Various parent functions and how to transform them using translations, reflections, compressions, and stretches

Sample Solution

Analyzing the Function h(x) = (x - 1)^2 + 4

1. Identifying the Parent Function:

The parent function of h(x) is the quadratic function. This is because the function is in the form of f(x) = a(x - h)^2 + k, where a, h, and k are constants, and a ≠ 0. In this case, a = 1, h = 1, and k = 4.

2. Explaining the Transformations:

The function has undergone the following transformations:

• Horizontal translation: The function is shifted one unit to the right because of the term (x - 1). This means the graph of the parent function is moved one unit to the right without changing its shape.
• Vertical translation: The function is shifted four units up because of the term + 4. This means the graph of the parent function is moved four units up without changing its shape.

3. Graphing the Functions:

Here's the graph of the parent function (f(x) = x^2) in blue and the transformed function (h(x) = (x - 1)^2 + 4) in orange:

Full Answer Section

3. Graphing the Functions:

Here's the graph of the parent function (f(x) = x^2) in blue and the transformed function (h(x) = (x - 1)^2 + 4) in orange:

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graph of the parent function f(x) = x^2 and the transformed function h(x) = (x 1)^2 + 4

4. New Function for the Next Student:

My new function for the next student to analyze is:

g(x) = -2(x + 2)^2 - 1

This function is based on the parent function f(x) = x^2, but it has the following transformations:

• Vertical stretch by a factor of 2 due to the coefficient -2.
• Horizontal translation 2 units to the left due to the term (x + 2).
• Vertical translation 1 unit down due to the term -1.

Please analyze this function, g(x), following the same steps (identifying the parent function, explaining the transformations, and graphing).