Alright, let's analyze the given function [tex]\( y - 1 = (4x)^2 + 7 \)[/tex] and compare it to the standard parabolic function [tex]\( y = x^2 \)[/tex].
1. Rewrite the Equation in Standard Form:
First, we rewrite the given function to a more familiar form:
[tex]\[
y - 1 = (4x)^2 + 7
\][/tex]
Simplifying the equation, we get:
[tex]\[
y = (4x)^2 + 7 + 1
\][/tex]
[tex]\[
y = 16x^2 + 8
\][/tex]
2. Analyze the Effect of the Coefficient:
The standard parabolic function is [tex]\( y = x^2 \)[/tex]. In this case, we have [tex]\( y = (4x)^2 \)[/tex].
3. Extracting the Transformation from [tex]\( (4x)^2 \)[/tex]:
- In the expression [tex]\( (4x)^2 \)[/tex], the coefficient 4 is within the parentheses and affects the input variable [tex]\( x \)[/tex].
- To see the effect, consider any value of [tex]\( x \)[/tex]. For example, if [tex]\( x = 1 \)[/tex] in [tex]\( y = x^2 \)[/tex], [tex]\( y = 1^2 = 1 \)[/tex]. In the transformed function, if [tex]\( x = 1 \)[/tex]:
[tex]\[
y = (4 \cdot 1)^2 = 16
\][/tex]
If [tex]\( x = 0.25 \)[/tex]:
[tex]\[
y = (4 \cdot 0.25)^2 = (1)^2 = 1
\][/tex]
4. Understanding the Horizontal Effect:
Notice that if [tex]\( x \)[/tex] changes by 0.25 in the transformed function, the output [tex]\( y \)[/tex] reaches the same value as when [tex]\( x \)[/tex] changes by 1 in the original function. This indicates that the graph is shrunk by a factor of 4 in the horizontal direction.
So, the number 4 applied to [tex]\( x \)[/tex] in [tex]\( (4x) \)[/tex] causes the graph to shrink horizontally. The horizontal distance is reduced to one-fourth of the original width.
Thus, the correct answer is:
D. It shrinks the graph horizontally to [tex]\( \frac{1}{4} \)[/tex] the original width.
