The function \( F(x) = x^2 + 2x - 3 \) is a quadratic function. Here's an analysis of the function:

### 1. **Standard Form**
The function is already in the standard quadratic form:
\[
F(x) = ax^2 + bx + c
\]
where:
- \( a = 1 \) (coefficient of \( x^2 \)),
- \( b = 2 \) (coefficient of \( x \)),
- \( c = -3 \) (constant term).

### 2. **Vertex**
The vertex of a parabola in standard form is given by:
\[
x = -\frac{b}{2a}
\]
Substituting \( a = 1 \) and \( b = 2 \):
\[
x = -\frac{2}{2 \cdot 1} = -1
\]
To find the \( y \)-coordinate of the vertex:
\[
F(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4
\]
Thus, the vertex is \( (-1, -4) \).

### 3. **Axis of Symmetry**
The axis of symmetry is the vertical line that passes through the vertex:
\[
x = -1
\]

### 4. **Roots (x-Intercepts)**
To find the roots, solve \( F(x) = 0 \):
\[
x^2 + 2x - 3 = 0
\]
Factorize:
\[
(x + 3)(x - 1) = 0
\]
Thus, the roots are:
\[
x = -3 \quad \text{and} \quad x = 1
\]

### 5. **y-Intercept**
The \( y \)-intercept is the value of \( F(x) \) when \( x = 0 \):
\[
F(0) = 0^2 + 2(0) - 3 = -3
\]
Thus, the \( y \)-intercept is \( (0, -3) \).

### Summary
- Vertex: \( (-1, -4) \)
- Axis of Symmetry: \( x = -1 \)
- Roots: \( x = -3, 1 \)
- \( y \)-Intercept: \( (0, -3) \)
- Direction: Opens upwards (since \( a > 0 \)).

Would you like a graph of this function or further details on any of the points above?

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### Related Questions:
1. How do you derive the vertex formula for a quadratic function?
2. What is the significance of the discriminant in determining the roots of a quadratic equation?
3. How do you complete the square for the function \( F(x) \)?
4. Can you explain the transformations if \( F(x) = (x + 1)^2 - 4 \) is rewritten?
5. How can the quadratic function be used in a real-world application, like physics or economics?

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**Tip:** Always check the sign of \( a \) in \( ax^2 \) to determine if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).

Explore the quadratic function F(x) = x^2 + 2x - 3, including its vertex, roots, axis of symmetry, and y-intercept. This comprehensive guide covers essential algebraic techniques, suitable for students in Grades 8-10.

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