Let's first simplify the given function:

$y = -4x^2 + 3(2x + 1) + 2$

Expanding the expression $3(2x + 1)$:

$y = -4x^2 + 6x + 3 + 2$

Now simplify the constant terms:

$y = -4x^2 + 6x + 5$

1. Reflection across the x-axis:

To reflect a function across the x-axis, we multiply the entire function by $-1$. So, the reflection of the function $y = -4x^2 + 6x + 5$ across the x-axis is:

$y = -(-4x^2 + 6x + 5) = 4x^2 - 6x - 5$

2. Reflection across the y-axis:

To reflect a function across the y-axis, we replace $x$ with $-x$ in the equation. Let's apply this transformation:

$y = -4(-x)^2 + 6(-x) + 5$

Since $(-x)^2 = x^2$, the equation becomes:

$y = -4x^2 - 6x + 5$

Final Reflected Functions:

• Reflection across the x-axis: $y = 4x^2 - 6x - 5$
• Reflection across the y-axis: $y = -4x^2 - 6x + 5$

Would you like to explore any of these reflections further, or visualize the transformations?

Here are 5 related questions:

1. What happens when reflecting a linear function over the axes?
2. How does reflection affect the symmetry of a quadratic function?
3. Can reflections change the roots of a quadratic equation?
4. What is the impact of reflections on the vertex of a parabola?
5. How do transformations affect the concavity of a function?

Tip: Reflections change the sign of specific terms but keep the degree of the polynomial the same.