Certainly! Let's analyze the function [tex]\( y = 0.25(x + 5)^2 \)[/tex] to identify its vertex and [tex]\( y \)[/tex]-intercept in a detailed, step-by-step manner.
### Step 1: Identifying the Vertex
The given function is in the vertex form of a quadratic equation:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Here, the function [tex]\( y = 0.25(x + 5)^2 \)[/tex] can be rewritten to match this form. Notice:
[tex]\[ y = 0.25(x - (-5))^2 \][/tex]
So, comparing this with [tex]\( y = a(x - h)^2 + k \)[/tex], we find:
- [tex]\(a = 0.25\)[/tex]
- [tex]\(h = -5\)[/tex] (from [tex]\( -(-5) \)[/tex])
- [tex]\(k = 0\)[/tex]
Thus, the vertex of the function is [tex]\( (-5, 0) \)[/tex].
### Step 2: Finding the [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept of a function is the point where the graph crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
To find the [tex]\( y \)[/tex]-intercept, substitute [tex]\( x = 0 \)[/tex] into the given function:
[tex]\[ y = 0.25(0 + 5)^2 \][/tex]
[tex]\[ y = 0.25(5)^2 \][/tex]
[tex]\[ y = 0.25 \times 25 \][/tex]
[tex]\[ y = 6.25 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the function is [tex]\( 6.25 \)[/tex].
### Conclusion
Based on our analysis, we can conclude that:
- The vertex of the function [tex]\( y = 0.25(x + 5)^2 \)[/tex] is [tex]\( (-5, 0) \)[/tex].
- The [tex]\( y \)[/tex]-intercept of the function is [tex]\( 6.25 \)[/tex].
From the given options, the correct answer is:
- Vertex: [tex]\((-5, 0)\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 6.25 \)[/tex]
Thus, the correct choice is:
[tex]\[ \text{vertex}, (-5,0) ; y\text{-intercept}, 6.25 \][/tex]
