To determine the horizontal and vertical shifts for the given absolute value function [tex]\( f(x) = |3x + 91| + 6 \)[/tex], let's go through the steps:
1. Vertical Shift: This is the simpler of the two shifts to determine. The term "+6" at the end of the function indicates a vertical shift. When you add a constant to a function, it shifts the graph vertically. In this case, the graph of the function is shifted 6 units upwards. Thus, the vertical shift is +6 units.
2. Horizontal Shift: To understand the horizontal shift, we need to look inside the absolute value function itself, specifically at [tex]\( |3x + 91| \)[/tex].
- Normally, an absolute value function without a coefficient and a constant will look like [tex]\( |x| \)[/tex], where the vertex or the "turning point" of the function is at [tex]\( x = 0 \)[/tex].
- For [tex]\( |3x + 91| \)[/tex], we need to determine the value of [tex]\( x \)[/tex] that makes the expression inside the absolute value equal to zero. This is the point where the "turning point" of the absolute value function occurs.
- Solve the equation [tex]\( 3x + 91 = 0 \)[/tex]:
[tex]\[
3x + 91 = 0 \\
3x = -91 \\
x = -\frac{91}{3}
\][/tex]
- Therefore, the horizontal shift is [tex]\( -\frac{91}{3} \)[/tex].
So, combining the results, the graph of the function [tex]\( f(x) = |3x + 91| + 6 \)[/tex] experiences:
- A horizontal shift of [tex]\( -\frac{91}{3} \)[/tex] units (approximately -30.33 units to the left).
- A vertical shift of +6 units upwards.
Therefore, the horizontal and vertical shifts for the given function are approximately -30.33 units to the left and 6 units upward, respectively.
