Hello I'm stuck with these two questions

The point (1,-2) is on the graph of . Describe the following transformations on , and determine the resulting point.

a) g(x)=f(x+1)-3

b) -f(2x)

to shift the graph of f(x) right by h and up by k,

y-k = f(x-h)

So, g(x) can be written

g+3 = f(x+1)

so, it is the graph of f shifted left 1 and down 3

-f(x) is f(x) reflected in the x-axis, since the sign of y changes.
f(2x) is the graph of f shrunk horizontally by a factor of 2, since 2x changes twice as fast as x. Everything shows up half as far away.

-f(x) is f(x)
reflected in the x-axis and shrunk horizontally by a factor of 2. Or, if you will, dilated by 1/2.

is the new point for a) (1,1)

Sure, I can help you with that!

To describe the transformations for each question, we need to understand the effects of each operation and then apply them to the given point (1,-2). Let's start with the first question:

a) g(x) = f(x+1) - 3

This can be broken down into two transformations applied successively: a horizontal shift and a vertical shift.

1. Horizontal shift: In f(x+1), the "+1" inside the parentheses represents a horizontal shift to the left by 1 unit. So, we shift the graph of f(x) one unit to the left.

2. Vertical shift: After the horizontal shift, we subtract 3, which indicates a vertical shift downwards by 3 units. So, we shift the graph of f(x+1) three units downwards.

Now let's determine the resulting point using the given point (1,-2) as our starting point:

a) g(x) = f(x+1) - 3

Horizontal shift: (1+1,-2) = (2,-2)
Vertical shift: (2,-2) - 3 = (2,-5)

Therefore, the resulting point on the graph of g(x) is (2,-5).

Now let's move on to the second question:

b) -f(2x)

This involves one transformation, which is a reflection followed by a horizontal compression.

1. Reflection: The "-" sign in front of f(2x) indicates a reflection across the x-axis. So, we reflect the graph of f(2x) upside down.

2. Horizontal compression: The "2" in f(2x) represents a horizontal compression. The compression factor is determined by taking the reciprocal of the coefficient, so the graph becomes twice as compressed.

Now let's determine the resulting point using the given point (1,-2) as our starting point:

b) -f(2x)

Reflection: (1,-2) reflected across the x-axis is (1,2)
Horizontal compression: (1/2,2)

Therefore, the resulting point on the graph of -f(2x) is (1/2,2).

I hope this helps! Let me know if you have any further questions.