Let's analyze the problem step-by-step.
First, let's consider the original function:
[tex]\[ y = 12x - 7 \][/tex]
Examining this equation, it's clear that it is a linear function in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the y-intercept.
- The slope [tex]\( m \)[/tex] of the original function is [tex]\( 12 \)[/tex].
- The y-intercept [tex]\( b \)[/tex] of the original function is [tex]\( -7 \)[/tex].
Next, let's consider the new function:
[tex]\[ y = 12x + 1 \][/tex]
Again, this is a linear function in the same form [tex]\( y = mx + b \)[/tex].
- The slope [tex]\( m \)[/tex] of the new function remains [tex]\( 12 \)[/tex], the same as in the original function.
- The y-intercept [tex]\( b \)[/tex] of the new function is [tex]\( 1 \)[/tex].
Now, let's compare the two functions:
1. The slope in both functions is [tex]\( 12 \)[/tex]. Since the slope remains unchanged, this means the steepness of the line does not change.
2. The y-intercept in the original function is [tex]\( -7 \)[/tex], while the y-intercept in the new function is [tex]\( 1 \)[/tex].
To understand how the graph changes:
- Since the slope does not change, we can disregard options A ("It would be less steep") and B ("It would be steeper").
- The change in the y-intercept from [tex]\( -7 \)[/tex] to [tex]\( 1 \)[/tex] means the graph of the new function is shifted vertically.
To specify the direction:
- The original y-intercept is at [tex]\( -7 \)[/tex].
- The new y-intercept is at [tex]\( 1 \)[/tex].
The y-intercept has increased from [tex]\( -7 \)[/tex] to [tex]\( 1 \)[/tex], which means the graph has moved upward.
Therefore, the correct answer is:
D. It would be shifted up.
