Identify the maximum or minimum value and the domain and range of the graph of the function
y=2(x-3)^2-4
Can someone please help me.
domain of any polynomial is all reals
this graph is a parabola, opening upwards.
The vertex is at (3,4), so the range is all reals y >= 4
Sure, I'd be happy to help! Let's tackle this question one step at a time.
First, let's find the minimum or maximum value of the function. We notice that the expression 2(x-3)^2 is always non-negative because it is a square, so no matter what x is, that term will be greater than or equal to 0. Therefore, the lowest value this function (y) can take is when that term is equal to 0, which occurs when (x-3)^2 = 0.
To find the maximum or minimum, we need to look at the coefficient in front of the squared term. In this case, the coefficient is positive 2. Since it's positive, we know that the parabola opens upwards, so the vertex represents the minimum value. The vertex occurs at x = 3, which we don't need to solve for because it's given in the equation. Plugging in x = 3 into the equation, we get:
y = 2(3-3)^2 - 4
y = 2(0)^2 - 4
y = 0 - 4
y = -4
So, the minimum value of the function is -4, and it occurs at x = 3.
Next, let's determine the domain and range of the graph. The domain is all possible x-values that the function can take. In this case, there are no restrictions on x, so the domain is all real numbers, or (-∞, ∞).
For the range, we know the minimum value is -4, but since the parabola opens upwards, there is no maximum value. Therefore, the range is y ≥ -4, meaning the function can take any value greater than or equal to -4.
In summary:
- The maximum or minimum value of the function is a minimum of y = -4 at x = 3.
- The domain is all real numbers, or (-∞, ∞).
- The range is y ≥ -4.
I hope that helps! Let me know if you have any other questions, and don't hesitate to ask if you want a little joke.
To find the maximum or minimum value of the graph of the function y = 2(x - 3)^2 - 4, we need to determine whether the parabola opens upward or downward.
The function is in the form of a quadratic equation, y = ax^2 + bx + c, where a = 2, b = -12, and c = -4.
Since the coefficient of x^2, a, is positive (a = 2 > 0), the parabola opens upward. This means that the graph will have a minimum value.
To find the vertex (minimum point) of the parabola, we can use the formula:
x = -b / (2a)
In this case, substituting the values of a and b, we get:
x = -(-12) / (2 * 2)
x = 12 / 4
x = 3
Thus, the x-coordinate of the vertex is x = 3.
To find the corresponding y-coordinate (which is the minimum value), we substitute this x-value back into the equation:
y = 2(3 - 3)^2 - 4 = -4
So, the minimum value of the function is -4, and it occurs at the vertex (3, -4).
Next, let's determine the domain and range of the graph:
The domain is the set of all possible x-values for the function. In this case, since it is a quadratic equation, the domain is all real numbers (-∞, ∞).
The range is the set of all possible y-values for the function. Since the parabola opens upward and has a minimum value of -4, the range will be from the minimum value (-4) to positive infinity (-4, ∞).
To summarize:
- The maximum or minimum value of the graph is -4, occurring at the vertex (3, -4).
- The domain is (-∞, ∞).
- The range is (-4, ∞).
Sure! I can help you with that. To find the maximum or minimum value, as well as the domain and range of the graph of the function, we can start by looking at the general form of the function, which is y = a(x - h)^2 + k.
In this case, your function is y = 2(x - 3)^2 - 4. Comparing it to the general form, we can see that a = 2, h = 3, and k = -4.
To find the vertex of the parabola (which represents the maximum or minimum value), we need to use the values of h and k. The vertex of the parabola is given by the coordinates (h, k). So, for your function, the vertex is (3, -4).
Since a = 2 is positive, the parabola will open upwards, which means it will have a minimum value at the vertex.
Now let's find the domain and range of the function:
- Domain: The domain of a function represents all the possible values of x. In this case, there are no restrictions on the value of x, so the domain is all real numbers. In interval notation, we can write the domain as (-∞, ∞).
- Range: The range of a function represents all the possible values of y. Since the parabola opens upwards (a > 0) and the vertex is at the minimum point, the minimum value of y occurs at the vertex. So, the range of the function is all real numbers greater than or equal to the y-value of the vertex. In interval notation, we can write the range as [-4, ∞).
To summarize:
- Maximum or minimum value: The minimum value is -4.
- Domain: (-∞, ∞)
- Range: [-4, ∞)
So, the graph of the function y = 2(x - 3)^2 - 4 has a minimum value of -4, a domain of all real numbers, and a range of [-4, ∞).