given the parent function f(x)=x^2, what does the 7 represent in g(x) = 7(x+3)^2-2

In the function \( g(x) = 7(x+3)^2 - 2 \), the number 7 is a **vertical stretch factor**. It scales the basic parabola \( f(x) = x^2 \) vertically by a factor of 7. Here's what that means:

- For the parent function \( f(x) = x^2 \), the points (1,1), (2,4), etc., are typical values.
- In \( g(x) \), the 7 multiplies the entire parabola, making it steeper. Every y-value of \( f(x) = x^2 \) gets multiplied by 7. For example, instead of (1,1), you’d have (1,7) and instead of (2,4), you’d have (2,28).

In general:
- When this coefficient is greater than 1, the parabola stretches vertically.
- When it's between 0 and 1, the parabola is compressed vertically.
- When it's negative, the parabola flips upside down.

Would you like more details on the transformation or a visual breakdown?

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Here are 5 related questions:
1. What does the -2 represent in \( g(x) = 7(x+3)^2 - 2 \)?
2. How would the graph change if the 7 were negative?
3. What would happen if the coefficient 7 were replaced with 1/2?
4. How does the transformation shift the vertex of the parabola?
5. What is the impact of the +3 inside the parentheses?

**Tip**: To understand transformations, always break them down into separate vertical and horizontal shifts, stretches, and reflections.

Given the parent function f(x)=x^2, what does the 7 represent in g(x) = 7(x+3)^2 - 2?

Explore the transformation of the parent function f(x) = x^2, focusing on the vertical stretch factor represented by the 7 in g(x) = 7(x+3)^2 - 2. This detailed explanation covers the impact of the coefficient 7 on the parabola, including how it affects the graph's shape. Suitable for Grades 9-10.

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