The given equation is:

$y = 2x^2 + 4x$

This equation represents a quadratic function in terms of $x$. Let's analyze it by finding a few key features, such as the vertex, axis of symmetry, and determining whether the parabola opens upwards or downwards.

1. Standard Form and Parabola Direction

The function is in the standard form $y = ax^2 + bx + c$, where:

• $a = 2$
• $b = 4$
• $c = 0$

Since $a > 0$, the parabola opens upwards.

2. Finding the Vertex

The vertex of a parabola given by $y = ax^2 + bx + c$ can be found using: $x = -\frac{b}{2a}$

For this equation: $x = -\frac{4}{2 \times 2} = -\frac{4}{4} = -1$

To find the $y$-coordinate of the vertex, substitute $x = -1$ into the equation: $y = 2(-1)^2 + 4(-1) = 2 \cdot 1 - 4 = 2 - 4 = -2$

Thus, the vertex is at $(-1, -2)$.

3. Axis of Symmetry

The axis of symmetry for a parabola in the form $y = ax^2 + bx + c$ is the line $x = -\frac{b}{2a}$. Therefore, the axis of symmetry here is: $x = -1$

4. Y-Intercept

The y-intercept occurs when $x = 0$: $y = 2(0)^2 + 4(0) = 0$

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

Summary

• Direction: Opens upwards
• Vertex: $(-1, -2)$
• Axis of Symmetry: $x = -1$
• Y-Intercept: $(0, 0)$

Would you like further details on graphing this function or solving related problems?

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Here are five questions that expand on this information:

1. How do we find the x-intercepts of this quadratic function?
2. How would changes in the coefficient $a$ affect the shape of the parabola?
3. What are the implications of the axis of symmetry on the function's values?
4. How can we convert this quadratic equation to vertex form?
5. How does this equation relate to other types of polynomial equations?

Tip: For any quadratic function, understanding the vertex is essential, as it represents either the maximum or minimum value depending on the parabola's direction.