Question

use the drawing tool(s) to form the correct answer on the provided graph.
the points in the table below are on the linear function f.

x	0	1	2	3	4
f(x)	-4	-2	0	2	4

function g is a transformation of function f using a horizontal shift 3 units left and a vertical compression by a factor of \\(\frac{1}{2}\\). plot the corresponding points on function g.
drawing tools
select
point
click on a tool to begin drawing.
undo
reset

Explanation:

Step1: Recall transformation rules

For a function \( y = f(x) \), a horizontal shift \( h \) units left is \( y = f(x + h) \), and a vertical compression by factor \( k \) is \( y = k \cdot f(x) \). Here, \( h = 3 \) (left shift) and \( k=\frac{1}{2} \), so \( g(x)=\frac{1}{2}f(x + 3) \).

Step2: Transform each point \((x,f(x))\)

• For \((0, - 4)\):
• Horizontal shift left 3: \( x'=0 - 3=-3 \)
• Vertical compression: \( y'=\frac{1}{2}\times(-4)=-2 \) → Point \((-3, -2)\)
• For \((1, - 2)\):
• \( x'=1 - 3=-2 \)
• \( y'=\frac{1}{2}\times(-2)=-1 \) → Point \((-2, -1)\)
• For \((2, 0)\):
• \( x'=2 - 3=-1 \)
• \( y'=\frac{1}{2}\times0 = 0 \) → Point \((-1, 0)\)
• For \((3, 2)\):
• \( x'=3 - 3=0 \)
• \( y'=\frac{1}{2}\times2 = 1 \) → Point \((0, 1)\)
• For \((4, 4)\):
• \( x'=4 - 3=1 \)
• \( y'=\frac{1}{2}\times4 = 2 \) → Point \((1, 2)\)

Answer:

The transformed points for \( g(x) \) are \((-3, -2)\), \((-2, -1)\), \((-1, 0)\), \((0, 1)\), and \((1, 2)\). These points should be plotted on the graph.