Question

identify the graph of $f(x)=3x^{2}+18x + 25$.

Explanation:

Step1: Find the vertex of the parabola.

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex is given by $x=-\frac{b}{2a}$. Here, $a = 3$, $b = 18$, so $x=-\frac{18}{2\times3}=- 3$.

Step2: Find the y - coordinate of the vertex.

Substitute $x=-3$ into $y = 3x^{2}+18x + 25$. Then $y=3\times(-3)^{2}+18\times(-3)+25=3\times9-54 + 25=27-54 + 25=-2$. So the vertex is $(-3,-2)$.

Step3: Determine the direction of the parabola.

Since $a = 3>0$, the parabola opens upwards.

Answer:

The graph of the function $y = 3x^{2}+18x + 25$ is a parabola that opens upwards with vertex at $(-3,-2)$. Among the given graphs, the one that matches this description is the one with the vertex at $(-3,-2)$ and opening upwards.