If f(x) =5(x) is reflected across the x-axis, then vertically shrunk by 2/3 then moved 3 units right, what is the new equation for g(x)?

Question

options to chose from:

A) g(x)= -3/2• 5(x-3)

B) g(x)= 2/3• 5(x-3)

C) g(x) = -2/3•5(x+3)

D) g(x)= -2/3• 5(x-3)

Answer 1

We start with the equation y=f(x)=5x, which is a straight line with a positive slope of +5, passing by the origin (0,0). Our line becomes zero at x=0

1.     The first step is to reflect the line across the x axis, this reflection means that for a certain value of x we correspond a negative value of y. Thus, y=-5x becomes the reflected line across the x axis, it is a line with a negative slope of -5, passing by the origin (0,0)

2.     Secondly, we are being asked to vertically shrink it by 2/3.  Scaled vertically by 2/3 means 2/3 times the y value for a given x.  Hence, the equation we obtained in step 1 becomes y=(2/3)(-5x). This becomes now a straight line passing by the origin (0,0) with a negative slope of -10/3

3.     Finally, they ask as to move the line 3 units to the right. Shifting to the right by 3 units, we substitute x by (x-3), because our new zero for the equation will be at the point x=3.  Therefore, our equation obtained in step 2 becomes y=g(x)=(2/3)[-5(x-3)], rewritten as

g(x)=-2/3 [5(x-3]. In conclusion, our correct answer is D.

This function g(x) can be rewritten as y=g(x)=-10x/3 +10. Notice that it is a straight line with a negative slope of -10/3. Also, g(3) =0, which satisfies that our zero is at x=3. In addition, g(0)=10, meaning that the line intercepts the y axis at y=10