Question

The graph is a translation 4 units down, followed by a vertical stretch by a factor of 6 and a reflection in the \(x\)-axis of the graph of the parent quadratic function.

Short answer

The transformed function is \(y = -6x^2 + 24\)

Step-by-step solution

Apply Translation

Start with the parent function \(y = x^2\). A translation 4 units down means subtracting 4 from the function: \(y = x^2 - 4\).

Apply Vertical Stretch

After translation, stretch the graph vertically by a factor of six. This is achieved by multiplying the function (without 'y') by 6: \(y = 6(x^2 - 4)\). This can be simplified to : \(y = 6x^2 - 24\).

Apply Reflection

Reflect the graph in the x-axis. This changes the sign of 'y' in our function: \(y = -(6x^2 - 24)\). Simplifying this gives: \(y = -6x^2 + 24\).

Key concepts

Translation

Translation in the context of quadratic functions involves shifting the graph of the function up, down, left, or right. In this exercise, the translation occurs 4 units down. To integrate this transformation, we begin with the parent function of a quadratic, which is \( y = x^2 \).

• To move the graph downward by 4 units, we subtract 4 from the original function: \( y = x^2 - 4 \).
• This manipulation adjusts the vertex of the parabola to sit 4 units below its initial position on the coordinate grid.

This transformation does not affect the shape or orientation of the parabola; it merely shifts its location vertically downward. Understanding this concept is key, as it allows one to adjust the starting point of a graph without altering its fundamental characteristics.

Vertical Stretch

A vertical stretch alters the "height" of a parabola, making it appear narrower or wider. This is achieved by multiplying the entire function (after any translations) by a constant factor. In this example, a vertical stretch by a factor of 6 follows the initial translation.

• The function prior to stretching is \( y = x^2 - 4 \).
• During stretching, multiply each term by 6, resulting in \( y = 6(x^2 - 4) \), which simplifies to \( y = 6x^2 - 24 \).
• This makes the parabola narrower, as it extends further vertically from the x-axis.

A larger stretch factor generally indicates a steeper curve, demonstrating how modifying coefficients profoundly impacts the graph's visual scale. It's essential to be comfortable with manipulating and interpreting these changes as they are often combined with other transformations.

Reflection

Reflection in quadratic functions typically involves flipping the graph over the x-axis, y-axis, or another line. When we reflect a quadratic function in the x-axis, we change the sign of the entire function. This means multiplying the result of any stretches or translations by -1.

• From our previous step, the function was \( y = 6x^2 - 24 \).
• Reflecting across the x-axis involves multiplying this by -1, giving us \( y = -(6x^2 - 24) \), which simplifies to \( y = -6x^2 + 24 \).
• This reflects the graph downwards, turning all upwards curves into downwards facing ones.

This transformation is crucial as it alters the direction in which the parabola opens, switching from concave up (shaped like a U) to concave down (shaped like an n). Reflections, like other transformations, do not initially change the position of the vertex but do significantly affect the orientation of the graph.