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: Rewrite the quadratic function in vertex - form

First, factor out the coefficient of $x^{2}$ from the first two terms of $y = 5x^{2}+50x + 125$. We have $y=5(x^{2}+10x)+125$. Completing the square inside the parentheses: $x^{2}+10x=(x + 5)^{2}-25$. So $y=5((x + 5)^{2}-25)+125=5(x + 5)^{2}-125 + 125=5(x + 5)^{2}$.

Step2: Analyze the properties of the quadratic function

For a quadratic function in the form $y=a(x - h)^{2}+k$, the vertex is $(h,k)$ and the sign of $a$ determines the direction of opening. Here $a = 5>0$, so the graph opens upward. The vertex of $y = 5(x + 5)^{2}$ is $(-5,0)$. Since the vertex lies on the $x$-axis and the function can be written as $y=5(x+5)(x + 5)$, $x=-5$ is a repeated root and the graph touches the $x$-axis once.

Answer:

Statement #3