Question

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

Short answer

The parental quadratic function, when reflected on the y-axis, vertically stretched by a factor of 6 and translated 4 units upward, becomes \(y = -6x^2 + 4\).

Step-by-step solution

Recognize Parent Function

Understand that the parent function is the quadratic function which is \(y = x^2\). This is the base equation you will be altering.

Apply Reflection Transformation

Recall that a reflection in the y-axis results in changing the sign of x in the equation. Thus, this transforms the equation to \(y = -x^2\).

Apply Vertical Stretch

A vertical stretch by a factor of 6 means multiplying the parent function by 6, resulting in \(y = -6x^2\).

Apply Upward Translation

A translation of 4 units up means adding 4 to the function. The final transformed function then becomes \(y = -6x^2 + 4\).

Key concepts

Parent Function

Understanding the concept of a parent function is crucial when dealing with transformations in algebra. The parent function serves as a template for transformations and helps students visualize changes more easily. For quadratic functions, the simplest form is given by the equation \(y = x^2\). This is the most basic version of a quadratic function and shows a U-shaped graph known as a parabola. The vertex of this parabola is at the origin \((0, 0)\), and the graph opens upwards with symmetry about the y-axis.

When working with parent functions, remember that all subsequent transformations will modify this original graph. However, the core shape of the U remains recognizable even after these transformations are applied. This characteristic makes it easier to predict and understand the effects of different types of transformations on the graph of the function.

Reflection in the Y-axis

A reflection in the y-axis is a specific type of transformation that flips the graph of a function over the y-axis. Visually, this means that the left side of the graph becomes the right side and vice versa. For the quadratic parent function \(y = x^2\), this reflection results in the change of the sign of the input value \(x\), leading to the new equation \(y = -x^2\).

It is important to understand that reflection does not affect the y-values or the vertical position of the graph. The peak or valley of the graph, which lies along the y-axis, remains stationary during a reflection. This can be compared to looking at the graph in a mirror placed along the y-axis, where only the sides are switched.

Vertical Stretch

A vertical stretch of a function's graph can be envisioned as pulling or stretching the graph away from the x-axis. When we say a graph undergoes a vertical stretch by a factor of 6, it means that the y-values of the parent function are multiplied by 6. This transformation can be expressed mathematically as changing the equation from \(y = x^2\) to \(y = 6x^2\), or in the case of our reflection, \(y = -6x^2\).

This stretching causes the graph to become narrower and taller, making it more steep. A vertical stretch does not affect the x-values or the horizontal position of the graph. Contrary to horizontal transformations, vertical stretches and compressions solely alter the height of the function's graph.

Translation of Functions

Translations shift the entire graph of a function in a certain direction without changing its shape. In the context of the exercise, we are looking at a translation 4 units up. This means that every point on the graph is moved 4 units higher on the y-axis. To apply this to the quadratic function \(y = -6x^2\), we simply add 4 to the entire function, producing the transformed equation \(y = -6x^2 + 4\).

Translations are the easiest transformation to identify on a graph because they move the graph intact, like sliding a piece of paper across a table. An upward translation increases the y-values of all points in the graph, effectively lifting the graph higher. This adjustment is reflected in the new position of the vertex, which in the case of the original parent function, would move from \((0, 0)\) to \((0, 4)\).