Question

sandra wrote $p(x) = 30x + 5x^2$ in vertex form. her work is below.

1. $p(x) = 5x^2 + 30x$
2. $p(x) = 5(x^2 + 6x)$
3. $left(\frac{6}{2}\

ight)^2 = 9$;

1. $p(x) = 5(x^2 + 6x + 9) - 5(9)$
2. $p(x) = 5(x + 3)^2 - 45$

describe sandra’s function.
what is the vertex of this function? is it a maximum or a minimum?
what is the axis of symmetry of this function?

Explanation:

Regarding the vertex and its nature:

Step1: Recall vertex form of quadratic

The vertex form of a quadratic function is \( p(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From Sandra's work, the vertex form is \( p(x) = 5(x + 3)^2 - 45 \), which can be rewritten as \( p(x) = 5(x - (-3))^2 + (-45) \).

Step2: Determine vertex and concavity

For a quadratic function \( y = ax^2 + bx + c \), if \( a>0 \), the parabola opens upward and the vertex is a minimum; if \( a<0 \), it opens downward and the vertex is a maximum. Here, \( a = 5 \) (from the original function or the vertex form, since the coefficient of \((x + 3)^2\) is 5), and \( 5>0 \), so the parabola opens upward. The vertex is \((-3, -45)\) and it is a minimum.

Regarding the axis of symmetry:

Step1: Recall axis of symmetry formula

For a quadratic function in vertex form \( p(x) = a(x - h)^2 + k \), the axis of symmetry is the vertical line \( x = h \).

Step2: Identify \( h \) from vertex form

From \( p(x) = 5(x + 3)^2 - 45 \), we have \( h = -3 \), so the axis of symmetry is \( x = -3 \).

Answer:

s:

• Vertex: \((-3, -45)\), it is a minimum.
• Axis of symmetry: \( x = -3 \)