# How do you graph the function, label the vertex, axis of symmetry, and x-intercepts. f(x)=3x^2-54x +241?

Jun 8, 2015

First, factor the leading coefficient 3 out of the first to terms to write the function as $f \left(x\right) = 3 \left({x}^{2} - 18 x\right) + 241$.

Now complete the square with ${x}^{2} - 18 x$ to write it as ${x}^{2} - 18 x + 81 - 81$ (take the $- 18$, divide it by 2 to get $- 9$, and square that to get $81$, which gets both added and subtracted).

Since $3 \cdot \left(- 81\right) = - 243$, we can say that $f \left(x\right) = 3 \left({x}^{2} - 18 x + 81\right) + 241 - 243 = 3 {\left(x - 9\right)}^{2} - 2$.

This means the vertex is at $\left(x , y\right) = \left(9 , - 2\right)$ and is the low point of the parabola (the parabola opens upward since the leading coefficient is positive) and the axis of symmetry is the vertical line $x = 9$.

The $x$-intercepts can be found either with the quadratic formula used on the equation or by solving $3 {\left(x - 9\right)}^{2} - 2 = 0$ to get $x - 9 = \setminus \pm \sqrt{\frac{2}{3}}$ so that $x = 9 \setminus \pm \sqrt{\frac{2}{3}} = \setminus \frac{27 \setminus \pm \setminus \sqrt{6}}{3} \setminus \approx 8.18 \setminus m b \otimes \left\{\mathmr{and}\right\} 9.82$.

To help graph the function, you can plot some other points. For example, $f \left(8\right) = f \left(10\right) = 3 - 2 = 1$ and $f \left(7\right) = f \left(11\right) = 3 \cdot 4 - 2 = 10$.