Question

COMPARING METHODS Let the graph of g be a translation 3 units up and 1 unit right followed by a vertical stretch by a factor of 2 of the graph of f(x) = x2. a. Identify the values of a, h, and k and use vertex form to write the transformed function. b. Use function notation to write the transformed function. Compare this function with your function in part (a). c. Suppose the vertical stretch was performed fi rst, followed by the translations. Repeat parts (a) and (b). d. Which method do you prefer when writing a transformed function? Explain.

Short answer

The transformed functions in both cases, regardless of the order of operations, are \( g(x) = 2(x+1)^2 + 3 \).

Step-by-step solution

Identify a, h, and k values

In this problem, it is mentioned that a vertical stretch by a factor of 2 is made on the graph of \( f(x) = x^2 \), so \( a=2 \). Furthermore, the graph is translated 3 units up and 1 unit right so \( k = 3 \) and \( h = -1 \) since rights shifts are represented as negative in a quadratic function.

Write the transformed function in vertex form

The vertex form of a function is \( g(x) = a(x-h)^2 + k \). Substituting the values from step 1, the transformed equation becomes \( g(x) = 2(x+1)^2 + 3 \).

Notate the function

Re-write the function in transformation notation as \( g(x) = 2f(x+1) + 3 \). This represents the original function \( f(x) = x^2 \) moved 1 unit right and stretched vertically by 2 units, then moved 3 units up.

Compare the transformed functions

The function from step 2 and step 3 represent the same graph. They both represent a stretch of \( f(x) = x^2 \) by a factor of 2, right by 1 unit and up by 3.

Change the order of transformation

As specified in the problem, the vertical stretch is performed first. This would give us the function \( f'(x) = 2f(x) = 2x^2 \). Then the translations are carried out which gives \( g(x) = 2f(x+1)+3 = 2(x+1)^2 + 3 \).

Express preference

At this point, one would express a preference for either method. However, the given task does not mention who is expressing a preference and so, an opinion cannot be stated.

Key concepts

Vertex Form

The vertex form of a quadratic function is a powerful tool that provides an easy way to understand transformations applied to a quadratic equation. It is expressed as \( g(x) = a(x-h)^2 + k \). Here,

• \( a \) signifies the vertical stretch or compression,
• \( h \) represents the horizontal shift, and
• \( k \) indicates the vertical shift.

In the original exercise, we transformed the function \( f(x) = x^2 \) using vertex form to find the new equation \( g(x) = 2(x+1)^2 + 3 \). This form is particularly useful because it immediately conveys how the graph is shifted in space, allowing for a visual understanding of the effect of each transformation. Not only does it simplify the comparison of equations, but it also makes plotting the transformed graph straightforward. Remember, the vertex of the graph is directly derived from the values \( (h, k) \), which is the turning point of the parabola.

Translation

Translation involves shifting a graph horizontally, vertically, or both, without altering its shape. In the context of quadratic functions, translation can be achieved by modifying the \( h \) and \( k \) values in the vertex form of an equation.

• A positive \( h \) shifts the graph to the left, while a negative \( h \) shifts it to the right.
• A positive \( k \) moves the graph up, whereas a negative \( k \) moves it down.

In our exercise, translating \( f(x) = x^2 \) by moving 3 units up and 1 unit right resulted in \( g(x) = 2(x+1)^2 + 3 \). The way we denote the shifts in the vertex form, \((x-h)\), helps maintain the form's consistency.
This means that each transformation step, when applied in the correct order, modifies the base function seamlessly, reflecting these shifts precisely and effectively on the graph.

Vertical Stretch

A vertical stretch occurs when a function is multiplied by a constant factor, making the graph taller or steeper while maintaining its general shape and orientation. This is represented by the parameter \( a \) in the vertex form of a quadratic equation.

• When \( |a| > 1 \), the graph stretches vertically, becoming narrower.
• If \( 0 < |a| < 1 \), it compresses, making the parabola wider.
• If \( a \) is negative, it reflects the graph over the x-axis along with the stretch or compression.

In the original problem, the vertical stretch is by a factor of 2, which means the function \( f(x) = x^2 \) becomes \( 2x^2 \). This stretch alters the graph but retains its basic quadrilateral shape.
Stretches can be visualized as pulling or pushing the function's graph along the y-axis and are essential for emphasizing the peaks and valleys of the parabolic curve.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, characterized by their distinctive parabolic shape. The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \).
These functions can be transformed and understood in various forms, with vertex form being one of the most illustrative for transformations.

• The graph of a quadratic function is a parabola, which opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
• The vertex form \( g(x) = a(x-h)^2 + k \) is especially useful for identifying and applying transformations as it reveals the vertex \((h, k)\) directly.

Understanding these properties allows students to manipulate and graph quadratic functions with ease, visualizing the effects of transformations such as translations and stretches on the overall shape of the parabola
By experimenting with various values of \( a \), \( h \), and \( k \), one can observe different structures of parabolas and their corresponding shifts and orientations on the coordinate plane.