desrribe what happens to the equation of a function when you make each change to its graph

a)compress horizontally by a factor of 1/4, and vertically by a factor of 1/2
b)expand horizontally by a factor of 2, then translate 3 units left

To understand what happens to the equation of a function when you make changes to its graph, you need to understand how each change transforms the original function. Let's go through each scenario and explain how it affects the equation:

a) Compress horizontally by a factor of 1/4 and vertically by a factor of 1/2:
When you compress a function horizontally by a factor of 1/4, it means the x-values get multiplied by 1/4. This results in the graph becoming narrower or squeezed in horizontally. Mathematically, if the original function is f(x), the compressed function would be f(4x).

Simultaneously, when you vertically compress a function by a factor of 1/2, it means the y-values get multiplied by 1/2. This results in the graph becoming shorter or squeezed in vertically. The vertical compression is represented by multiplying the entire function by 1/2. So, if the original function is f(x), the compressed function would be (1/2) * f(4x).

In summary, when you compress a function horizontally by 1/4 and vertically by 1/2, the equation transforms from f(x) to (1/2) * f(4x).

b) Expand horizontally by a factor of 2 and translate 3 units left:
When you expand a function horizontally by a factor of 2, it means the x-values get divided by 2. This results in the graph becoming wider or stretched out horizontally. Mathematically, if the original function is f(x), the expanded function would be f(x/2).

Additionally, when you translate a function 3 units to the left, it means you subtract 3 from the x-values. This shift moves the graph horizontally to the left. The horizontal translation is represented by subtracting 3 from the x-values. So, if the original function is f(x), the translated function would be f(x - 3).

In summary, when you expand a function horizontally by a factor of 2 and translate it 3 units to the left, the equation transforms from f(x) to f((x/2) - 3).

By understanding how these transformations affect the original function, you can adjust the equation accordingly to match the resulting graph.