The function given is:

$f(x) = 2x^2 + 2x$

This is a quadratic function in terms of $x$. Let's analyze its key properties:

1. Standard Form:

The function is already in standard form for a quadratic equation:

$f(x) = ax^2 + bx + c$

where $a = 2$, $b = 2$, and $c = 0$.

2. Vertex Form:

To find the vertex form of the quadratic function, we can complete the square:

$f(x) = 2x^2 + 2x$

Factor out the 2 from the first two terms:

$f(x) = 2(x^2 + x)$

To complete the square inside the parentheses, take half of the coefficient of $x$ (which is 1), square it (resulting in $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$), and add and subtract it inside the parentheses:

$f(x) = 2\left(x^2 + x + \frac{1}{4} - \frac{1}{4}\right)$

This simplifies to:

$f(x) = 2\left( (x + \frac{1}{2})^2 - \frac{1}{4} \right)$

Distribute the 2:

$f(x) = 2(x + \frac{1}{2})^2 - \frac{1}{2}$

The vertex form is:

$f(x) = 2(x + \frac{1}{2})^2 - \frac{1}{2}$

Here, the vertex of the parabola is at $\left(-\frac{1}{2}, -\frac{1}{2}\right)$.

3. Axis of Symmetry:

The axis of symmetry for a parabola in the form $ax^2 + bx + c$ is given by:

$x = -\frac{b}{2a}$

Substituting the values of $a = 2$ and $b = 2$:

$x = -\frac{2}{2 \cdot 2} = -\frac{1}{2}$

So, the axis of symmetry is $x = -\frac{1}{2}$.

4. Y-Intercept:

The y-intercept occurs when $x = 0$:

$f(0) = 2(0)^2 + 2(0) = 0$

So, the y-intercept is at $(0, 0)$.

5. Roots (X-Intercepts):

To find the roots, set $f(x) = 0$:

$2x^2 + 2x = 0$

Factor out the common term:

$2x(x + 1) = 0$

Set each factor to zero:

1. $2x = 0 \implies x = 0$
2. $x + 1 = 0 \implies x = -1$

So, the roots are $x = 0$ and $x = -1$.

Would you like more details on any of these points, or do you have any questions?

Relative Questions:

1. What is the significance of the vertex of a parabola in a quadratic function?
2. How do you derive the vertex form of a quadratic function from its standard form?
3. How can the roots of a quadratic function be interpreted graphically?
4. What is the role of the axis of symmetry in a quadratic function?
5. How does the coefficient $a$ affect the shape and direction of a parabola?

Tip:

When converting a quadratic function to vertex form, remember that completing the square is a powerful technique for identifying the vertex and understanding the graph's properties better.