Quadratic graphs and their properties:
Identify the domain and range of each function:
y = 4x^2 - 3
I have no idea how to find the range and domain. Would you please explain in detail how I can determine the domain and range in the quadratic problems?
IDK
Certainly! To find the domain and range of a quadratic function, you can follow these steps:
1. Domain: The domain refers to all the possible input values for the function. In the case of a quadratic function, there are no restrictions on the values of x. Therefore, the domain is all real numbers (-∞, +∞).
2. Range: The range represents all the possible output values for the function. In the case of a quadratic function in the form y = ax^2 + bx + c, where a, b, and c are constants, we can determine the range by considering the coefficient of the x^2 term (in this case, 4).
a. If the coefficient a is positive, the parabola opens upwards and has a minimum value, which is the y-coordinate of the vertex. In this case, since a = 4 > 0, the parabola opens upwards.
b. Since the parabola opens upwards, it means that the vertex is the lowest point on the graph. The x-coordinate of the vertex can be found using the formula x = -b/2a. Here, a = 4 and b = 0 (since there is no x term).
c. Substituting the values into the formula, we get x = -0 / (2*4) = 0.
d. To find the y-coordinate of the vertex, substitute the x-value back into the equation: y = 4(0)^2 - 3 = -3.
e. The vertex of this parabola is (0, -3). Since the parabola opens upwards, the range is all y-values greater than or equal to the y-coordinate of the vertex. So the range is [-3, +∞).
In summary, for the function y = 4x^2 - 3:
- The domain is all real numbers (-∞, +∞).
- The range is [-3, +∞).
Remember, these steps can be applied to any quadratic function to find the domain and range.
surely your class text has a discussion of the topic, and examples.
the domain is the set of values for which the function is defined. For all polynomials, that would be all real numbers. There is no value of x for which you cannot determine a value for y: just take 4x^2-3!
For the range, it's the set of values for y which can be produced by applying the function to x. Note that for 4x^2-3, x^2 is always positive, and is zero only for x=0. Thus, the minimum value for y is -3. Pick any x you want, and you will always get a value greater than -3. And, the; larger x gets, the larger y gets. So, the range is all reals at least -3: y>=-3.
an excellent graphing resource is
http://rechneronline.de/function-graphs/
Just enter your function and you can see what it looks like.