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 ($a < 0$).