Which graph is a possible sketch of the polynomial f(x)=x3+5x2+6x?(1 point)
Responses
An upward-opening parabola is plotted on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
Image with alt text: An upward-opening parabola is plotted on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
A curve passes through quadrants 3, 2, and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
Image with alt text: A curve passes through quadrants 3, 2, and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
A curve passes through quadrants 3, 1, and 4 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
Image with alt text: A curve passes through quadrants 3, 1, and 4 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
An upward line and an upward-opening parabola are plotted on a coordinate plane with the x-axis ranging from negative 8 to 4 in increments of 1.
The correct response is:
A curve passes through quadrants 3, 2, and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
The possible sketch of the polynomial f(x) = x^3 + 5x^2 + 6x is:
A curve passes through quadrants 3, 2, and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
Image with alt text: A curve passes through quadrants 3, 2, and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
To determine which graph is a possible sketch of the polynomial f(x) = x^3 + 5x^2 + 6x, we need to understand the behavior of the function.
A polynomial of degree 3, like the one given, is called a cubic polynomial. Cubic polynomials can have various shapes and behaviors depending on their coefficients. In general, the behavior of a cubic polynomial can be described by looking at the leading coefficient (the coefficient of the highest power of x) and the signs of the other coefficients.
In this case, the leading coefficient is 1, which means that the graph opens upwards on both ends. Additionally, since the coefficient of x^3 is positive, the graph will be "upside-down U" shaped (like an upward-opening parabola) rather than "U" shaped (like a downward-opening parabola).
Now let's examine the given options:
1. An upward-opening parabola is plotted on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
This option does not match the behavior of a cubic polynomial, which should have two "arms" rather than just one.
2. A curve passes through quadrants 3, 2, and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
This option suggests that the graph passes through quadrants 2 and 3, which is not possible for the given cubic polynomial. The graph should only pass through quadrant 1.
3. A curve passes through quadrants 3, 1, and 4 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
This option correctly suggests that the graph passes through quadrant 1 and quadrant 3. However, it also passes through quadrant 4, which is not possible for the given cubic polynomial. The graph should not dip into quadrant 4.
4. An upward line and an upward-opening parabola are plotted on a coordinate plane with the x-axis ranging from negative 8 to 4 in increments of 1.
This option suggests that the graph consists of both a line and an upward-opening parabola. Since we know that the cubic polynomial should resemble an upward-opening parabola, this option seems to be the most plausible.
Therefore, option 4 - an upward line and an upward-opening parabola plotted on a coordinate plane with the x-axis ranging from negative 8 to 4 in increments of 1 - is likely a possible sketch of the polynomial f(x) = x^3 + 5x^2 + 6x.