5 questions on graphs and functions

Question

1. If a function is multiplied by a coefficient, what will happen to the graph of the function?

2. What does multiplying x by a number greater than one create?

3. What happens when we multiply x by a number between 0 and 1?

4. In order to obtain a reflection over the y axis what do we have to do to x?

5. How do we obtain a reflection over the x axis?

Answer 1

1. If a function is multiplied by a coefficient, what will happen to the graph of the function?

Examples: g(x) = 4f(x)...function g's y values will be uniformly multiplied by 4, accomplishing a vertical stretch.

               g(x) = -3f(x)...this will do a vertical stretch and reflection in x-axis, i.e. formerly positive y values become negative                                     & vice versa.

2. What does multiplying x by a number greater than one create?  Poorly phrased.  One example: y = sin 2x takes a normal sine graph that has a period of two pi, and speeds it up, so to speak, so that the new period is pi; this is a horizontal compression.  At a more elementary level, comparing f(x) = x and g(x) = 2x, that's also a horizontal compression (due to increased slope for the original line).

3. What happens when we multiply x by a number between 0 and 1? The product is < the original number. Example:

  g(x) = 0.3f(x) is a vertical compression of original f(x).

4. In order to obtain a reflection over the y axis what do we have to do to x? Replace x by its opposite: g(x) = f(-x).

5. How do we obtain a reflection over the x axis? original y values are replaced by their opposites, i.e. g(x) = -f(x)

Answer 2

1)

 

You get a vertical stretch if the coefficient if greater than zero.  Horizontal stretch if the coefficient is less than zero.

 

 

2)

 

You get a positive slope of the line.

 

 

3)

 

You get a line between 0 degrees and 45 degrees.

 

 

4)

 

Negate x.

 

 

5)

 

Negate the entire function.  Or simply change the signs of each term of the function.  You may need to perform a vertical shift as a second step if the key part of the graph is not on the x-axis.