Math Problem Statement

The graph above is a transformation of the function x^2. Write an equation for the function graphed above.

Solution

Let's analyze the given graph, which is a transformation of the parent function x2x^2. The parent function of a parabola, f(x)=x2f(x) = x^2, has a vertex at (0,0)(0, 0) and opens upwards. From the graph you provided, we can identify the following transformations:

  1. Vertical shift: The vertex of the parabola has moved to (2,3)(-2, 3) from the origin. This indicates a horizontal shift of 2 units to the left and a vertical shift of 3 units up.

  2. Reflection: The parabola opens downward, suggesting a reflection over the x-axis.

  3. Vertical stretch or compression: There seems to be no vertical stretch or compression based on the scale of the graph, so the vertical scaling factor remains unchanged.

Given these transformations, the general form for a transformed quadratic function is: g(x)=a(xh)2+kg(x) = a(x - h)^2 + k where (h,k)(h, k) is the vertex, and aa controls the vertical stretch and the direction of the parabola.

  • The vertex is (2,3)(-2, 3), so h=2h = -2 and k=3k = 3.
  • Since the parabola is reflected over the x-axis, a=1a = -1.

Thus, the equation for the function is: g(x)=(x+2)2+3g(x) = -(x + 2)^2 + 3

Would you like further details on this or have any questions? Here are some related questions for deeper understanding:

  1. How does changing the value of aa affect the shape of a parabola?
  2. How would the equation change if the parabola were stretched vertically?
  3. What happens to the vertex of the parabola if you change the value of hh?
  4. Can you describe what happens if only kk changes in the equation of a parabola?
  5. How does the horizontal shift impact the graph of a quadratic function?

Tip: A reflection over the x-axis occurs when the leading coefficient aa in the quadratic equation is negative.

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Math Problem Analysis

Mathematical Concepts

Algebra
Transformations of Functions
Quadratic Functions

Formulas

General form of a quadratic function: f(x) = a(x - h)^2 + k
Transformations of parabolas: horizontal shift, vertical shift, reflection

Theorems

Quadratic Function Vertex Form

Suitable Grade Level

Grades 9-11