Math Problem Statement

Use properties of functions to match each of the following functions with its graph without using a calculator. The functions are: 1. f(x) = x^2 + 2 2. f(x) = -x^2 + 2 3. f(x) = x^2 - 2 4. f(x) = -x^2 - 2 The available graphs (A, B, C, D) show various parabolas with different orientations and vertex positions.

Solution

To solve this problem, let's analyze each function and identify its graph by observing the transformations of the standard quadratic function f(x)=x2f(x) = x^2. The transformations involve vertical shifts and reflections. Here's the approach:

  1. f(x)=x2+2f(x) = x^2 + 2:

    • This function represents an upward-opening parabola (positive coefficient of x2x^2) shifted up by 2 units.
    • Look for a graph with a vertex at (0,2)(0, 2) and opening upwards.
  2. f(x)=x2+2f(x) = -x^2 + 2:

    • This function represents a downward-opening parabola (negative coefficient of x2x^2) shifted up by 2 units.
    • Look for a graph with a vertex at (0,2)(0, 2) and opening downwards.
  3. f(x)=x22f(x) = x^2 - 2:

    • This is an upward-opening parabola shifted down by 2 units.
    • Look for a graph with a vertex at (0,2)(0, -2) and opening upwards.
  4. f(x)=x22f(x) = -x^2 - 2:

    • This is a downward-opening parabola shifted down by 2 units.
    • Look for a graph with a vertex at (0,2)(0, -2) and opening downwards.

Now, let's match each function to the graphs (A, B, C, D) based on these observations:

  • Graph A: Upward-opening parabola with a vertex at (0,2)(0, -2), matching f(x)=x22f(x) = x^2 - 2.
  • Graph B: Upward-opening parabola with a vertex at (0,2)(0, 2), matching f(x)=x2+2f(x) = x^2 + 2.
  • Graph C: Downward-opening parabola with a vertex at (0,2)(0, 2), matching f(x)=x2+2f(x) = -x^2 + 2.
  • Graph D: Downward-opening parabola with a vertex at (0,2)(0, -2), matching f(x)=x22f(x) = -x^2 - 2.

Final Matching:

  1. f(x)=x2+2f(x) = x^2 + 2 matches Graph B
  2. f(x)=x2+2f(x) = -x^2 + 2 matches Graph C
  3. f(x)=x22f(x) = x^2 - 2 matches Graph A
  4. f(x)=x22f(x) = -x^2 - 2 matches Graph D

Would you like further details on any specific step?


Here are five questions to deepen understanding:

  1. How would you determine the vertex of a quadratic function in general?
  2. What happens to the graph of f(x)=x2f(x) = x^2 if you add or subtract a constant?
  3. How does the sign of the coefficient of x2x^2 affect the direction in which a parabola opens?
  4. Can you describe the effect of horizontal shifts on the graph of a parabola?
  5. How would you graphically identify a reflection in the x-axis?

Tip: Remember, a positive coefficient of x2x^2 means the parabola opens upward, while a negative coefficient means it opens downward.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Graphing Parabolas
Transformations

Formulas

f(x) = ax^2 + bx + c for a quadratic function
Vertex form: f(x) = a(x - h)^2 + k where (h, k) is the vertex

Theorems

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Suitable Grade Level

Grades 9-12