Use the zeros to determine which of the following graphs is a sketch of the polynomial f(x)=−x(x+2)2.(1 point)
Responses
A curve passes through quadrants 2 and 4 on a coordinate plane with the x-axis ranging from negative 2 to 6 in unit increments.
Image with alt text: A curve passes through quadrants 2 and 4 on a coordinate plane with the x-axis ranging from negative 2 to 6 in unit increments.
A curve with arrows at both the ends is plotted on a coordinate plane. The x-axis ranges from negative 6 to 2 in unit increments. The y-axis does not show any values.
Image with alt text: A curve with arrows at both the ends is plotted on a coordinate plane. The x-axis ranges from negative 6 to 2 in unit increments. The y-axis does not show any values.
A curve passes through quadrants 2 and 4 on a coordinate plane with the x-axis ranging from negative 6 to 2 in unit increments.
Image with alt text: A curve passes through quadrants 2 and 4 on a coordinate plane with the x-axis ranging from negative 6 to 2 in unit increments.
A downward-opening parabola is plotted on a coordinate plane with the x-axis ranging from negative 4 to 2 in unit increments.
The correct answer is the first option: "A curve passes through quadrants 2 and 4 on a coordinate plane with the x-axis ranging from negative 2 to 6 in unit increments."
This is because the zeros of the polynomial, which are the x-values where the polynomial equals zero, are -2 and 0. The graph of the polynomial will intersect the x-axis at -2 and 0. The other options do not accurately represent this information.
To determine which graph is a sketch of the polynomial f(x) = -x(x+2)^2, we need to analyze the zeros.
The zeros of a polynomial are the values of x for which f(x) = 0. In this case, we set f(x) = -x(x+2)^2 equal to zero and solve for x.
-x(x+2)^2 = 0
Setting each factor equal to zero, we have two cases:
Case 1: x = 0
Case 2: (x+2)^2 = 0
Solving case 2, we take the square root of both sides:
x + 2 = 0
x = -2
Therefore, the zeros of f(x) are x = 0 and x = -2.
Analyzing the options given:
A. This option mentions quadrants 2 and 4, but it does not specify the zeros. Therefore, it cannot be determined if this option is correct or not.
B. This option does not provide any information about the zeros, so it cannot be determined if this option is correct or not.
C. This option mentions quadrants 2 and 4 and specifies the x-axis ranging from -6 to 2, which includes both zeros (x = 0 and x = -2). Therefore, this option is a potential sketch of the polynomial f(x) = -x(x+2)^2.
D. This option does not mention the zeros or the x-axis ranging from -6 to 2, so it cannot be determined if this option is correct or not.
Based on the analysis, the correct graph sketch of the polynomial f(x) = -x(x+2)^2 is likely option C.
To determine which graph is a sketch of the polynomial f(x) = -x(x+2)^2, we need to analyze the zeros of this function.
The polynomial given has a zero of x = 0, which is obtained by setting -x = 0 or x + 2 = 0. This means that the graph will intersect the x-axis at x = 0.
To determine the behavior of the graph near x = -2, we can substitute values slightly larger and smaller than -2 into the function. For example, substituting x = -3 and x = -1, we find:
f(-3) = -(-3)(-3+2)^2 = 3(1)^2 = 3
f(-1) = -(-1)(-1+2)^2 = 1(1)^2 = 1
Since f(-3) > 0 and f(-1) > 0, the graph of the function will be above the x-axis in the region of x < -2, indicating a curve passing through quadrant 2.
Now let's analyze the overall behavior of the graph by examining the values of the function for larger x-values. Substituting x = 1 and x = 3, we find:
f(1) = -(1)(1+2)^2 = -(1)(3)^2 = -9
f(3) = -(3)(3+2)^2 = -(3)(5)^2 = -75
Since f(1) < 0 and f(3) < 0, the graph of the function will be below the x-axis in the region of x > 0, indicating a curve passing through quadrant 4.
Considering all the information obtained, the only graph that matches the description is the one described as "A curve passes through quadrants 2 and 4 on a coordinate plane with the x-axis ranging from negative 2 to 6 in unit increments."