https://file.aiquickdraw.com/mathbot/file/1727275202483a49c4fdl.jpg

From the image, we have two types of questions:

1. **Graphs (33 and 34):** These display parabolas, and you're likely being asked to describe characteristics of the parabola, such as vertex, axis of symmetry, and domain/range.

2. **Exercises 35-42:** The goal is to rewrite each function in the form \( f(x) = a(x - h)^2 + k \), which is the vertex form of a parabola. This involves completing the square and identifying the vertex.

I'll walk through one example of completing the square for these exercises.

### Example: Exercise 35

We are given:  
\[ f(x) = x^2 - 3x \]

To convert this to vertex form, we complete the square:

1. Factor out the coefficient of \(x^2\), but here it's 1, so we skip this step.
2. Take half the coefficient of \(x\) (which is -3), square it, and add and subtract it inside the equation:
   \[ x^2 - 3x = \left( x^2 - 3x + \left( \frac{-3}{2} \right)^2 \right) - \left( \frac{-3}{2} \right)^2 \]
   \[ = (x - \frac{3}{2})^2 - \frac{9}{4} \]

Thus, the vertex form is:
\[ f(x) = (x - \frac{3}{2})^2 - \frac{9}{4} \]

The vertex is \( \left( \frac{3}{2}, -\frac{9}{4} \right) \).

Would you like me to solve the rest or provide more details on completing the square?

### Relative Questions:
1. What is the process for finding the axis of symmetry for a parabola?
2. How do you determine if a parabola opens upwards or downwards?
3. What is the relationship between the vertex form of a quadratic and its graph?
4. How do you calculate the y-intercept of a quadratic function?
5. What are the key differences between the standard form and vertex form of a quadratic function?

**Tip:** In vertex form \( f(x) = a(x - h)^2 + k \), the vertex is \( (h, k) \), and the value of \( a \) tells you whether the parabola opens up or down (positive opens upwards, negative opens downwards).

Write the given expression in the form f(x) = a(x - h)^2 + k. Identify the vertex. Exercises 35-42.

Learn how to convert quadratic functions to vertex form and identify the vertex for exercises 35-42. This involves completing the square and understanding the vertex formula for parabolas. Suitable for high school students in Grades 9-12.

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