Determine the domain of the quadratic function.
f(x) = -6x^2 - 3x + 3
Determine the range of the quadratic function.
f(x) = 3x^2 - 8x + 3
How do I look for the domain and range in these cases?
Have you
the domain of all polynomials is all real numbers
#2.
for the range, you know the parabola opens upwards, because the leading coefficient is positive. So, find the vertex, and the range is all real numbers greater than or equal to y at the vertex.
I got -7/3
How would I express it as an inequality?
To determine the domain of a quadratic function, you need to consider the values of x for which the function is defined. In other words, you need to find the set of all real numbers for which the function output is a real number.
For quadratic functions in general, such as f(x) = ax^2 + bx + c, the domain is always all real numbers unless there are specific restrictions mentioned.
In your first example, f(x) = -6x^2 - 3x + 3, there are no restrictions mentioned. Therefore, the domain of this function is all real numbers (-∞, +∞).
To determine the range of a quadratic function, you need to consider the set of all possible values that the function can output. For quadratic functions in general, the range depends on the shape of the parabola.
If the coefficient 'a' in the quadratic function is positive (a > 0), the parabola opens upward, and the range is the set of all real numbers greater than or equal to the y-coordinate of the vertex.
If the coefficient 'a' is negative (a < 0), the parabola opens downward, and the range is the set of all real numbers less than or equal to the y-coordinate of the vertex.
In your second example, f(x) = 3x^2 - 8x + 3, the coefficient 'a' is positive. To find the range, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a = 3 and b = -8.
x = -(-8) / (2 * 3) = 8 / 6 = 4 / 3.
Substituting this x-coordinate back into the equation, we can find the y-coordinate of the vertex:
f(4/3) = 3(4/3)^2 - 8(4/3) + 3 = 16/3 - 32/3 + 3 = -13/3.
Since 'a' is positive, the parabola opens upward, and the range is all real numbers greater than or equal to -13/3.
So, the range of the quadratic function f(x) = 3x^2 - 8x + 3 is (-∞, -13/3] for positive 'a' values.
ever hear of ≥
??