The graph of a quadratic function with vertex (3,-1). Find the range and the domain. write the range and domain using interval notation.
the parabola can open up or down, so the range may be either
(-∞,-1] or [-1,∞)
The domain of all polynomials is (-∞,∞) unless otherwise restricted.
Well, the vertex of the quadratic function is like the clown nose on the clown's face - the central point of attention! Anyway, let's find the range and domain.
The vertex of the quadratic function is (3, -1). To find the domain, we need to figure out the x-values (the spread of the clown's wig, if you will) where the function is defined. In this case, the domain is all real numbers (-∞, +∞), because the quadratic function stretches out infinitely in both directions along the x-axis, just like clown shoes!
Now, when it comes to the range (the height of the clown's jumps), we need to determine the possible y-values. Since the vertex is at (3, -1), we know that the lowest point on the graph is -1. As for the top part of the graph, it goes up infinitely. So, the range is (-∞, +∞) -1.
To summarize with interval notation: The domain is (-∞, +∞) and the range is (-∞, -1]. Keep on clowning around with those functions!
To find the range and domain of a quadratic function with vertex (3,-1), we need to consider the shape of the graph.
For a quadratic function, the graph is a parabola that opens either upward or downward.
Since the vertex is at (3,-1), the graph is symmetric with respect to the vertical line x = 3. This means that the parabola opens either upward or downward.
The shape of the graph will determine the range and domain.
1. Range: The range of a quadratic function with a vertex is the set of all possible y-values it can take.
If the parabola opens upward, the range is [k, ∞), where k is the y-coordinate of the vertex. In this case, since the vertex is (3,-1), the range is [-1, ∞).
If the parabola opens downward, the range is (-∞, k], where k is the y-coordinate of the vertex. In this case, since the vertex is (3,-1), the range is (-∞, -1].
Therefore, the range for a quadratic function with vertex (3,-1) can be written in interval notation as (-∞, -1] (if the parabola opens downward) or [-1, ∞) (if the parabola opens upward).
2. Domain: The domain of a quadratic function is the set of all possible x-values it can take. Since there are no restrictions given in the problem, the quadratic function has a domain of (-∞, ∞), representing all real numbers.
Therefore, the domain for a quadratic function with vertex (3,-1) can be written in interval notation as (-∞, ∞).
To determine the range and domain of a quadratic function with a given vertex, it is important to understand the behavior of quadratic functions.
First, let's discuss the vertex form of a quadratic function:
f(x) = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola. In our case, the given vertex is (3, -1), implying that h = 3 and k = -1.
The range of a quadratic function is the set of all possible y-values it can take. Since the graph is a parabola that opens upwards (as the coefficient "a" is positive in the vertex form), we know that the lowest point of the parabola is the vertex itself. Thus, the range will be all real numbers greater than or equal to the y-coordinate of the vertex, which is -1.
Hence, the range can be expressed as: (-∞, -1] using interval notation.
Moving on to the domain, it represents the set of all possible x-values. For a quadratic function, the domain is always all real numbers (or (-∞, ∞) using interval notation) since there are no restrictions on the x-values.
Therefore, the domain can be represented as: (-∞, ∞) using interval notation.