Given the function $c(x)=3 x^2-18 x+23$, write your answers in exact form.

The function $c(x) = 3x^2 - 18x + 23$ is a quadratic function of the form $f(x) = ax^2 + bx + c$, where the coefficients are $a = 3$, $b = -18$, and $c = 23$. Since $a = 3 > 0$, the graph of this function is a parabola that opens upwards. To analyze this function fully, we will determine its vertex, axis of symmetry, y-intercept, and its roots.

Finding the Vertex and Axis of Symmetry

The vertex $(h, k)$ of a parabola given by $ax^2 + bx + c$ can be found using the formula $h = -\frac{b}{2a}$.
Substituting the coefficients:
\[h = -\frac{-18}{2 \times 3} = \frac{18}{6} = 3\]
Now, we find the y-coordinate $k$ by evaluating the function at $h = 3$:
\[k = c(3) = 3(3)^2 - 18(3) + 23 = 3(9) - 54 + 23 = 27 - 54 + 23 = -4\]
Thus, the vertex is $(3, -4)$. The axis of symmetry is the vertical line passing through the vertex, which is $x = 3$.

Determining the y-intercept

The y-intercept occurs where the graph crosses the y-axis, which is at $x = 0$. Evaluating the function at $x = 0$:
\[c(0) = 3(0)^2 - 18(0) + 23 = 23\]
So, the y-intercept is at the point $(0, 23)$.

Calculating the Roots

To find the roots of the function, we set $c(x) = 0$:
\[3x^2 - 18x + 23 = 0\]
Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:
\[x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(3)(23)}}{2(3)}\]
\[x = \frac{18 \pm \sqrt{324 - 276}}{6}\]
\[x = \frac{18 \pm \sqrt{48}}{6}\]
Since $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$, we have:
\[x = \frac{18 \pm 4\sqrt{3}}{6} = 3 \pm \frac{2\sqrt{3}}{3}\]
Thus, the roots are $3 + \frac{2\sqrt{3}}{3}$ and $3 - \frac{2\sqrt{3}}{3}$.

Examples

In physics, the trajectory of a projectile is often modeled by a quadratic function. If you were to throw a ball, the height of the ball at time $t$ could be represented by a quadratic equation. Knowing the vertex allows you to determine the maximum height the ball reaches and the time at which it happens. The roots of the equation tell you exactly when the ball is at ground level (at the start and when it hits the ground). Solving these quadratics is essential for predicting motion and designing systems that rely on projectile dynamics.