Which of the following graphs is a possible sketch of the polynomial f(x)=2x^3(x−2)^2 ?(1 point)
Without any provided graphs to choose from, it is not possible to determine which graph is a possible sketch of the polynomial f(x)=2x^3(x−2)^2.
To determine which graph is a possible sketch of the polynomial f(x) = 2x^3(x - 2)^2, we can examine the properties of the function and compare them to the given graphs.
1. Start by analyzing the degree of the polynomial. In this case, the highest power of x is 3. Therefore, the graph should have a general shape of a cubic curve.
2. Next, consider the behavior of the function as x approaches infinity. Since the leading term of the polynomial is 2x^3, the graph should have one arm pointing up to positive infinity on one side.
3. On the other side, as x approaches negative infinity, the graph should behave symmetrically and have another arm pointing upwards.
4. Additionally, the polynomial has a factor of (x - 2)^2, which means it has a double root at x = 2. This means the graph should touch or bounce off the x-axis (y = 0) at x = 2.
5. Finally, considering the multiplicities of the roots, since (x - 2)^2 is squared in the expression, the graph should only touch the x-axis at x = 2 without crossing it.
By analyzing these characteristics of the polynomial, you can compare them to the given graphs and choose the one that matches the behavior described above.
To determine which of the given graphs is a possible sketch of the polynomial f(x) = 2x^3(x-2)^2, we need to analyze the behavior of the polynomial and identify key features.
First, let's look at the degree of the polynomial. The highest exponent in this polynomial is 3 due to the term 2x^3. This tells us that the graph will have a maximum of three x-intercepts.
Next, we consider the factors in the polynomial. The factor (x-2)^2 indicates that the point x = 2 is a double root or a point of inflection.
Based on these insights, we can analyze the given graphs and eliminate the ones that do not satisfy these conditions. The correct plot will have three x-intercepts and exhibit a point of inflection or a curve change at x = 2.
By evaluating the given graphs, we can deduce the correct sketch by applying the following steps:
1. Evaluate the x-intercepts: Substitute x = 0 in f(x) and count the number of times the graph intersects the x-axis.
- For example, if f(0) = 0, it implies that 0 is an x-intercept.
2. Verify the degree of the polynomial: Identify the highest exponent in the polynomial.
- Count the number of times the graph crosses the x-axis. It should match the degree.
3. Determine the behavior at x = 2: Check if the graph shows a point of inflection or a curve change at x = 2.
By analyzing each given graph using these steps, you can identify the sketch that matches the polynomial f(x) = 2x^3(x-2)^2.