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. y = 7x^2 + 14x + 6.
V(h , k),
h = Xv = -b / 2a = -14 / 14 = -1.
Substitute Xv (h) in the given Eq to
get Yv(k):
k = Yv = 7(-1)^2 + 14(-1) + 6,
= 7 - 14 + 6 = -1.
V(h , k) = V(-1 , -1).
1. To determine the vertex of the parabola given by the equation y = 7x^2 + 14x + 6, we can use the formula for the x-coordinate of the vertex, which is given by -b/2a.
In this case, the coefficient of x^2 is 7 (a) and the coefficient of x is 14 (b). Plugging these values into the formula gives:
x-coordinate of vertex = -14 / (2 * 7) = -1
To find the y-coordinate of the vertex, substitute the x-coordinate back into the original equation:
y = 7(-1)^2 + 14(-1) + 6 = 7 - 14 + 6 = -1
Therefore, the vertex of the parabola is (-1, -1).
2. a) To identify the horizontal and vertical asymptotes of the function f(x) = 3x(x-2) / (x(x-1)), we will analyze the behavior of the function as x approaches infinity and negative infinity, as well as when x is very large positive or negative.
Horizontal asymptote: To find the horizontal asymptote, we will check the limit of the function as x approaches infinity and negative infinity. We divide the coefficients of the highest power terms in the numerator and denominator:
lim (x -> ±∞) f(x) = lim (x -> ±∞) (3x(x-2)) / (x(x-1))
As x approaches infinity or negative infinity, the highest power terms dominate the function. The highest power terms in the numerator and denominator are both x^2, so the limit simplifies to:
lim (x -> ±∞) f(x) = lim (x -> ±∞) (3x^2) / (x^2)
Since the degree of the highest power terms is the same, the ratio of their coefficients will determine the horizontal asymptote. In this case, the coefficient ratio is 3/1, which means the horizontal asymptote is y = 3.
Vertical asymptotes: To find the vertical asymptotes, we set the denominator equal to zero and solve for x:
x(x-1) = 0
This gives us two values of x that will make the denominator equal to zero: x = 0 and x = 1. Therefore, there are two vertical asymptotes: x = 0 and x = 1.
b) For the function g(x) = x / (x+7)^2, we will again determine the horizontal and vertical asymptotes.
Horizontal asymptote: We will calculate the limit of the function as x approaches infinity and negative infinity:
lim (x -> ±∞) g(x) = lim (x -> ±∞) x / (x+7)^2
As x approaches infinity or negative infinity, the highest power term (x) in the numerator and the denominator will dominate the function. Therefore, the function simplifies to:
lim (x -> ±∞) g(x) = lim (x -> ±∞) (x) / (x+7)^2
Since the degree of the highest power terms is the same, the ratio of their coefficients will determine the horizontal asymptote. In this case, the ratio is 1/1, meaning the horizontal asymptote is y = 1.
Vertical asymptote: To find the vertical asymptote, we set the denominator equal to zero and solve for x:
(x+7)^2 = 0
Taking the square root of both sides gives:
x + 7 = 0
Solving for x gives us x = -7. Therefore, the vertical asymptote is x = -7.
1. To determine the vertex of a parabola in the form y = ax^2 + bx + c, you can use the formula for the x-coordinate of the vertex, which is given by x = -b/2a. In this case, the equation is y = 7x^2 + 14x + 6.
First, identify a, b, and c from the equation: a = 7, b = 14, and c = 6.
Next, substitute these values into the formula x = -b/2a:
x = -(14)/(2 * 7)
x = -14/14
x = -1
Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting this value 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).
2. For the function f(x) = 3x(x-2)/x(x-1):
a) To find the horizontal asymptote, check the powers of x in the numerator and the denominator. In this case, they are both x^1. Since the degrees are equal, the horizontal asymptote is y = a/b, where a and b are the leading coefficients of the numerator and denominator, respectively.
Here, the leading coefficient of the numerator is 3 and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 3/1 = 3.
b) To find the vertical asymptote, check if there are any values of x that make the denominator equal to zero. In this case, x = 0 and x = 1 are the values that will make the denominator zero. Therefore, the vertical asymptotes are x = 0 and x = 1 for the function f(x).
For the function g(x) = x/(x+7)^2:
a) Similar to the previous question, check the powers of x in the numerator and the denominator. In this case, the numerator has a degree of x^1, while the denominator has a degree of (x+7)^2, which is also x^2. Since the degree of the numerator is lesser than the degree of the denominator, there is no horizontal asymptote for this function.
b) To find the vertical asymptote, check if there are any values of x that make the denominator equal to zero. In this case, x = -7 is the value that will make the denominator zero. Therefore, the vertical asymptote is x = -7 for the function g(x).