. What is the reflections in the axes of this function y=-4x^{2}+3\left(2x+1\right)+2


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.

What is the reflections in the axes of this function y=-4x^{2}+3\left(2x+1\right)+2?

Understand the reflections of the quadratic function y = -4x^2 + 6x + 5 across the x-axis and y-axis with step-by-step explanations. This problem covers key algebraic concepts and is suitable for students in Grades 9-12.

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