Math Problem Statement

graph f(x)=-3|1/2x-1| -11 give me ponits

Solution

To graph the function f(x)=312x111f(x) = -3 \left| \frac{1}{2}x - 1 \right| - 11, we proceed by analyzing and selecting appropriate points.

Step-by-Step Breakdown:

  1. Vertex Calculation:
    The expression inside the absolute value becomes zero at 12x1=0\frac{1}{2}x - 1 = 0. Solving for xx,
    x=2x = 2
    So, the vertex of the graph is at (2,11)(2, -11).

  2. Behavior Left and Right of the Vertex:

    • For x<2x < 2, 12x1=(12x1)\left| \frac{1}{2}x - 1 \right| = -\left( \frac{1}{2}x - 1 \right)
      Therefore, f(x)=3(12x+1)11f(x) = -3 \left( -\frac{1}{2}x + 1 \right) - 11
      Simplifying,
      f(x)=32x14f(x) = \frac{3}{2}x - 14

    • For x2x \geq 2, 12x1=12x1\left| \frac{1}{2}x - 1 \right| = \frac{1}{2}x - 1
      Therefore, f(x)=3(12x1)11f(x) = -3 \left( \frac{1}{2}x - 1 \right) - 11
      Simplifying,
      f(x)=32x8f(x) = -\frac{3}{2}x - 8

Selected Points:

  1. Vertex: (2,11)(2, -11)

  2. To the left of the vertex:

    • At x=0x = 0,
      f(0)=32(0)14=14f(0) = \frac{3}{2}(0) - 14 = -14
      So, (0,14)(0, -14)
    • At x=1x = 1,
      f(1)=32(1)14=12.5f(1) = \frac{3}{2}(1) - 14 = -12.5
      So, (1,12.5)(1, -12.5)
  3. To the right of the vertex:

    • At x=3x = 3,
      f(3)=32(3)8=12.5f(3) = -\frac{3}{2}(3) - 8 = -12.5
      So, (3,12.5)(3, -12.5)
    • At x=4x = 4,
      f(4)=32(4)8=14f(4) = -\frac{3}{2}(4) - 8 = -14
      So, (4,14)(4, -14)

Table of Points:

xxf(x)f(x)
0-14
1-12.5
2-11
3-12.5
4-14

Would you like a plot of this graph or further analysis?


Follow-up Questions:

  1. What are the steps for finding the vertex of any absolute value function?
  2. How does the slope change before and after the vertex?
  3. Why does the graph form a "V" shape instead of a straight line?
  4. How would shifting the function vertically affect the graph?
  5. Can you explain the role of the negative sign in front of the absolute value here?

Tip:

Remember, the vertex of an absolute value graph is always located where the inside of the absolute value equals zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Functions
Graphing Piecewise Functions

Formulas

Vertex of absolute value function: x = h when |ax - h| = 0
Piecewise function for absolute value: f(x) = a|bx - c| + d
Slope-intercept form: y = mx + b

Theorems

Definition of Absolute Value
Transformation of Absolute Value Functions

Suitable Grade Level

Grades 9-11