Answer :
Sure, let's describe the transformation process step-by-step for mapping [tex]\( f(x) = |x| \)[/tex] onto [tex]\( g(x) = 5\left|\frac{1}{4}(x+3)\right|-2 \)[/tex].
### Step-by-Step Transformation Process:
1. Horizontal Stretch by a Factor of 4:
- The expression inside the absolute value is [tex]\(\frac{1}{4}(x+3)\)[/tex]. The coefficient [tex]\(\frac{1}{4}\)[/tex] indicates a horizontal stretch.
- Since it's [tex]\(\frac{1}{4}\)[/tex] of [tex]\(x+3\)[/tex], it stretches the graph horizontally by a factor of [tex]\(4\)[/tex] (reciprocal of [tex]\(\frac{1}{4}\)[/tex]).
2. Shift Left by 3 Units:
- The term [tex]\(x + 3\)[/tex] inside the absolute value indicates a horizontal shift.
- Adding [tex]\(3\)[/tex] to [tex]\(x\)[/tex] shifts the graph to the left by [tex]\(3\)[/tex] units.
3. Vertical Stretch by a Factor of 5:
- The coefficient [tex]\(5\)[/tex] outside the absolute value function stretches the graph vertically by a factor of [tex]\(5\)[/tex].
- This means the y-values are multiplied by [tex]\(5\)[/tex].
4. Shift Down by 2 Units:
- The term [tex]\(-2\)[/tex] outside the absolute value function indicates a vertical shift.
- Subtracting [tex]\(2\)[/tex] moves the entire graph down by [tex]\(2\)[/tex] units.
5. Reflection:
- There is no negative sign in front of the absolute value function, so there is no reflection over the x-axis.
### Summary of Transformations:
Here is the list of transformations matched to their values as described above:
- Horizontal Stretch: by a factor of [tex]\(4\)[/tex].
- Shift Left: by [tex]\(3\)[/tex] units.
- Vertical Stretch: by a factor of [tex]\(5\)[/tex].
- Shift Down: by [tex]\(2\)[/tex] units.
- Reflection: None.
Thus, the function [tex]\( g(x) = 5\left|\frac{1}{4}(x+3)\right| - 2 \)[/tex] is obtained by applying these transformations to the base function [tex]\( f(x) = |x| \)[/tex].
### Step-by-Step Transformation Process:
1. Horizontal Stretch by a Factor of 4:
- The expression inside the absolute value is [tex]\(\frac{1}{4}(x+3)\)[/tex]. The coefficient [tex]\(\frac{1}{4}\)[/tex] indicates a horizontal stretch.
- Since it's [tex]\(\frac{1}{4}\)[/tex] of [tex]\(x+3\)[/tex], it stretches the graph horizontally by a factor of [tex]\(4\)[/tex] (reciprocal of [tex]\(\frac{1}{4}\)[/tex]).
2. Shift Left by 3 Units:
- The term [tex]\(x + 3\)[/tex] inside the absolute value indicates a horizontal shift.
- Adding [tex]\(3\)[/tex] to [tex]\(x\)[/tex] shifts the graph to the left by [tex]\(3\)[/tex] units.
3. Vertical Stretch by a Factor of 5:
- The coefficient [tex]\(5\)[/tex] outside the absolute value function stretches the graph vertically by a factor of [tex]\(5\)[/tex].
- This means the y-values are multiplied by [tex]\(5\)[/tex].
4. Shift Down by 2 Units:
- The term [tex]\(-2\)[/tex] outside the absolute value function indicates a vertical shift.
- Subtracting [tex]\(2\)[/tex] moves the entire graph down by [tex]\(2\)[/tex] units.
5. Reflection:
- There is no negative sign in front of the absolute value function, so there is no reflection over the x-axis.
### Summary of Transformations:
Here is the list of transformations matched to their values as described above:
- Horizontal Stretch: by a factor of [tex]\(4\)[/tex].
- Shift Left: by [tex]\(3\)[/tex] units.
- Vertical Stretch: by a factor of [tex]\(5\)[/tex].
- Shift Down: by [tex]\(2\)[/tex] units.
- Reflection: None.
Thus, the function [tex]\( g(x) = 5\left|\frac{1}{4}(x+3)\right| - 2 \)[/tex] is obtained by applying these transformations to the base function [tex]\( f(x) = |x| \)[/tex].
