Please check my answers.
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
1.f(x) = (x - 4)2 - 4
-I got (4,-4)
Find the axis of symmetry of the parabola defined by the given quadratic function.
3.f(x) = (x + 2)2 + 7
-I got: x=-2
Find the range of the quadratic function.
5.f(x) = 7 - (x + 4)2
I got: (-inifinity,7]
Let's verify your answers together.
1. To find the coordinates of the vertex for the parabola defined by the quadratic function f(x) = (x - 4)^2 - 4, we can observe that the equation is already written in vertex form (h, k). In this case, the vertex form is f(x) = (x - h)^2 + k.
The vertex form indicates that the vertex of the parabola is at the point (h, k). By comparing the given function f(x) = (x - 4)^2 - 4, we can see that h = 4 and k = -4. Therefore, the coordinates of the vertex are (4, -4).
So, your answer for (1) is correct: (4, -4).
Moving on to the next question:
3. To find the axis of symmetry of the parabola defined by the quadratic function f(x) = (x + 2)^2 + 7, we need to use the standard form of the quadratic equation: f(x) = ax^2 + bx + c.
The axis of symmetry is given by the equation x = -b/2a, where a, b, and c come from the standard form of the quadratic equation.
Comparing the given function f(x) = (x + 2)^2 + 7 with the standard form, we can see that a = 1, b = 2, and c = 7.
Substituting these values into the formula for the axis of symmetry, we get x = -2/(2 * 1), x = -2/2, x = -1.
So, your answer for (3) is incorrect. The correct answer is x = -1.
Lastly:
5. To find the range of the quadratic function f(x) = 7 - (x + 4)^2, we need to determine the possible values of the function.
One approach is to analyze the graph of the quadratic function. We can observe that the coefficient of the squared term is negative, indicating that the vertex is the maximum point. Since the vertex is given by (-4, 7), the maximum value of the function is 7.
Therefore, the range of the quadratic function is (-∞, 7].
Your answer for (5) is correct.
To summarize:
1. The correct answer is (4, -4).
3. The correct answer is x = -1.
5. The correct answer is (-∞, 7].
Your answers are correct.
1. The vertex form of the quadratic function is f(x) = (x - h)^2 + k, where (h, k) represents the vertex of the parabola. By comparing the given function f(x) = (x - 4)^2 - 4 with the vertex form, we can see that the vertex is (4, -4).
3. The axis of symmetry of a parabola is a vertical line that passes through its vertex. Since you correctly identified the vertex as (h, k) = (-2, 7), the equation of the axis of symmetry is x = -2.
5. To find the range of the quadratic function f(x) = 7 - (x + 4)^2, we need to determine the maximum or minimum value it can take. In this case, since the coefficient of the (x + 4)^2 term is negative, the parabola opens downwards and has a maximum value. By evaluating the function at the vertex (h, k) = (-4, 7), we can see that the maximum value is 7. Therefore, the range of the function is (-∞, 7].