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Use the equation of the polynomial function ​f(x)equals5 x cubed minus x to complete the following.

The curve opens
▼
to the right because the leading coefficient is
▼
negative
positive
. Because the polynomial is
▼
quartic,
cubic,
the graph has end behaviors in the
▼
same
opposite
​direction, so the other end opens
▼
up
down
to the left.

1 answer

To analyze the polynomial function \( f(x) = 5x^3 - x \), let's break down the statements given in the prompt.

  1. The curve opens: Since the leading term is \( 5x^3 \), which has a positive coefficient, the curve opens to the right.

  2. Because the leading coefficient is: The leading coefficient is positive.

  3. Because the polynomial is: This polynomial is cubic (the highest degree of \( x \) is 3).

  4. The graph has end behaviors in the: For a cubic function, the end behaviors are in opposite directions. Specifically, as \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.

  5. So the other end opens: Since one end approaches positive infinity and the other end approaches negative infinity, the other end opens down.

So, putting it all together, we have:

  • The curve opens to the right because the leading coefficient is positive.
  • Because the polynomial is cubic, the graph has end behaviors in the opposite direction, so the other end opens down.