Math Problem Statement

Plot the points on graph

Plot the points on the graph of the function f(x) = |x - 3|.

Solution

To plot the graph of the function f(x)=x3f(x) = |x - 3|, let's break it down step-by-step.

Understanding the Function:

The given function is an absolute value function, which shifts the standard absolute value graph f(x)=xf(x) = |x|.

  • The parent function f(x)=xf(x) = |x| forms a "V" shape, with the vertex at the origin.
  • The function f(x)=x3f(x) = |x - 3| translates this graph horizontally by 3 units to the right. This is because the expression (x3)(x - 3) shifts the graph right by 3 units.

Key Points to Plot:

  1. Vertex: The vertex is at (3,0)(3, 0), which is the point where the graph changes direction.
  2. Other Points: Since the graph is symmetric, choose values for xx and find the corresponding yy-values:
  • For x=0x = 0, f(0)=03=3f(0) = |0 - 3| = 3. So, (0,3)(0, 3) is a point.
  • For x=1x = 1, f(1)=13=2f(1) = |1 - 3| = 2. So, (1,2)(1, 2) is a point.
  • For x=2x = 2, f(2)=23=1f(2) = |2 - 3| = 1. So, (2,1)(2, 1) is a point.
  • For x=4x = 4, f(4)=43=1f(4) = |4 - 3| = 1. So, (4,1)(4, 1) is a point.
  • For x=5x = 5, f(5)=53=2f(5) = |5 - 3| = 2. So, (5,2)(5, 2) is a point.

Plot:

  • Start by plotting the vertex at (3,0)(3, 0).
  • Then plot the points (0,3)(0, 3), (1,2)(1, 2), (2,1)(2, 1), (4,1)(4, 1), and (5,2)(5, 2).
  • Connect these points, and remember the graph forms a V-shape with symmetry about the line x=3x = 3.

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Math Problem Analysis

Mathematical Concepts

Absolute Value Functions
Graphing Functions

Formulas

f(x) = |x - h| where h shifts the graph horizontally

Theorems

Translation Theorem (Horizontal Shifts in Graphs)

Suitable Grade Level

Grades 8-10