Let's solve the problem step by step based on the given quadratic function $f(x) = x^2 + 4x - 1$:

a) Vertex

The vertex of a parabola defined by $y = ax^2 + bx + c$ can be found using the formula: $x = -\frac{b}{2a}$ Here, $a = 1$ and $b = 4$: $x = -\frac{4}{2(1)} = -2$

To find the $y$-coordinate of the vertex, substitute $x = -2$ into the equation: $f(-2) = (-2)^2 + 4(-2) - 1 = 4 - 8 - 1 = -5$ Thus, the vertex is: $\text{Vertex: } (-2, -5)$

b) Axis of Symmetry

The axis of symmetry is the vertical line that passes through the $x$-coordinate of the vertex: $\text{Axis of symmetry: } x = -2$

c) Opening of the Parabola

The sign of $a$ determines the direction of the opening:

• If $a > 0$, the parabola opens upward.
• If $a < 0$, the parabola opens downward.

Here, $a = 1 > 0$, so: $\text{The parabola opens upward.}$

Final Answers:

• a) Vertex: $(-2, -5)$
• b) Axis of symmetry: $x = -2$
• c) Opening: Upward

Would you like a detailed graph or further explanations?

Here are 5 related questions to expand on this topic:

1. How do you derive the formula for the vertex of a quadratic function?
2. What is the significance of the axis of symmetry in a parabola?
3. How does changing the value of $a$ in a quadratic equation affect the graph's shape?
4. Can you determine the vertex form of the quadratic equation $f(x) = x^2 + 4x - 1$?
5. What happens to the parabola if we change the constant term $c$?

Tip: To convert a quadratic equation into vertex form, complete the square.