Let the graph of g be a horizontal shrink by a factor of 1/3, followed by a translation 1 unit up of the graph of f(x)= x^2. Write a rule for g

To find the rule for the function \( g(x) \), which is a transformation of \( f(x) = x^2 \), we need to apply the given transformations in the correct order.

### Step 1: Horizontal Shrink by a Factor of 1/3
A horizontal shrink by a factor of \( 1/3 \) is applied by replacing \( x \) with \( 3x \). So, we first transform the function \( f(x) = x^2 \) to:
\[
f(3x) = (3x)^2 = 9x^2
\]
This gives the horizontally shrunk version of the function.

### Step 2: Translation 1 Unit Up
To translate the graph 1 unit upward, we add 1 to the function:
\[
g(x) = 9x^2 + 1
\]

Thus, the rule for \( g(x) \) is:
\[
g(x) = 9x^2 + 1
\]

Would you like further explanation or have any questions?

Here are 5 related questions:
1. What effect does a horizontal shrink have on a graph?
2. How would the rule change if the translation was 2 units up instead?
3. What if instead of a shrink, there was a horizontal stretch by a factor of 3?
4. How does translating a graph vertically differ from horizontal translation?
5. How would the rule for \( g(x) \) change with a reflection over the x-axis?

**Tip:** When applying multiple transformations to a function, follow the sequence: horizontal changes (shrink/stretch), reflections, and then translations.

Let the graph of g be a horizontal shrink by a factor of 1/3, followed by a translation 1 unit up of the graph of f(x)= x^2. Write a rule for g.

Learn how to transform the quadratic function f(x) = x^2 by applying a horizontal shrink by a factor of 1/3 and a translation of 1 unit up. The final transformed function is g(x) = 9x^2 + 1. This detailed guide covers key algebraic concepts and function transformations, suitable for students in Grades 9-11.

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