Question

Determine the quadratic function of the form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k whose graph is given on the right.
Question content area bottom left
Part 1
​f(x)equals
  
enter your response here
​(Do not​ simplify.)
.
.
.
Question content area right
Part 1
-5
5
-20
-15
-10
-5
5
10
x
y
(2,6)
(0,-6)

Answer 1

To determine the quadratic function of the form \( f(x) = a(x - h)^2 + k \), we need to identify the values of \( a \), \( h \), and \( k \) using the information given.

1. **Identify Vertex**: In the form \( f(x) = a(x - h)^2 + k \), the vertex is given by the coordinates \( (h, k) \). Since the graph of the function is not visible in this scenario, and specific points are provided, we typically assume the vertex is at the maximum or minimum value visible, likely near \( (2, 6) \) given that it seems higher than \( (0, -6) \).

2. **Assign h and k**:
We assume the vertex is at \( (h, k) = (2, 6) \). Therefore, \( h = 2 \) and \( k = 6 \).

3. **Determine a**:
Now, we need to use another point to determine the value of \( a \).

We have the point \( (0, -6) \). Substituting this point into the vertex form of the equation gives us:
\[
f(0) = a(0 - 2)^2 + 6
\]
Plugging in \( f(0) = -6 \):
\[
-6 = a(2^2) + 6
\]
\[
-6 = 4a + 6
\]
\[
-12 = 4a
\]
\[
a = -3
\]

4. **Final function**:
Now we can substitute \( a \), \( h \), and \( k \) into the equation. Therefore, the function is:
\[
f(x) = -3(x - 2)^2 + 6
\]

So the final answer is:
\[
f(x) = -3(x - 2)^2 + 6
\]