The equation y = -4log(9(x + 12)) + 11 accurately represents the logarithmic function after all the specified transformations.
Here's the equation of the transformed logarithmic function, along with justifications for each transformation:
Equation: y = -4log(9(x + 12)) + 11
Justifications:
1. Vertical Stretch by 4:
- The coefficient "-4" outside the logarithm function stretches the graph vertically by a factor of 4.
- It makes the graph 4 times taller compared to the original logarithmic function.
2. Reflection in the x-axis:
- The negative sign in front of the logarithm function reflects the graph across the x-axis.
- It flips the graph upside down.
3. Horizontal Compression by 1/9:
- The coefficient "9" inside the logarithm function compresses the graph horizontally by a factor of 1/9.
- It makes the graph 9 times narrower compared to the original function.
- The parentheses around "x + 12" ensure that the compression affects the entire x-expression.
4. Vertical Shift Up 11 Units:
- The constant "11" added to the end of the equation shifts the graph 11 units upward.
- It moves the entire graph vertically upwards.
5. Horizontal Shift Left 12 Units:
- The "-12" inside the parentheses with "x" shifts the graph 12 units to the left.
- It moves the entire graph horizontally to the left.
Complete the question:
Write the equation of a logarithmic function that has been vertically stretched by 4, reflected in the x-axis, horizontally compressed by 1/9, moved 11 units up and 12 units left. Justify your answer.