The graph of a quadratic function f(x) is shown above. It has a vertex at (2,4) and passes the point ( 0,2) . Find the quadratic function.
the vertex tells you that
f(x) = a(x-2)^2 + 4
use the point to find a:
a(0-2)^2 + 4 = 2
Well, isn't this graph being quite the shape-shifter? Let's see if we can figure out its true identity.
We know that the vertex of this quadratic function is at (2,4). That means, the equation for the graph can be written in vertex form as f(x) = a(x - h)^2 + k, where (h, k) is the vertex.
So, substituting the given vertex into the equation, we have f(x) = a(x - 2)^2 + 4.
Now, we need another point to solve for "a." Fortunately, the graph also passes through the point (0,2). Plugging in these coordinates, we get 2 = a(0 - 2)^2 + 4.
Simplifying the equation, we have 2 = 4a + 4.
Subtracting 4 from both sides, we get -2 = 4a.
Finally, dividing both sides by 4, we find that a = -1/2.
So, the quadratic function is f(x) = -1/2(x - 2)^2 + 4. Now that we've cracked the code, this quadratic function might not be as hilarious as a clown, but it sure does know how to shape-shift!
To find the quadratic function, we need to use the vertex form of a quadratic equation, which is given by:
f(x) = a(x-h)^2 + k
where (h, k) is the vertex of the parabola.
Given that the vertex is (2,4), we have h = 2 and k = 4. Now we can plug in the vertex values into the vertex form equation:
f(x) = a(x-2)^2 + 4
Next, we can use the given point (0,2) to find the value of "a". Plugging in the coordinates of the point, we have:
2 = a(0-2)^2 + 4
Simplifying this equation:
2 = a(-2)^2 + 4
2 = 4a + 4
-2 = 4a
a = -1/2
Now, we can substitute the value of "a" back into the vertex form equation:
f(x) = (-1/2)(x-2)^2 + 4
So, the quadratic function is f(x) = (-1/2)(x-2)^2 + 4.
To find the quadratic function that corresponds to the given information, we need to use the standard form of a quadratic function: f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
In this case, we are given that the vertex is at (2, 4), so h = 2 and k = 4. Substituting these values into the equation, we have:
f(x) = a(x - 2)^2 + 4
Now, we need to determine the value of 'a' to complete the quadratic function. To do this, we can use the given point (0, 2) which lies on the graph. Substituting these values into the equation, we have:
2 = a(0 - 2)^2 + 4
Simplifying this equation, we get:
2 = 4a + 4
Subtracting 4 from both sides, we have:
-2 = 4a
Dividing both sides by 4, we have:
a = -1/2
Now that we have the value of 'a', we can substitute it back into the equation:
f(x) = (-1/2)(x - 2)^2 + 4
Therefore, the quadratic function that corresponds to the given information is:
f(x) = (-1/2)(x - 2)^2 + 4