The equation given in the image is:

$y = 3x^2 - 6x - 1$

This is a quadratic equation of the form $y = ax^2 + bx + c$, where $a = 3$, $b = -6$, and $c = -1$.

Steps to graph the function:

1. Find the vertex: The vertex of a parabola $y = ax^2 + bx + c$ can be found using the formula for the x-coordinate of the vertex: $x = \frac{-b}{2a}$ Substituting $a = 3$ and $b = -6$: $x = \frac{-(-6)}{2(3)} = \frac{6}{6} = 1$ Now, substitute $x = 1$ back into the equation to find the y-coordinate of the vertex: $y = 3(1)^2 - 6(1) - 1 = 3 - 6 - 1 = -4$ The vertex is at $(1, -4)$.

2. Determine the direction: Since $a = 3$ is positive, the parabola opens upwards.

3. Find the y-intercept: The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the equation: $y = 3(0)^2 - 6(0) - 1 = -1$ The y-intercept is $(0, -1)$.

4. Find the x-intercepts (if any): The x-intercepts occur when $y = 0$. Solve the equation $3x^2 - 6x - 1 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substituting $a = 3$, $b = -6$, and $c = -1$: $x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(-1)}}{2(3)}$ $x = \frac{6 \pm \sqrt{36 + 12}}{6} = \frac{6 \pm \sqrt{48}}{6} = \frac{6 \pm 6.93}{6}$ So, the two solutions are: $x_1 = \frac{6 + 6.93}{6} \approx 2.16, \quad x_2 = \frac{6 - 6.93}{6} \approx -0.16$ The x-intercepts are approximately $(2.16, 0)$ and $(-0.16, 0)$.

Plotting the graph:

• The vertex is at $(1, -4)$.
• The parabola opens upwards.
• The y-intercept is $(0, -1)$.
• The x-intercepts are approximately $(2.16, 0)$ and $(-0.16, 0)$.

Would you like a detailed graph or further explanation?

Related Questions:

1. How do you find the axis of symmetry of a quadratic function?
2. Can a parabola ever open downward, and how does that affect its vertex?
3. How does the value of $a$ in the quadratic equation affect the shape of the graph?
4. What are the implications of complex roots in a quadratic equation?
5. How can the quadratic formula be derived from completing the square?

Tip: Always check the sign of $a$ in the quadratic equation to determine if the parabola opens upwards or downwards.