Answer
See explanation
Work Step by Step
Finding the quadratic function in standard form $f(x)=a(x-h)^2+k$:
$$f(x)=x^2+3x+\frac{1}{4}$$ $$f(x)=x^2+3x+\left(\frac{3}{2}\right)^2+\frac{1}{4}-\left(\frac{3}{2}\right)^2$$ $$f(x)=x^2+3x+\frac{9}{4}+\frac{1}{4}-\frac{9}{4}$$ $$f(x)=\left(x+\frac{3}{2}\right)^2-2$$
The sketch of the graph of the function is as shown.
Notice that the vertex is:
$$\left(-\frac{3}{2},-2\right)$$
The axis of symmetry is: $$x=-\frac{3}{2}$$
The $x$-intercepts are:
$$(-2.91,0),(-0.09,0)$$
