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.

Explore the quadratic function f(x) = 2x^2 + 2x, including how to convert it to vertex form, determine the axis of symmetry, and find its roots. This step-by-step solution is ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation