Let's carefully analyze the function [tex]\(h(x) = -\frac{3}{4}(x+7)^2 + 5\)[/tex].
### Step-by-Step Explanation:
1. Finding the Vertex:
The given function is in the vertex form [tex]\( h(x) = a(x - h)^2 + k \)[/tex], where [tex]\( a = -\frac{3}{4} \)[/tex], [tex]\( h = -7 \)[/tex], and [tex]\( k = 5 \)[/tex].
The vertex of the parabola [tex]\( h(x) \)[/tex] is given by the point [tex]\((-h, k)\)[/tex].
Here, [tex]\((-h, k)\)[/tex] translates to [tex]\((-(-7), 5) = (-7, 5)\)[/tex].
True Statement: The vertex is [tex]\((-7, 5)\)[/tex].
False Statement: The vertex is [tex]\((-7, -5)\)[/tex].
2. Axis of Symmetry:
The axis of symmetry for a parabola of the form [tex]\( h(x) = a(x - h)^2 + k \)[/tex] is the vertical line [tex]\( x = h \)[/tex].
For our function, [tex]\( h = -7 \)[/tex].
True Statement: The axis of symmetry is [tex]\( x = -7 \)[/tex].
3. Maximum Value:
Since the coefficient of [tex]\( (x+7)^2 \)[/tex] is negative ([tex]\( a = -\frac{3}{4} \)[/tex]), the parabola opens downwards.
The maximum value of [tex]\( h(x) \)[/tex] is at the vertex, which is [tex]\( k = 5 \)[/tex].
True Statement: The maximum value is 5.
4. Range:
For a downward-opening parabola [tex]\( h(x) = -\frac{3}{4}(x+7)^2 + 5 \)[/tex], the highest value is at the vertex, which is 5.
The function takes all values less than or equal to 5.
True Statement: The range is [tex]\((-\infty, 5]\)[/tex].
False Statement: The range is [tex]\([5, \infty)\)[/tex].
5. Domain:
Quadratic functions of the form [tex]\( h(x) = a(x - h)^2 + k \)[/tex] have all real numbers as their domain.
True Statement: The domain is [tex]\((-\infty, \infty)\)[/tex].
6. Minimum Value:
Since the parabola opens downwards, it does not have a minimum value within the real number range; it keeps decreasing indefinitely.
False Statement: The minimum value is 5.
### Summary of True Statements:
- The axis of symmetry is [tex]\( x = -7 \)[/tex].
- The maximum value is 5.
- The range is [tex]\((-\infty, 5]\)[/tex].
- The domain is [tex]\((-\infty, \infty)\)[/tex].
### Summary of False Statements:
- The vertex is [tex]\((-7, -5)\)[/tex].
- The range is [tex]\([5, \infty)\)[/tex].
- The minimum value is 5.
Based on these steps, we have determined the correctness of each statement about the function [tex]\( h(x) \)[/tex].
