Question
If Tomás is trying to model data using the polynomial equation y = (1/6)x3 + (1/2)x2 + (1/3)x, how many times will the function cross the x-axis when he graphs it?
A. zero
B. one
C. three
D. two
A. zero
B. one
C. three
D. two
Answers
GPT 3.5
The function will cross the x-axis when the y-value is equal to zero. Therefore, to find how many times the function will cross the x-axis when graphed, Tomás needs to determine the number of real roots of the polynomial equation y = (1/6)x^3 + (1/2)x^2 + (1/3)x.
By analyzing the equation, we see that it is a cubic polynomial. Cubic functions can have 0, 1, 2, or 3 real roots. To determine the number of real roots of a cubic equation, it is helpful to examine the discriminant of the polynomial. The discriminant of a cubic polynomial in the form ax^3 + bx^2 + cx + d is given by the expression Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2.
In this case, the coefficients of the polynomial are a = 1/6, b = 1/2, c = 1/3, and d = 0. Let's calculate the discriminant:
Δ = 18(1/6)(1/2)(1/3)(0) - 4(1/2)^3(0) + (1/2)^2(1/3)^2 - 4(1/6)(1/3)^3 - 27(1/6)^2(0)^2
= 0 - 0 + (1/4)(1/9) - (1/6)(1/27) - 0
= 1/36 - 1/162
= 3/162 - 1/162
= 2/162
= 1/81
Since the discriminant is positive (Δ > 0), the function will have three different real roots. Therefore, when Tomás graphs the function, it will cross the x-axis three times.
The answer is C. three.
By analyzing the equation, we see that it is a cubic polynomial. Cubic functions can have 0, 1, 2, or 3 real roots. To determine the number of real roots of a cubic equation, it is helpful to examine the discriminant of the polynomial. The discriminant of a cubic polynomial in the form ax^3 + bx^2 + cx + d is given by the expression Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2.
In this case, the coefficients of the polynomial are a = 1/6, b = 1/2, c = 1/3, and d = 0. Let's calculate the discriminant:
Δ = 18(1/6)(1/2)(1/3)(0) - 4(1/2)^3(0) + (1/2)^2(1/3)^2 - 4(1/6)(1/3)^3 - 27(1/6)^2(0)^2
= 0 - 0 + (1/4)(1/9) - (1/6)(1/27) - 0
= 1/36 - 1/162
= 3/162 - 1/162
= 2/162
= 1/81
Since the discriminant is positive (Δ > 0), the function will have three different real roots. Therefore, when Tomás graphs the function, it will cross the x-axis three times.
The answer is C. three.