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 check your answers step by step.
1. To find the coordinates of the vertex of the parabola defined by the quadratic function f(x) = (x - 4)^2 - 4, we can see that the vertex form of a parabola is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
Comparing this with the given equation, we have a = 1, h = 4, and k = -4. Therefore, the vertex is located at (h, k) = (4, -4).
Your answer of (4, -4) is correct.
2. To find the axis of symmetry of the parabola defined by the quadratic function f(x) = (x + 2)^2 + 7, we can again refer to the vertex form of a parabola, f(x) = a(x - h)^2 + k. The axis of symmetry is given by the equation x = h.
Comparing this with the given equation, we have h = -2. Therefore, the axis of symmetry is x = -2.
Your answer of x = -2 is correct.
3. To find the range of the quadratic function f(x) = 7 - (x + 4)^2, we need to determine the possible output values of the function.
The given function is a downward-opening parabola with the vertex at (-4, 7). Since the parabola opens downward, the range will be from the vertex to negative infinity (or the lowest point of the parabola).
Therefore, the range of the quadratic function f(x) = 7 - (x + 4)^2 is (-∞, 7].
Your answer of (-∞, 7] is correct.
To check your answers, let's go through each question one by one:
1. Find the coordinates of the vertex for the parabola defined by the given quadratic function f(x) = (x - 4)^2 - 4. You correctly identified that the vertex form of a quadratic function is given by (h, k), where h represents the x-coordinate of the vertex and k represents the y-coordinate of the vertex.
The given quadratic function is f(x) = (x - 4)^2 - 4. By comparing this equation with the vertex form of a quadratic function, we can see that h = 4 and k = -4. Therefore, the coordinates of the vertex are (4, -4). Your answer is correct.
2. Find the axis of symmetry of the parabola defined by the given quadratic function f(x) = (x + 2)^2 + 7. The axis of symmetry for a parabola is a vertical line that passes through the vertex of the parabola.
Using the same equation f(x) = (x + 2)^2 + 7, we can see that the vertex has the form (-h, k). In this case, h = -2 and k = 7. So, the x-coordinate of the vertex is -(-2) = 2. Therefore, the equation of the axis of symmetry is x = 2. It seems like you made a mistake in your answer, which should be x = 2, not x = -2.
3. Find the range of the quadratic function f(x) = 7 - (x + 4)^2. The range of a function refers to the set of possible values for the output (y-values). In this case, we need to determine the possible range values for f(x).
To find the range, let's analyze the quadratic function f(x) = 7 - (x + 4)^2. Notice that the term (x + 4)^2 is always greater than or equal to 0 because it represents a square term. Therefore, the maximum value for (x + 4)^2 is 0.
Substituting this maximum value into the function, we get f(x) = 7 - 0 = 7. Hence, the range of the quadratic function is (-∞, 7]. Your answer is correct.
Great job overall!