To determine the effect of the number 4 in the function [tex]\( y - 1 = (4x)^2 + 7 \)[/tex] compared to the base function [tex]\( v = x^2 \)[/tex], we need to analyze the transformation steps.
1. Standard Function Comparison:
We start with the standard parabola function [tex]\( v = x^2 \)[/tex].
2. Identify Transformation Form:
The given function is [tex]\( y - 1 = (4x)^2 + 7 \)[/tex]. Let's rearrange it to see the transformation more clearly:
[tex]\[
y - 1 = 16x^2 + 7
\][/tex]
Simplifying this, we get:
[tex]\[
y = 16x^2 + 8
\][/tex]
3. Understand Coefficient Inside the Function:
- In [tex]\( y = v(x) \)[/tex], where [tex]\( v(x) = x^2 \)[/tex], any modification within the argument of the function [tex]\( x \)[/tex] (that is, [tex]\( 4x \)[/tex] in this case) implies a horizontal transformation.
- [tex]\( 4x \)[/tex] indicates that the [tex]\( x \)[/tex] value is being scaled.
4. Horizontal Transformations:
Transformations inside the function's argument, like [tex]\( (4x) \)[/tex], affect the graph horizontally. Specifically:
- If we have [tex]\( f(x) = v(kx) \)[/tex] where [tex]\( k > 1 \)[/tex], it shrinks the graph horizontally by a factor of [tex]\( \frac{1}{k} \)[/tex].
5. Apply the Shrink Factor:
- Here, [tex]\( k = 4 \)[/tex], so [tex]\( (4x) \)[/tex] shrinks the graph horizontally by a factor of [tex]\( \frac{1}{4} \)[/tex].
Thus, the number 4 in [tex]\( (4x)^2 \)[/tex] shrinks the graph horizontally to [tex]\( \frac{1}{4} \)[/tex] the original width of the graph of [tex]\( v = x^2 \)[/tex].
Therefore, the correct answer is:
D. It shrinks the graph horizontally to [tex]\( \frac{1}{4} \)[/tex] the original width.
