, for this problem f(x)=3x^2-6x-7,what is the minimum value and where do it occurs, and what is this functions domain and range?
for any parabola, the min/max occurs where x = -b/2a
In your case, at x=6/6 = 1
domain for any polynomial is all reals
f(x) = 3x^2 - 6x - 7
= 3(x-1)^2 - 10
from this you can easily see that the min at x=1 is -10
so the range is all reals >= 10
By completing square,
f(x) = 3[x^2 -2x - 7/3]
= 3[x^2 - 2(x)(1) + (1)^2 - (1)^1 - 7/3]
= 3[(x - 1)^2 - 1 - 7/3]
= 3(x - 1)^2 - 3 - 7
= 3(x - 1)^2 - 10
To find the minimum value and its location, we can use the formula for the x-coordinate of the vertex of a quadratic function, given by x = -b/2a.
For the given function f(x) = 3x^2 - 6x - 7, we can identify a = 3 and b = -6.
Using the formula x = -b/2a, we can calculate:
x = -(-6) / (2*3)
= 6 / 6
= 1
So the x-coordinate of the vertex is x = 1.
To find the y-coordinate of the vertex, we substitute this x-value back into the original function:
f(1) = 3(1)^2 - 6(1) - 7
= 3 - 6 - 7
= -10
So the vertex is at (1, -10) and the minimum value of the function is -10.
Now let's determine the domain and range of the function:
The domain of a quadratic function is all real numbers, so for this function, the domain is (-∞, ∞).
To find the range, we can consider the graph of a quadratic function. Since the coefficient of x^2 is positive (3 > 0), the parabola opens upward, and its vertex, (-10), is the minimum point.
Therefore, the range of the function is (-∞, -10]. It includes all real numbers less than or equal to -10.
To find the minimum value and where it occurs for the function f(x) = 3x^2 - 6x - 7, we can follow these steps:
Step 1: Determine the vertex of the parabolic function.
The vertex of a parabolic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/(2a). For this problem, a = 3 and b = -6. Plugging these values into the formula, we have:
x = -(-6) / (2*3)
x = 6 / 6 = 1
Step 2: Find the corresponding y-coordinate of the vertex.
To find the y-coordinate, substitute the x-value we found (x = 1) into the function:
f(1) = 3(1)^2 - 6(1) - 7
f(1) = 3 - 6 - 7
f(1) = -10
Therefore, the minimum value of f(x) = 3x^2 - 6x - 7 is -10, and it occurs when x = 1.
Now, let's determine the domain and range of the function:
- Domain: The domain represents the set of all possible input values (x-values) for the function. In this case, since f(x) = 3x^2 - 6x - 7 is a quadratic function, there are no specific restrictions on the domain. Therefore, the domain is all real numbers, denoted as (-∞, ∞).
- Range: The range represents the set of all possible output values (y-values) for the function. Since the coefficient a of the quadratic term is positive (a = 3), the parabola opens upwards, indicating that its vertex is a minimum point. Therefore, the range of this function goes from the y-coordinate of the vertex (-10) and up towards positive infinity. Hence, the range is (-∞, -10].
In summary, the minimum value of the function f(x) = 3x^2 - 6x - 7 is -10, occurring at x = 1. The domain is all real numbers, and the range is (-∞, -10].