Find the vertical transformation constant in a quadratic function

Q: How do you write g(x) in terms of f(x) when g(x)=4(x-4)^2-4 and f(x)=(x-4)^2-4?

Since f(x)=(x-4)^2-4, we get (x-4)^2=f(x)+4, so g(x)=4(f(x)+4)-4=4f(x)+12.

Q: What transformations does g(x)=4f(x)+12 represent?

It represents a vertical dilation by factor 4 and a vertical shift up by 12 units.

Question

7. The function f(x) is defined by
f(x) = (x-4)^2 - 4
The function g(x) is defined by
g(x) = 4(x-4)^2 - 4

4[(x-4)^2 - 4] + k
[a = 4]
4(x-4)^2 - 16 + k
-16 + k = -4
k = -4 + 16
k = 12

• Vertical dilation by a factor of 4.
• Vertical translation towards the positive y-axis by 12 units

g(x) = 4f(x) + 12

Acemy AI · Accepted Answer

> **Key concept**: This is a good example of matching a transformed graph to an algebraic form. The vertical factor and vertical shift can be read from the structure of the function.

### Step by step
We are given
$$
f(x)=(x-4)^2-4
$$
and
$$
g(x)=4(x-4)^2-4.
$$

Write $g(x)$ in terms of $f(x)$:

$$
f(x)=(x-4)^2-4
\Rightarrow (x-4)^2=f(x)+4.
$$

Substitute into $g(x)$:
$$
g(x)=4\bigl((x-4)^2\bigr)-4
=4\bigl(f(x)+4\bigr)-4
=4f(x)+16-4
=4f(x)+12.
$$

So the transformation is:
- vertical dilation by factor $4$
- vertical translation up by $12$

$$
\boxed{g(x)=4f(x)+12}
$$

The constant $12$ appears after rewriting the shifted square in terms of $f(x)$.

### Pitfall alert
A frequent slip is to think the $-4$ at the end means the graph is shifted down 4 only. Here the multiplication by 4 changes the constant term first, so you must rewrite $(x-4)^2$ using $f(x)+4$ before simplifying.

### Try different conditions
If the coefficient had been $2$ instead of $4$, the same method would give
$$
g(x)=2f(x)+k.
$$
Then you would solve for $k$ by matching the constant term. The algebra is identical; only the multiplier changes.

### Further reading
vertical dilation, quadratic vertex form, function composition

Question

Find the vertical transformation constant in a quadratic function

Expert Verified Solution

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FAQ

How do you write g(x) in terms of f(x) when g(x)=4(x-4)^2-4 and f(x)=(x-4)^2-4?

What transformations does g(x)=4f(x)+12 represent?