Math Snap

STEP 1

1. The function given is a quadratic function of the form $f(x) = ax^2 + bx + c$.
2. The vertex of a quadratic function in standard form can be found using the formula $x = -\frac{b}{2a}$.
3. The $x$-intercepts (roots) of the quadratic function can be found using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

STEP 2

1. Identify the coefficients $a$, $b$, and $c$.
2. Calculate the vertex of the quadratic function.
3. Calculate the $x$-intercepts using the quadratic formula.

STEP 3

Identify the coefficients from the quadratic function $f(x) = -3x^2 + 2x + 3$.
Here, $a = -3$, $b = 2$, and $c = 3$.

STEP 4

Calculate the $x$-coordinate of the vertex using the formula $x = -\frac{b}{2a}$.
$$x = -\frac{2}{2(-3)} = \frac{-2}{-6} = \frac{1}{3}$$

STEP 5

Substitute $x = \frac{1}{3}$ into the function to find the $y$-coordinate of the vertex.
$$f\left(\frac{1}{3}\right) = -3\left(\frac{1}{3}\right)^2 + 2\left(\frac{1}{3}\right) + 3$$ $$= -3\left(\frac{1}{9}\right) + \frac{2}{3} + 3$$ $$= -\frac{1}{3} + \frac{2}{3} + 3$$ $$= \frac{1}{3} + 3$$ $$= \frac{1}{3} + \frac{9}{3}$$ $$= \frac{10}{3}$$ The vertex is $\left(\frac{1}{3}, \frac{10}{3}\right)$.

STEP 6

Calculate the $x$-intercepts using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
First, calculate the discriminant:
$$b^2 - 4ac = 2^2 - 4(-3)(3)$$ $$= 4 + 36$$ $$= 40$$

Substitute the discriminant into the quadratic formula:
$$x = \frac{-2 \pm \sqrt{40}}{2(-3)}$$ $$= \frac{-2 \pm \sqrt{40}}{-6}$$ Simplify $\sqrt{40}$:
$$\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$$ $$x = \frac{-2 \pm 2\sqrt{10}}{-6}$$ $$= \frac{-2}{-6} \pm \frac{2\sqrt{10}}{-6}$$ $$= \frac{1}{3} \mp \frac{\sqrt{10}}{3}$$ The $x$-intercepts are:
$$x = \frac{1 + \sqrt{10}}{3} \quad \text{and} \quad x = \frac{1 - \sqrt{10}}{3}$$ Calculate these values to the nearest hundredth:
$$x \approx \frac{1 + 3.16}{3} \approx \frac{4.16}{3} \approx 1.39$$ $$x \approx \frac{1 - 3.16}{3} \approx \frac{-2.16}{3} \approx -0.72$$ The vertex is $\left(\frac{1}{3}, \frac{10}{3}\right) \approx (0.33, 3.33)$.
The $x$-intercepts are approximately $1.39$ and $-0.72$.