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The problem shown in the image asks to graph the parabola defined by the quadratic equation \( y = 3x^2 + 5x - 3 \), and to identify key points on the parabola, including the vertex and the points where the parabola intersects the x-axis (roots).

### Let's break down the steps to solve this:

1. **Equation Analysis**:
   - The equation is \( y = 3x^2 + 5x - 3 \).
   - This is a standard quadratic equation of the form \( y = ax^2 + bx + c \), where \( a = 3 \), \( b = 5 \), and \( c = -3 \).

2. **Finding the Vertex**:
   - The vertex of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula:
     \[
     x_{\text{vertex}} = -\frac{b}{2a}
     \]
     Substituting the values of \( a \) and \( b \):
     \[
     x_{\text{vertex}} = -\frac{5}{2(3)} = -\frac{5}{6}
     \]
   - To find the corresponding y-coordinate, substitute \( x_{\text{vertex}} \) back into the equation:
     \[
     y_{\text{vertex}} = 3\left(-\frac{5}{6}\right)^2 + 5\left(-\frac{5}{6}\right) - 3
     \]
     Simplifying this:
     \[
     y_{\text{vertex}} = \frac{75}{36} - \frac{150}{36} - 3 = -\frac{75}{36} - 3 = -\frac{75}{36} - \frac{108}{36} = -\frac{183}{36} = -\frac{61}{12}
     \]
   - Therefore, the vertex is \( \left(-\frac{5}{6}, -\frac{61}{12}\right) \).

3. **Finding the Roots (x-intercepts)**:
   - The x-intercepts are found by setting \( y = 0 \) and solving the quadratic equation \( 3x^2 + 5x - 3 = 0 \).
   - The roots can be found using the quadratic formula:
     \[
     x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
     \]
     Substituting the values of \( a \), \( b \), and \( c \):
     \[
     x = \frac{-5 \pm \sqrt{5^2 - 4(3)(-3)}}{6} = \frac{-5 \pm \sqrt{25 + 36}}{6} = \frac{-5 \pm \sqrt{61}}{6}
     \]
   - So the two roots are:
     \[
     x_1 = \frac{-5 + \sqrt{61}}{6}, \quad x_2 = \frac{-5 - \sqrt{61}}{6}
     \]

4. **Graphing the Parabola**:
   - The vertex \( \left(-\frac{5}{6}, -\frac{61}{12}\right) \) is the lowest point since the parabola opens upwards (because \( a > 0 \)).
   - The x-intercepts are at \( x_1 \) and \( x_2 \).

### Summary:
- **Vertex**: \( \left(-\frac{5}{6}, -\frac{61}{12}\right) \)
- **Roots**: \( \frac{-5 + \sqrt{61}}{6} \) and \( \frac{-5 - \sqrt{61}}{6} \)

Would you like any additional details or clarification on the steps? 

### Related Questions:
1. How does the value of \( a \) in a quadratic equation affect the shape of the parabola?
2. What are the properties of the discriminant in determining the nature of the roots?
3. How would you determine the axis of symmetry for a given parabola?
4. What happens to the graph if the value of \( c \) changes in the quadratic equation?
5. How do you calculate the y-intercept for the quadratic equation?

### Tip:
Always verify your vertex and roots by substituting them back into the original equation to ensure they satisfy the equation correctly.

Graph the parabola y = 3x^2 + 5x - 3 and identify the vertex and roots.

This lesson explains how to graph the parabola defined by y = 3x^2 + 5x - 3. Learn how to find the vertex, x-intercepts (roots), and plot the graph step by step. Suitable for students in Grades 9-10.

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