Let's go through Sandra's work to understand how she rewrote the function [tex]\( p(x) = 30x + 5x^2 \)[/tex] in vertex form and then describe the function in detail.
1. Sandra starts with the quadratic function in standard form:
[tex]\[
p(x) = 30x + 5x^2
\][/tex]
2. She rearranges the terms to put it in a clearer standard form:
[tex]\[
p(x) = 5x^2 + 30x
\][/tex]
3. Next, she factors out the coefficient of [tex]\( x^2 \)[/tex], which is 5:
[tex]\[
p(x) = 5(x^2 + 6x)
\][/tex]
4. To complete the square, she calculates [tex]\(\left(\frac{6}{2}\right)^2 = 9\)[/tex]:
[tex]\[
\left(\frac{6}{2}\right)^2 = 9
\][/tex]
5. She then adds and subtracts this perfect square inside the parentheses:
[tex]\[
p(x) = 5(x^2 + 6x + 9 - 9)
\][/tex]
6. Sandra rewrites the expression by factoring the perfect square trinomial and simplifying:
[tex]\[
p(x) = 5((x + 3)^2 - 9)
\][/tex]
[tex]\[
p(x) = 5(x + 3)^2 - 45
\][/tex]
Now the function is in vertex form [tex]\( p(x) = a(x - h)^2 + k \)[/tex], where [tex]\( a = 5 \)[/tex], [tex]\( h = -3 \)[/tex], and [tex]\( k = -45 \)[/tex].
Vertex Form Description:
- The vertex form of the quadratic function is [tex]\( p(x) = 5(x + 3)^2 - 45 \)[/tex].
Vertex of the Function:
- The vertex of the function is the point [tex]\( (h, k) \)[/tex], which is [tex]\( (-3, -45) \)[/tex].
Nature of the Vertex:
- Since the coefficient of the [tex]\((x + 3)^2\)[/tex] term, which is 5, is positive, the parabola opens upwards. This means the vertex represents a minimum point.
Axis of Symmetry:
- The axis of symmetry of the function is the vertical line [tex]\( x = h \)[/tex], which is:
[tex]\[
x = -3
\][/tex]
To summarize:
- The vertex of Sandra’s function is [tex]\((-3, -45)\)[/tex].
- The vertex represents a minimum.
- The axis of symmetry is [tex]\( x = -3 \)[/tex].
