https://file.aiquickdraw.com/mathbot/file/17297820333370fk45nzz.jpg

Let's work through each of the equations and fill in the missing forms.

### 1. \( y = x^2 + 2x - 35 \)

- **Intercept Form**: To find the intercept form, we factor the quadratic. Factoring gives us:
  \[
  y = (x - 5)(x + 7)
  \]
- **Vertex Form**: We can complete the square to rewrite it in vertex form.
  \[
  y = (x + 1)^2 - 36
  \]

### 2. \( y = x^2 + 8x \)
- **Standard Form**: The missing value for the standard form is 0, so the equation is:
  \[
  y = x^2 + 8x + 0
  \]
- **Intercept Form**: Factoring this gives:
  \[
  y = x(x + 8)
  \]
- **Vertex Form**: Given that the vertex form is provided as \( y = (x + 4)^2 - 1 \), no further work is needed here.

### 3. \( y = x^2 - 8x \)
- **Standard Form**: The missing value is 0, so the equation is:
  \[
  y = x^2 - 8x + 0
  \]
- **Intercept Form**: Factor the quadratic:
  \[
  y = (x - 4)(x - 4)
  \]
- **Vertex Form**: Completing the square gives:
  \[
  y = (x - 4)^2
  \]

### 4. \( y = -x^2 + 8x \)
- **Standard Form**: The missing value is 0, so the equation is:
  \[
  y = -x^2 + 8x + 0
  \]
- **Intercept Form**: Factor the quadratic:
  \[
  y = -x(x - 8)
  \]
- **Vertex Form**: Completing the square gives:
  \[
  y = -(x - 4)^2 + 1
  \]

### 5. \( y = -x^2 - 6x + 16 \)
- **Intercept Form**: Factoring the quadratic:
  \[
  y = -(x + 8)(x - 2)
  \]
- **Vertex Form**: Completing the square gives:
  \[
  y = -(x + 3)^2 + 25
  \]

### 6. \( y = x^2 \)
- **Intercept Form**: The quadratic already shows its simplest form:
  \[
  y = (x)(x)
  \]
- **Vertex Form**: The vertex form is:
  \[
  y = (x - 0)^2
  \]

### 7. \( y = -x^2 - 8x \)
- **Standard Form**: The missing value is 0, so the equation is:
  \[
  y = -x^2 - 8x + 0
  \]
- **Intercept Form**: There are no roots as this is a downward-facing parabola.

### 8. \( y = x^2 - 8x + 15 \)
- **Intercept Form**: Factor the quadratic:
  \[
  y = (x - 3)(x - 5)
  \]
- **Vertex Form**: Completing the square gives:
  \[
  y = (x - 4)^2 - 1
  \]

Let me know if you have any questions or need more details. Here are five follow-up questions to deepen your understanding:

1. How do you factor a quadratic equation?
2. What are the steps to complete the square for a quadratic function?
3. How can you determine the vertex of a parabola from its equation?
4. Why is it important to understand the different forms of a quadratic equation?
5. How can the graph of a quadratic function help in identifying its equation?

**Tip:** Always check the sign of the leading coefficient in quadratic equations, as it determines whether the parabola opens upward or downward.

Complete the table by converting each quadratic equation into Standard, Intercept, and Vertex forms.

Learn how to convert quadratic equations into different forms: Standard, Intercept, and Vertex forms. This worksheet provides step-by-step methods for factoring, completing the square, and identifying key points of the graph. Suitable for students in Grades 9-10.

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