Write The Function For S(X) If The Absolute Value Parent Function F(X)- Lx Is Reflected Across The X-Axis, Horizontally Stretched By A Factor Of 4, And Then Translated Horizontally 3 Units Right And Vertically 7 Units Up. Function: F(X)=4] X- +1
The Correct Answer and Explanation is :
To derive the function ( S(x) ) from the absolute value parent function ( F(x) = |x| ), we will apply the specified transformations step by step:
- Reflection Across the X-Axis: Reflecting ( F(x) = |x| ) across the x-axis changes the function to ( -|x| ). This transformation inverts the graph, flipping it upside down.
- Horizontal Stretch by a Factor of 4: To horizontally stretch the function by a factor of 4, we replace ( x ) with ( \frac{x}{4} ). This results in the function ( -\left| \frac{x}{4} \right| ). Horizontal stretches affect the x-values, making the graph wider.
- Translation 3 Units to the Right: Translating the function 3 units to the right involves replacing ( x ) with ( x – 3 ). The function becomes ( -\left| \frac{x – 3}{4} \right| ). Horizontal translations shift the graph along the x-axis.
- Translation 7 Units Up: Finally, translating the function 7 units up adds 7 to the entire expression, resulting in ( S(x) = -\left| \frac{x – 3}{4} \right| + 7 ). Vertical translations move the graph along the y-axis.
Thus, the transformed function is:
[ S(x) = -\left| \frac{x – 3}{4} \right| + 7 ]
Explanation of Transformations:
- Reflection Across the X-Axis: Reflecting a function across the x-axis changes the sign of all its output values. For the absolute value function ( F(x) = |x| ), this reflection results in ( -|x| ), flipping the graph upside down.
- Horizontal Stretch by a Factor of 4: A horizontal stretch by a factor of 4 means that for every x-value, the corresponding y-value remains the same, but the graph becomes wider. This is achieved by replacing ( x ) with ( \frac{x}{4} ), which scales the x-values by a factor of 4.
- Translation 3 Units to the Right: Translating a function to the right by 3 units involves replacing ( x ) with ( x – 3 ). This shifts the entire graph 3 units along the x-axis, moving all points to the right.
- Translation 7 Units Up: Translating a function upward by 7 units adds 7 to the entire function. This shifts the graph 7 units along the y-axis, moving all points up.
By applying these transformations in sequence, we obtain the final function ( S(x) = -\left| \frac{x – 3}{4} \right| + 7 ), which represents the absolute value parent function reflected across the x-axis, horizontally stretched by a factor of 4, and translated 3 units to the right and 7 units up.