A quadratic function y, equals, f, of, xy=f(x) is plotted on a graph and the vertex of the resulting parabola is left parenthesis, minus, 3, comma, 3, right parenthesis(−3,3). What is the vertex of the function defined as g, of, x, equals, f, of, x, plus, 3g(x)=f(x+3)?

The problem involves the transformation of the quadratic function \( f(x) \) to a new function \( g(x) = f(x+3) \). To determine the vertex of \( g(x) \), we can use the transformation properties of parabolas.

The vertex of the original quadratic function \( f(x) \) is at \( (-3, 3) \). When you replace \( x \) with \( x+3 \) in the function, the effect is a **horizontal shift** to the left by 3 units.

### Here's how the transformation works:
- The general rule is that replacing \( x \) with \( x + h \) in a function shifts the graph horizontally by \( h \) units.
- In this case, \( g(x) = f(x+3) \), so this shifts the graph 3 units to the **left**.

Since the vertex of \( f(x) \) is originally at \( (-3, 3) \), shifting it 3 units to the left moves it to \( (-6, 3) \).

### Therefore, the vertex of the function \( g(x) = f(x+3) \) is:
\[
(-6, 3)
\]

Would you like a more detailed explanation, or do you have any additional questions?

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Here are 5 related questions to explore further:
1. What is the equation of the parabola if the function \( f(x) \) is given explicitly?
2. How do vertical shifts affect the vertex of a quadratic function?
3. What other transformations could we apply to the function, such as reflections or stretches?
4. How does the form \( f(x) = a(x-h)^2 + k \) help in identifying the vertex easily?
5. What happens if we combine horizontal and vertical shifts in the same transformation?

**Tip**: Remember, horizontal transformations affect the \( x \)-coordinate, while vertical transformations affect the \( y \)-coordinate of the vertex!

A quadratic function y = f(x) is plotted on a graph and the vertex of the resulting parabola is (-3, 3). What is the vertex of the function defined as g(x) = f(x + 3)?

Explore how to transform the vertex of a quadratic function. Given f(x) with a vertex at (-3, 3), learn how the vertex shifts with the function g(x) = f(x + 3), moving it to (-6, 3). Understand the horizontal shift in quadratic graph transformations. Ideal for students in Grades 9-11.

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