Math Problem Statement

Graph the following function using transformations. y equals 4 left parenthesis x minus 1 right parenthesis squared minus 3

Solution

To graph the function y=4(x1)23y = 4(x - 1)^2 - 3 using transformations, we can break it down step by step. This function is a transformed version of the basic quadratic function y=x2y = x^2.

Step 1: Start with the parent function

The parent function is: y=x2y = x^2 This is a basic parabola that opens upward with its vertex at (0,0)(0, 0).

Step 2: Apply horizontal shift

The term (x1)(x - 1) indicates a horizontal shift of the graph to the right by 1 unit. Now, the vertex moves from (0,0)(0, 0) to (1,0)(1, 0).

Step 3: Apply vertical stretch

The coefficient 4 in front of (x1)2(x - 1)^2 means there is a vertical stretch by a factor of 4. This makes the parabola narrower compared to the standard y=x2y = x^2.

Step 4: Apply vertical shift

The term 3-3 at the end of the function represents a vertical shift. The graph is shifted downward by 3 units. Now, the vertex moves from (1,0)(1, 0) to (1,3)(1, -3).

Summary of Transformations:

  • Horizontal shift: Right 1 unit.
  • Vertical stretch: Stretched by a factor of 4.
  • Vertical shift: Down 3 units.

The final graph is a parabola with the vertex at (1,3)(1, -3), opening upwards, and narrower than the standard parabola due to the vertical stretch.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Graph Transformations

Formulas

y = a(x - h)^2 + k

Theorems

Transformations of quadratic functions

Suitable Grade Level

Grades 9-11