Given the function $f(x) = x^2 + 2x - 3$, let's analyze it:

Key Characteristics of the Function

1. Type: This is a quadratic function, so its graph will be a parabola.

2. Standard Form: The function is in the standard form $f(x) = ax^2 + bx + c$, where:

   • $a = 1$
   • $b = 2$
   • $c = -3$

3. Direction of the Parabola: Since $a = 1 > 0$, the parabola opens upwards.

Finding the Vertex

The vertex form of a parabola $y = ax^2 + bx + c$ can be found at $x = -\frac{b}{2a}$: $x = -\frac{2}{2 \cdot 1} = -1$ Substituting $x = -1$ back into the function to find $f(-1)$: $f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4$ So, the vertex is at $(-1, -4)$.

X-Intercepts (Roots)

To find the x-intercepts, set $f(x) = 0$: $x^2 + 2x - 3 = 0$ Factoring, we get: $(x + 3)(x - 1) = 0$ Therefore, the solutions are $x = -3$ and $x = 1$. These are the x-intercepts.

Y-Intercept

The y-intercept occurs when $x = 0$: $f(0) = (0)^2 + 2(0) - 3 = -3$ So, the y-intercept is $(0, -3)$.

Summary

• Vertex: $(-1, -4)$
• X-Intercepts: $x = -3$ and $x = 1$
• Y-Intercept: $(0, -3)$
• Opens: Upwards

Would you like more details or have any questions?

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1. How do we convert a quadratic function to vertex form?
2. What is the process for completing the square for a quadratic function?
3. How can we determine the axis of symmetry for a parabola?
4. What does the sign of "a" in a quadratic function tell us about its graph?
5. How do we calculate the maximum or minimum value of a quadratic function?

Tip: Factoring is a useful method for finding x-intercepts but may not always work if the quadratic doesn’t factor nicely. In such cases, the quadratic formula is helpful.