To describe Sandra's function, let's analyze it step by step based on the work provided.
1. Initial Function: [tex]\( p(x) = 5x^2 + 30x \)[/tex]
2. Factor Out the Coefficient of [tex]\( x^2 \)[/tex]:
Sandra factored out the coefficient of [tex]\(x^2\)[/tex], which is 5:
[tex]\[ p(x) = 5(x^2 + 6x) \][/tex]
3. Complete the Square:
To complete the square for [tex]\( x^2 + 6x \)[/tex], we need to add and subtract the square of half the coefficient of [tex]\( x \)[/tex]:
[tex]\[ \left( \frac{6}{2} \right)^2 = 3^2 = 9 \][/tex]
4. Add and Subtract 9 Inside the Parentheses:
[tex]\[ p(x) = 5(x^2 + 6x + 9 - 9) \][/tex]
Simplifies to:
[tex]\[ p(x) = 5((x + 3)^2 - 9) \][/tex]
5. Distribute 5 and Simplify:
Finally, distribute the 5 and simplify:
[tex]\[ p(x) = 5(x + 3)^2 - 45 \][/tex]
Now, we have rewritten Sandra's function in vertex form, which is:
[tex]\[ p(x) = 5(x + 3)^2 - 45 \][/tex]
### Description of the Function:
- Vertex Form:
The function [tex]\( p(x) = 5(x + 3)^2 - 45 \)[/tex] is a quadratic function in vertex form: [tex]\( y = a(x - h)^2 + k \)[/tex], where:
- [tex]\( a = 5 \)[/tex]
- [tex]\( h = -3 \)[/tex]
- [tex]\( k = -45 \)[/tex]
### Vertex of the Function:
The vertex of this function is [tex]\((h, k)\)[/tex]:
[tex]\[ \text{Vertex} = (-3, -45) \][/tex]
### Maximum or Minimum:
Since the coefficient of [tex]\((x - h)^2\)[/tex] (which is 5) is positive, the parabola opens upwards. Therefore, the vertex represents a minimum point.
### Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex:
[tex]\[ x = -3 \][/tex]
### Summary:
- Vertex: [tex]\((-3, -45)\)[/tex]
- Nature of Vertex: Minimum
- Axis of Symmetry: [tex]\( x = -3 \)[/tex]
This analysis describes Sandra's function completely based on the given steps and results.
