Math Problem Statement

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

Solution

The graph shown represents a transformation of the basic quadratic function f(x)=x2f(x) = x^2.

Analyzing the Transformation:

  1. Reflection: The graph opens downward, indicating a reflection over the x-axis. This means the leading coefficient is negative.

  2. Vertical Stretch/Compression: The graph appears to be stretched vertically since it is narrower than the graph of x2x^2.

  3. Horizontal Translation: The vertex of the graph is located at (2,4)(2, 4), indicating that the graph has been translated 2 units to the right.

  4. Vertical Translation: The vertex also indicates a translation 4 units up.

Formulating the Equation:

Given the transformations, the equation can be written as: g(x)=a(xh)2+kg(x) = a(x-h)^2 + k where (h,k)(h, k) is the vertex of the parabola.

Substituting the values h=2h = 2 and k=4k = 4, and considering the reflection over the x-axis (which makes aa negative), the equation becomes: g(x)=a(x2)2+4g(x) = a(x-2)^2 + 4

To determine the value of aa, observe that when x=0x = 0, g(x)=5g(x) = -5: 5=a(02)2+4-5 = a(0-2)^2 + 4 5=4a+4-5 = 4a + 4 4a=9a=944a = -9 \quad \Rightarrow \quad a = -\frac{9}{4}

Thus, the equation of the transformed function is: g(x)=94(x2)2+4g(x) = -\frac{9}{4}(x-2)^2 + 4

Would you like more details on any part of the process? Here are five related questions you might find useful:

  1. What are the effects of different values of aa on the shape of a parabola?
  2. How does the vertex form of a quadratic function help in identifying transformations?
  3. What are other possible transformations of a quadratic function?
  4. Can this method be applied to other types of functions, like cubic or exponential?
  5. How would the graph change if the quadratic function had a different vertex?

Tip: When identifying transformations, always start by locating the vertex and determining if the parabola opens upwards or downwards!

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Math Problem Analysis

Mathematical Concepts

Transformation of Functions
Quadratic Functions
Vertex Form of a Parabola

Formulas

g(x) = a(x-h)^2 + k
Quadratic Formula

Theorems

Vertex form of quadratic functions

Suitable Grade Level

Grades 9-12