https://file.aiquickdraw.com/mathbot/file/172808787273824x0g9zq.jpg

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.

Learn how to graph the quadratic function y = 3x^2 - 6x - 1 by finding its vertex, intercepts, and shape. Understand key concepts such as the quadratic formula, vertex formula, and parabola symmetry. Ideal for students in Grades 9-11.

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