The vertex of a parabola is at (-2,4), and the x-intercepts are at -6 and 2. Determine the domain and range of the function.
a.
D: all real numbers
R: all real numbers
b.
D: all real numbers
R: y ≤ 4
c.
D: -6 ≤ x ≤ 2
R: y ≤ 4
d.
D: x ≤ 4
R: all real numbers
domain is all reals for all polynomials. So, C and D are out.
since a parabola has a vertex, it has a maximum or a minimum, so the range cannot be all reals. So, A is out.
B must be the answer.
The only confusion might be whether the vertex is a max or a min. If B is correct, it must be a maximum.
Since there are real roots, the vertex at (-2,4), which is above the x-axis, must be a maximum, so y <= 4.
To determine the domain and range of the function, we need to understand the characteristics of a parabola.
The vertex form of a parabola is given by: y = a(x - h)^2 + k, where (h, k) represents the vertex coordinates.
From the given information, we have the vertex coordinates as (-2, 4). So, h = -2 and k = 4.
The formula for finding the x-intercepts of a parabola is given by: x = h ± √(k/a), where a is the coefficient of x^2.
Given that the x-intercepts are at -6 and 2, we can set up two equations based on this formula:
-6 = -2 ± √(4/a)
2 = -2 ± √(4/a)
Simplifying these equations, we get:
-4 = ± √(4/a)
4 = ± √(4/a)
Now, we can solve for a by squaring both sides of the equations:
16 = 4/a
a = 4/16
a = 1/4
Now, we can substitute the values of h, k, and a into the vertex form of the parabola:
y = (1/4)(x + 2)^2 + 4
To determine the domain, we need to find all possible values of x for which the function is defined. In this case, the parabola is a quadratic function, meaning it is defined for all real numbers. So, the domain is all real numbers, which is represented by D: all real numbers.
To determine the range, we need to find all possible values of y that the function can take. Since the coefficient of the x^2 term is positive (1/4 > 0), the parabola opens upward, and its vertex is the lowest point. Therefore, the lowest point is ( -2, 4), which means that y can be any real number greater than or equal to 4, or y ≤ 4. So, the range is represented by R: y ≤ 4.
Hence, the correct answer is b. D: all real numbers and R: y ≤ 4.