Math Problem Statement

If f(x) = x^2, what is the equation of the graphed function?

Solution

The given function is f(x)=x2f(x) = x^2, and the problem asks for the equation of the graphed function shown in the image. The options provided are:

A. y=f(x3)2y = f(x - 3) - 2
B. y=f(x+3)2y = f(x + 3) - 2
C. y=f(x+2)3y = f(x + 2) - 3
D. y=f(x2)+3y = f(x - 2) + 3

Let's analyze the transformations step-by-step:

  1. Horizontal Shifts:
    A shift left or right is represented by f(x±h)f(x \pm h). If the graph shifts left by hh units, it becomes f(x+h)f(x + h), and if it shifts right by hh units, it becomes f(xh)f(x - h).

  2. Vertical Shifts:
    A vertical shift is represented by adding or subtracting a constant kk outside the function: f(x)±kf(x) \pm k.

Analysis of the Graph:

  1. Horizontal Shift:
    The graph of f(x)=x2f(x) = x^2 is typically centered at the origin (0,0)(0, 0). The graph in the image appears to have been shifted right by 2 units. This corresponds to the transformation f(x2)f(x - 2).

  2. Vertical Shift:
    The vertex of the parabola f(x)=x2f(x) = x^2 normally starts at (0,0)(0, 0). In the graph, it has moved up by 3 units. This corresponds to adding 3 to the function.

Combining Transformations:

Combining the horizontal and vertical transformations, the equation of the transformed function is:
y=f(x2)+3=(x2)2+3y = f(x - 2) + 3 = (x - 2)^2 + 3

This corresponds to option D: y=f(x2)+3y = f(x - 2) + 3.

Would you like more details or have any questions?

Related Questions:

  1. How do vertical transformations affect the graph of a quadratic function?
  2. How can you identify horizontal shifts in polynomial functions?
  3. What are the effects of reflecting a function over the x-axis or y-axis?
  4. How do you find the vertex of a transformed quadratic function?
  5. What is the general form of a transformed quadratic function?

Tip: Always check both the horizontal and vertical shifts independently to accurately determine the function's transformation.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Function Transformations

Formulas

f(x) = (x-h)^2 + k

Theorems

Quadratic Transformation Theorem

Suitable Grade Level

Grades 9-11