Which of the following graphs is a possible sketch of the polynomial f(x)=2x3(x−2)2?(1 point)
Responses
A curve passes through quadrants 3 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 and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
A curve passes through quadrants 2, 3, 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 2, 3, 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 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 and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
An upward-opening parabola is plotted on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
The correct graph is the first option: "A curve passes through quadrants 3 and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments."
The correct response is: A curve passes through quadrants 3 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) = 2x^3(x-2)^2, we can analyze the given options and compare them to the characteristics of the polynomial.
First, let's consider the degree of the polynomial, which is 3. This means that the graph of the polynomial will have one less than the degree number of turning points.
Now let's analyze the factors of the polynomial: 2x^3 and (x-2)^2.
The term 2x^3 implies that the polynomial has an odd degree, leading to a graph that extends to infinity on both ends. Therefore, the graph should cross both the positive and negative x-axes.
The term (x-2)^2 indicates a double root or zero at x = 2. This means that the graph will touch the x-axis at x = 2, but not cross it.
Based on these characteristics, we can eliminate options that contradict them.
Option 1: A curve passes through quadrants 3 and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
This option does not satisfy the requirement of crossing the positive x-axis. Therefore, it is not a possible sketch.
Option 2: A curve passes through quadrants 2, 3, and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
This option violates the property of having a double root at x = 2, as it shows a curve that crosses the x-axis at x = 2. Thus, it is not a possible sketch.
Option 3: A curve passes through quadrants 3 and 1 on a coordinate plane with the x-axis ranging from negative 5 to 5 in unit increments.
This option also does not satisfy the requirement of crossing the positive x-axis. Therefore, it is not a possible sketch.
Option 4: 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 represent a cubic polynomial. It is an upward-opening parabola, which is not consistent with the given polynomial. Thus, it is not a possible sketch.
Based on the analysis, none of the given options represents a possible sketch of the polynomial f(x) = 2x^3(x-2)^2.