Question

which of the following accurately describes what the graph of $y = 5x^2 + 50x + 125$ looks like, with the repeated root of $x = -5$?

statement #1: the graph opens downward

statement #2: the graph has a vertex to the right of the x - axis

statement #3: the graph touches the x - axis once

(1 point)

statement #

Explanation:

Step1: Analyze the coefficient of \(x^2\)

The quadratic function is \(y = 5x^2+50x + 125\). For a quadratic function \(y=ax^2+bx + c\), the sign of \(a\) determines the direction the parabola opens. Here, \(a = 5>0\), so the parabola opens upward. Thus, Statement #1 is false.

Step2: Find the x - coordinate of the vertex

The x - coordinate of the vertex of a quadratic function \(y=ax^2+bx + c\) is given by \(x=-\frac{b}{2a}\). For \(y = 5x^2+50x + 125\), \(a = 5\) and \(b = 50\). So \(x=-\frac{50}{2\times5}=-\frac{50}{10}=- 5\). The x - coordinate of the vertex is \(x=-5\), which is to the left of the x - axis (since \(x=-5<0\)). Thus, Statement #2 is false.

Step3: Analyze the number of x - intercepts

A quadratic function \(y = ax^2+bx + c\) has a discriminant \(D=b^2 - 4ac\). For \(y = 5x^2+50x + 125\), \(a = 5\), \(b = 50\), \(c = 125\). Then \(D=(50)^2-4\times5\times125=2500 - 2500=0\). When the discriminant \(D = 0\), the quadratic function has exactly one real root (a repeated root), which means the graph of the quadratic function touches the x - axis at exactly one point. So Statement #3 is true.

Answer:

Statement #3: The graph touches the x - axis once.