Math Problem Statement

The graph of g is a vertical shrink by a factor of 1/3 and a reflection in the y-axis, followed by a translation 3 units right of the graph of f(x) = x^2. Write a rule for g. Then identify the vertex.

Solution

The given problem asks for the transformation of the function f(x)=x2f(x) = x^2 into a new function g(x)g(x), involving three transformations:

  1. Vertical shrink by a factor of 13\frac{1}{3}.
  2. Reflection in the y-axis.
  3. Translation 3 units to the right.

Let's break down the transformation step by step:

Step 1: Vertical Shrink by 13\frac{1}{3}

A vertical shrink multiplies the entire function by 13\frac{1}{3}, so the function becomes: g(x)=13f(x)=13x2g(x) = \frac{1}{3} f(x) = \frac{1}{3} x^2

Step 2: Reflection in the y-axis

A reflection in the y-axis replaces xx with x-x. Applying this reflection: g(x)=13(x)2=13x2g(x) = \frac{1}{3} (-x)^2 = \frac{1}{3} x^2 Note that reflecting x2x^2 about the y-axis doesn’t change the function because (x)2=x2(-x)^2 = x^2.

Step 3: Translation 3 units to the right

To translate the function 3 units to the right, replace xx with x3x - 3. The equation now becomes: g(x)=13(x3)2g(x) = \frac{1}{3} (x - 3)^2

Vertex

The vertex of f(x)=x2f(x) = x^2 is at (0,0)(0, 0). After translating 3 units to the right, the new vertex is at (3,0)(3, 0).

Final Answer

The rule for g(x)g(x) is: g(x)=13(x3)2g(x) = \frac{1}{3} (x - 3)^2 The vertex is (3,0)(3, 0).

Would you like further details on any of the steps?

Here are 5 related questions to extend your understanding:

  1. How does a vertical shrink differ from a vertical stretch?
  2. What is the effect of translating a function vertically instead of horizontally?
  3. How do you find the vertex of any quadratic function in the form a(xh)2+ka(x - h)^2 + k?
  4. What happens to the graph of f(x)=x2f(x) = x^2 when reflected over the x-axis instead of the y-axis?
  5. Can a function be reflected over both axes simultaneously?

Tip: When reflecting over the y-axis, quadratic functions often remain unchanged because x2=(x)2x^2 = (-x)^2.

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Math Problem Analysis

Mathematical Concepts

Quadratic Transformations
Function Reflection
Function Translation

Formulas

g(x) = a(x - h)^2 + k

Theorems

Transformation Theorems for Quadratic Functions

Suitable Grade Level

Grades 9-11