91Ó°ÊÓ

To understand function transformation, let's start with the basics. Mathematically, a function represents a relationship between inputs (x-values) and outputs (y-values). When we transform a function, we change how the input values affect the output.
In the given exercise, we have a quadratic function: \( f(x) = -3x^2 - 5x + 1 \).
A common type of transformation is substituting \(-x\) for \(x\) in the function. This is known as a reflection over the y-axis.
When \(x\) is replaced with \(-x\), the resulting function \(f(-x)\) shows how the output changes when the input is reflected. Here's what this looks like: \[ f(-x) = -3(-x)^2 - 5(-x) + 1 = -3x^2 + 5x + 1 \]
Notice that the coefficient of \(x^2\) stays the same, because \((-x)^2 = x^2\). However, the linear term's sign changes from \(-5x\) to \(5x\), showing the reflection.

Quadratic functions are polynomial functions of degree 2. They have the general form: \[ f(x) = ax^2 + bx + c \].
In our example: \[ f(x) = -3x^2 - 5x + 1 \], \[ \] where \(a = -3, b = -5,\) and \(c = 1\).
Key features of quadratic functions:

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• **Vertex**: The highest or lowest point on the graph.
• **Axis of Symmetry**: A vertical line that runs through the vertex.
• **Direction of Opening**: Determined by the sign of \(a\). If \(a < 0\), the parabola opens downwards. If \(a > 0\), it opens upwards.
  In the given function, \(a = -3\), so it opens downwards.

When performing transformations on quadratic functions, like substituting \(-x\), it's important to remember that it affects symmetry and direction but keeps the vertex position intact relative to other transformations.

The substitution method involves replacing a variable with another expression. This is common in function transformations.

• **Step-by-Step Process**:
• Identify the pattern. For instance, substituting \(x\) with \(-x\) transforms the input.
• Apply substitution inside the function. Follow the order of operations (PEMDAS/BODMAS).
  In our example: Given \( f(x) = -3x^2 - 5x + 1 \):
• Substitute \(-x\) for \(x\): \(-3(-x)^2 - 5(-x) + 1 \)
• Simplify: \[ -3x^2 + 5x + 1 \]

The transformed function after substitution is \(f(-x) = -3x^2 + 5x + 1\).
Through substitution, we alter the structure of the function systematically. This is crucial for understanding how functions behave and interact under different conditions.