Let's analyze the given function [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex] and evaluate each statement step by step.
1. The vertex is [tex]\((1, -9)\)[/tex]:
- The function [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex] is in the vertex form [tex]\( a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex.
- For the given function, [tex]\( (x + 1) \)[/tex] indicates [tex]\( h = -1 \)[/tex] and the constant term [tex]\(-9\)[/tex] indicates [tex]\( k = -9 \)[/tex].
- Hence, the vertex of the graph is actually [tex]\((-1, -9)\)[/tex], not [tex]\((1, -9)\)[/tex].
- False.
2. The graph opens upward:
- The coefficient of the squared term [tex]\((x+1)^2\)[/tex] is 6, which is positive.
- A positive coefficient means the parabola opens upward.
- True.
3. The graph is obtained by shifting the graph of [tex]\( f(x)=6(x+1)^2 \)[/tex] up 9 units:
- The given function [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex] can be compared with [tex]\( f(x) = 6(x+1)^2 \)[/tex].
- The term [tex]\(-9\)[/tex] actually represents a downward shift by 9 units, not an upward shift.
- False.
4. The graph is narrower than the graph of [tex]\( f(x) = x^2 \)[/tex]:
- The given function is [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex], which has a coefficient of 6 for the squared term.
- The graph of [tex]\( f(x) = x^2 \)[/tex] has a coefficient of 1 for the squared term.
- Since 6 > 1, the parabola [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex] is narrower than the parabola [tex]\( f(x) = x^2 \)[/tex].
- True.
5. The graph is the same as the graph of [tex]\( f(x) = 6x^2 + 12x - 3 \)[/tex]:
- We can expand the original function [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex]:
[tex]\[
f(x) = 6(x^2 + 2x + 1) - 9 = 6x^2 + 12x + 6 - 9 = 6x^2 + 12x - 3.
\][/tex]
- Thus, the function [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex] is equivalent to [tex]\( f(x) = 6x^2 + 12x - 3 \)[/tex].
- True.
In summary, the following statements are true about the graph of [tex]\( f(x) = 6(x+1)^2 - 9 \)[/tex]:
- The graph opens upward.
- The graph is narrower than the graph of [tex]\( f(x) = x^2 \)[/tex].
- The graph is the same as the graph of [tex]\( f(x) = 6x^2 + 12x - 3 \)[/tex].
