Determine the quadratic function f, whose graph is given. **The graph given is (0,-2) and (3,-11)
The vertex is (3,11) and the y-intercept is -2
So, f(x)=
**Please help me understand how to do this question. Could you please show work?**
come on, guy. You should be getting the hang of these by now.
the vertex says
f(x) = a(x-3)^2 + 11
the point (0,-2) says
-2 = 9a+11
a = -13/9
f(x) = -13/9 (x-3)^2 + 11
would have been nicer with a y-intercept of +2.
To determine the quadratic function f(x) using the given graph, we need to use the vertex form of a quadratic equation.
The vertex form of a quadratic equation is given by:
f(x) = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
We have been given the vertex (3, 11). So, we substitute these values into the equation:
f(x) = a(x - 3)^2 + 11
To find the value of "a," we can use one of the given points on the graph. Let's choose the point (0, -2).
Substituting the values from this point into the equation, we get:
-2 = a(0 - 3)^2 + 11
-2 = a(9) + 11
-2 = 9a + 11
Now, let's solve for "a."
-2 - 11 = 9a
-13 = 9a
a = -13/9
Substituting the value of "a" back into the equation, we get:
f(x) = (-13/9)(x - 3)^2 + 11
So, the quadratic function f(x) whose graph is given by (0, -2) and (3, -11) is:
f(x) = (-13/9)(x - 3)^2 + 11
To determine the quadratic function f, you need to know the general form of a quadratic function, which is f(x) = ax^2 + bx + c. To find the specific values for a, b, and c, you can use the given information about the graph.
1. Given the vertex (3, 11):
- The x-coordinate of the vertex is the average of the x-values of the two given points on the graph: (0 + 3) / 2 = 3.
- Therefore, the x-coordinate of the vertex is 3.
- The y-coordinate of the vertex is the value of f(x) at x = 3, which is 11.
- So, the vertex of the quadratic function is (3, 11).
2. Given the y-intercept (-2):
- The y-intercept is the value of f(x) when x = 0.
- So, f(0) = -2.
Using these pieces of information, we can substitute the vertex coordinates into the general form of a quadratic function to find the specific function f(x).
1. Substituting the vertex coordinates, we get:
f(x) = a(x - 3)^2 + 11.
2. Using the y-intercept, we substitute x = 0:
-2 = a(0 - 3)^2 + 11.
-2 = 9a + 11.
-2 - 11 = 9a.
-13 = 9a.
a = -13/9.
Now, we have the value of a. Substituting this back into the equation with the vertex coordinates, we find the quadratic function f(x):
f(x) = (-13/9)(x - 3)^2 + 11.
Therefore, the quadratic function f(x) is (-13/9)(x - 3)^2 + 11.