomial represented by the graph? 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. 

What is the y-intercept of \[f(x) = \frac{5x}{x + 2}?\] The x-axis spans from below negative 10 to just above 5, and the y-axis spans from just below negative 20 to above 20. The x-axis has a scale of 5 in increments of 1, and the y-axis has a scale of 20 in increments of 5. The convex curve is in the second quadrant, passing through the points (negative 3, 15) and (negative 6, 7.5). The curve starts from positive infinity above the vertical asymptote near x= negative 2, then decreases, approaching the horizontal asymptote near y = 5. The concave curve spans the third and the first quadrants, passing through the approximate points (negative 1.5, negative 15) and (3, 2.5). It starts from negative infinity approaching the vertical asymptote near x= negative 2, then rises approaching the horizontal asymptote near y = 5. 

Which system of inequalities represents the constraints shown in the graph? “Area graph. A gray-shaded area and a purple-shaded area intersect on a coordinate plane, forming a grayish purple area. The x-axis spans from negative 5 to just above 5, and the y-axis spans from below 0 to above 5. Both axes show intervals of 5 with grid lines in increments of 1. A solid black diagonal line with a negative slope has an x-intercept roughly halfway between (2, 0) and (3, 0) and a y-intercept at (0, 7), shading the area below in dark gray. A dashed purple diagonal line with a positive slope has an x-intercept at (negative 3, 0) and a y-intercept at (0, 3), shading the area above in light purple. The two shaded areas overlap, forming a grayish purple region. The overlapping area spans most of the second quadrant and some portion of the first and third quadrants.”

Which transformation occurs when \( g(x) = f(-x) \)?