1. Determine the vertex of the parabola:
y= 7x^2+14x+6
2. For the following 2 functions,identify the horizontal and vertical asymptotes.
a)f(x)= 3x(x-2)/x(x-1)
b)g(x)= x/(x+7)^2
1. To determine the vertex of a parabola given its equation in the form y = ax^2 + bx + c, you can use the formula x = -b / (2a). In this case, the equation is y = 7x^2 + 14x + 6. So, a = 7, b = 14, and c = 6.
First, calculate -b / (2a):
-14 / (2 * 7) = -14 / 14 = -1
The x-coordinate of the vertex is -1. To find the y-coordinate, substitute this value of x back into the original equation:
y = 7(-1)^2 + 14(-1) + 6
y = 7 - 14 + 6
y = -1
Therefore, the vertex of the parabola is (-1, -1).
2a. To determine the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0.
In the function f(x) = 3x(x-2) / x(x-1), both the numerator and denominator are of degree 2. Therefore, the horizontal asymptote is the ratio of the leading coefficients, which is y = 3/1 = 3.
For the vertical asymptote, set the denominator equal to zero and solve for x. In this case, x(x-1) = 0. Thus, the vertical asymptotes are x = 0 and x = 1.
2b. For the function g(x) = x / (x+7)^2, the degree of the numerator is 1, and the degree of the denominator is 2. Therefore, the horizontal asymptote is y = 0.
To find the vertical asymptote for this function, set the denominator equal to zero and solve for x. In this case, x + 7 = 0. Thus, the vertical asymptote is x = -7.
So, for function g(x), the horizontal asymptote is y = 0, and the vertical asymptote is x = -7.