What kind of transformation converts the graph of 

f

(

x

)

=

7

(

x

–

10

)

2

+

1

 into the graph of 

g

(

x

)

=

7

x

2

–

7

?

To solve the problem of transforming the graph of \( f(x) = 7(x - 10)^2 + 1 \) into the graph of \( g(x) = 7x^2 - 7 \), let's analyze the given functions.

### Step 1: Analyze the original function \( f(x) = 7(x - 10)^2 + 1 \)
- \( (x - 10)^2 \) indicates that the graph is a parabola shifted **10 units to the right**.
- The factor of 7 in front stretches the parabola vertically by a factor of 7.
- The "+1" shifts the entire graph **1 unit up**.

### Step 2: Analyze the transformed function \( g(x) = 7x^2 - 7 \)
- \( x^2 \) indicates that the graph is a parabola centered at the origin (no horizontal shift).
- The factor of 7 again stretches the parabola vertically by a factor of 7, just like in \( f(x) \).
- The "-7" shifts the entire graph **7 units down**.

### Step 3: Determine the transformation
- To go from \( (x - 10)^2 \) to \( x^2 \), the function must be shifted **10 units to the left** (which is the reverse of the initial shift to the right by 10 units).
- To go from \( +1 \) to \( -7 \), the graph must be shifted **8 units down** (since \( 1 - 8 = -7 \)).

### Conclusion
The transformation that converts \( f(x) = 7(x - 10)^2 + 1 \) into \( g(x) = 7x^2 - 7 \) involves:
- **Shifting the graph 10 units to the left** (horizontally).
- **Shifting the graph 8 units down** (vertically).

So, the correct transformation is a **horizontal shift of 10 units left** and a **vertical shift of 8 units down**.

Would you like to explore more about this transformation, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you describe the effect of the coefficient "7" on the shape of the parabola?
2. What transformation would be required to move \( f(x) = 7(x + 4)^2 + 3 \) to \( g(x) = 7x^2 + 9 \)?
3. How does shifting a graph horizontally differ from shifting it vertically in terms of the equation?
4. If \( f(x) = 7(x + 2)^2 - 5 \), what would be the new equation after shifting it 6 units up?
5. What would happen to the graph of \( f(x) = 7x^2 + 1 \) if you shifted it 3 units right and 4 units down?

**Tip:** Always start by identifying the transformations for horizontal shifts (inside the function) and vertical shifts (outside the function) separately before combining them.

Explore how to transform the quadratic function f(x) = 7(x - 10)^2 + 1 into g(x) = 7x^2 - 7 using horizontal and vertical shifts. Understand the effects of these transformations on the graph and equations. Suitable for advanced high school students.

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