Math Problem Statement

a quadratic parent function is vertically compresed by two thrids, reflected across the x axis, translated left three and up 7, write the equation in vertex form

Solution

To write the equation of the described quadratic function in vertex form, follow the steps:

Vertex form:

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

Where:

  • aa determines the vertical stretch/compression and reflection.
  • (h,k)(h, k) is the vertex of the parabola.

Step-by-step solution:

  1. Start with the parent quadratic function: y=x2y = x^2

  2. Apply the vertical compression by 23\frac{2}{3}: y=23x2y = \frac{2}{3}x^2

  3. Reflect across the x-axis (negate aa): y=23x2y = -\frac{2}{3}x^2

  4. Translate left by 3 units (replace xx with x+3x+3): y=23(x+3)2y = -\frac{2}{3}(x+3)^2

  5. Translate up by 7 units (add 7 to the entire equation): y=23(x+3)2+7y = -\frac{2}{3}(x+3)^2 + 7

Final equation in vertex form:

y=23(x+3)2+7y = -\frac{2}{3}(x+3)^2 + 7


Would you like additional clarification or details? Here are 5 related questions to deepen your understanding:

  1. What would happen to the graph if the reflection across the x-axis was removed?
  2. How does a vertical compression differ from a vertical stretch?
  3. Can you describe the role of the vertex (3,7)(-3, 7) in the equation?
  4. What are some examples of transformations that don’t affect the vertex position?
  5. How would the equation change if the parabola was translated to the right instead of the left?

Tip: To visualize transformations, sketch each step incrementally to see how the graph evolves!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Transformations

Formulas

Vertex form of quadratic equation: y = a(x - h)^2 + k

Theorems

Transformation of quadratic functions

Suitable Grade Level

Grades 8-10