Let's break down the solution step-by-step for the given function [tex]\( f(x) = 6(x - 6)^2 - 3 \)[/tex].
1. Identify the Vertex Form of the Quadratic Function:
The given function is in the form [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] represents the vertex of the parabola.
2. Determine the Coefficient [tex]\(a\)[/tex]:
Here, the coefficient [tex]\( a \)[/tex] is 6, which is positive, indicating that the parabola opens upwards.
- Answer: The parabola opens upwards.
3. Find the Vertex [tex]\((h, k)\)[/tex]:
Compare the given function [tex]\( f(x) = 6(x - 6)^2 - 3 \)[/tex] with the standard form [tex]\( f(x) = a(x - h)^2 + k \)[/tex].
- We see that [tex]\( h = 6 \)[/tex] and [tex]\( k = -3 \)[/tex].
- Therefore, the vertex of the parabola is at [tex]\( (x, y) = (6, -3) \)[/tex].
4. Calculate [tex]\( f(6) \)[/tex]:
To find the value of the function at [tex]\( x = 6 \)[/tex]:
[tex]\( f(6) = 6(6 - 6)^2 - 3 \)[/tex]
Simplify inside the parentheses first:
[tex]\( 6 - 6 = 0 \)[/tex]
Then square the result:
[tex]\( 0^2 = 0 \)[/tex]
Multiply by 6:
[tex]\( 6 \times 0 = 0 \)[/tex]
Finally, subtract 3:
[tex]\( 0 - 3 = -3 \)[/tex]
- Therefore, [tex]\( f(6) = -3 \)[/tex].
5. Compile the Results:
- The vertex of the parabola is at [tex]\( (6, -3) \)[/tex].
- The value of the function at [tex]\( x = 6 \)[/tex] is [tex]\( -3 \)[/tex].
Putting this all together in the final blanks:
- The graph of [tex]\( f(x) = 6(x - 6)^2 - 3 \)[/tex] is a parabola that opens upwards, with its vertex at [tex]\( (x, y) = (6, -3) \)[/tex], and [tex]\( f(6) = -3 \)[/tex] is the minimum value of [tex]\( f \)[/tex].
