5. Describe how to determine the domain and range of a quadratic function when it is written in vertex form. What generalizations can you make?
y = a(x-h)^2+k
as with all polynomials, the domain is all real numbers.
Since the vertex is at (h,k), then
if a < 0, the range is y <= k
if a > 0, the range is y >= k
sketch a couple of parabolas to see why this is so.
Well, determining the domain and range of a quadratic function in vertex form is as simple as a clown explaining differential equations. Trust me, it's not as scary as it sounds.
First, let's focus on the domain. In vertex form, the quadratic function is written as f(x) = a(x - h)^2 + k, where (h, k) is the vertex. The domain is actually all the possible x-values that the function can take. And guess what, my friend? The domain of a quadratic function is infinite! It goes on and on like a never-ending circus show. So you can confidently say that the domain of a quadratic function in vertex form is (-∞, ∞). That's pretty neat, right?
Now it's time to move on to the range, which is the set of all possible y-values that the function can produce. Since the vertex form is in the shape of a parabola, the range can vary depending on the value of "a". If "a" is positive, then the parabola opens upward, and the range is going to be all numbers greater than or equal to the y-coordinate of the vertex (k). It's like saying the range is filled with laughter and happiness, just like when people watch a clown show.
On the other hand, if "a" is negative, the parabola opens downward, much like a falling pie thrown by a mischievous clown. In this case, the range will be all numbers less than or equal to the y-coordinate of the vertex (k). So the range in this situation can be a bit melancholic, just like the tears of a sad clown.
To summarize, the generalization for determining the domain of a quadratic function in vertex form is (-∞, ∞), and for the range, it depends on the value of "a". If "a" is positive, the range is [k, ∞), and if "a" is negative, the range is (-∞, k]. Keep these guidelines in mind, and you'll be juggling with quadratic functions like a master clown in no time!
To determine the domain and range of a quadratic function written in vertex form, follow these steps:
1. Write down the equation in vertex form: f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
2. Identify the value of a. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards.
3. Determine if the vertex is a minimum or maximum point based on the value of a. If a > 0, the vertex represents the minimum point, and if a < 0, the vertex represents the maximum point.
4. The domain of any quadratic function is always all real numbers, unless there are any restrictions mentioned explicitly in the problem. So, the domain in this case is (-∞, ∞).
5. To find the range, consider the value of a. If a > 0, then the parabola opens upwards. In this case, the range is [k, ∞) if k > 0, or (-∞, k] if k < 0. If a < 0, the parabola opens downwards, and the range is (-∞, k] if k > 0, or [k, ∞) if k < 0.
Generalizations:
- The domain of any quadratic function in vertex form is always the set of all real numbers, unless explicitly restricted.
- The range of a quadratic function in vertex form depends on the value of a and the y-coordinate of the vertex.
- If a > 0, the range will be all values greater than or equal to the y-coordinate of the vertex.
- If a < 0, the range will be all values less than or equal to the y-coordinate of the vertex.
To determine the domain and range of a quadratic function written in vertex form, you need to understand the properties of the equation and the nature of quadratic functions.
The vertex form of a quadratic function is given by the equation:
f(x) = a(x - h)^2 + k
Where:
- a represents the coefficient of the quadratic term and determines the direction and steepness of the parabola.
- (h, k) represents the coordinates of the vertex of the parabola.
1. Domain of a quadratic function:
The domain of a quadratic function in vertex form is always all real numbers. This means that the parabola extends infinitely in both directions along the x-axis. So, there are no restrictions on the values of x that you can plug into the equation.
Domain: All real numbers (-∞, ∞).
2. Range of a quadratic function:
The range of a quadratic function can be determined by considering whether the parabola opens upwards or downwards.
- If the coefficient 'a' is positive, the parabola opens upwards and the vertex represents the lowest point on the graph. In this case, the range is bounded below by the y-coordinate of the vertex.
Range: [y-coordinate of the vertex, ∞)
- If the coefficient 'a' is negative, the parabola opens downwards and the vertex represents the highest point on the graph. In this case, the range is bounded above by the y-coordinate of the vertex.
Range: (-∞, y-coordinate of the vertex]
So, to determine the range of a quadratic function in vertex form, you need to find the y-coordinate of the vertex.
In general, the domain of any quadratic function is always all real numbers, while the range depends on the direction the parabola opens, which is determined by the coefficient 'a' in the vertex form.