Find the equation that results after the parent function, $y=\log _7 x$, is shifted 4 units up, 3 units to the right and reflected over the $x$-axis.

The final transformed equation after shifting 4 units up, 3 units to the right, and reflecting over the x-axis is y = − lo g 7 ​ ( x − 3 ) − 4 . These transformations are applied in a specific order to achieve the correct result.
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Answered by Anonymous | 2025-07-07

Apply horizontal shift: y = lo g 7 ​ ( x − 3 ) .
Apply vertical shift: y = lo g 7 ​ ( x − 3 ) + 4 .
Reflect over the x-axis: y = − ( lo g 7 ​ ( x − 3 ) + 4 ) .
Simplify: y = − lo g 7 ​ ( x − 3 ) − 4 .
The final equation is y = − lo g 7 ​ ( x − 3 ) − 4 ​ .

Explanation

Understanding the Problem We are given the parent function y = lo g 7 ​ x . We need to apply the following transformations in order: a horizontal shift of 3 units to the right, a vertical shift of 4 units up, and a reflection over the x-axis.

Applying Horizontal Shift First, we apply the horizontal shift of 3 units to the right. This means we replace x with ( x − 3 ) in the original equation: y = lo g 7 ​ ( x − 3 ) .

Applying Vertical Shift Next, we apply the vertical shift of 4 units up. This means we add 4 to the equation: y = lo g 7 ​ ( x − 3 ) + 4 .

Applying Reflection over x-axis Finally, we reflect the equation over the x-axis. This means we multiply the entire equation by -1: y = − ( lo g 7 ​ ( x − 3 ) + 4 ) . Distributing the negative sign, we get: y = − lo g 7 ​ ( x − 3 ) − 4 .

Final Equation Therefore, the final equation after all the transformations is y = − lo g 7 ​ ( x − 3 ) − 4 .


Examples
Understanding transformations of functions is crucial in many fields. For example, in seismology, analyzing seismic waves involves understanding shifts and reflections of wave functions. Similarly, in image processing, transformations like shifting and reflecting are used to manipulate images. In finance, understanding how changes in parameters affect a model can be seen as transformations of a function. These transformations help us understand and predict the behavior of systems in various real-world applications.

Answered by GinnyAnswer | 2025-07-07