Math Problem Statement

Graph the following: y = (x - 3)^2, y = (x + 2)^2 + 4

Solution

The two equations you provided are quadratic functions, and both represent parabolas. Let's analyze and describe how to graph each one.

1. Equation 1: y=(x3)2y = (x - 3)^2

  • This is a standard form of a parabola, y=(xh)2+ky = (x - h)^2 + k, where:
    • h=3h = 3 (shifts the graph horizontally to the right by 3 units),
    • k=0k = 0 (no vertical shift).
  • The vertex of the parabola is at (3,0)(3, 0), and since the coefficient of (x3)2(x - 3)^2 is positive, the parabola opens upwards.
  • The axis of symmetry is x=3x = 3.
  • This graph is symmetrical around x=3x = 3, and its shape is the same as the standard parabola y=x2y = x^2, just shifted.

2. Equation 2: y=(x+2)2+4y = (x + 2)^2 + 4

  • This is also in the form y=(xh)2+ky = (x - h)^2 + k, where:
    • h=2h = -2 (shifts the graph horizontally to the left by 2 units),
    • k=4k = 4 (shifts the graph vertically up by 4 units).
  • The vertex of this parabola is at (2,4)(-2, 4), and it also opens upwards because the coefficient of (x+2)2(x + 2)^2 is positive.
  • The axis of symmetry is x=2x = -2.

Both graphs represent parabolas that open upwards, but their vertices are different due to horizontal and vertical shifts.

Would you like a graphical representation or more details on how to draw the graph on the grid?

5 Questions for Expansion:

  1. How do we determine the direction of a parabola (up or down)?
  2. What is the effect of changing the sign of the square term, for example y=(x3)2y = -(x - 3)^2?
  3. How can we find the points where these parabolas intersect the x-axis or y-axis?
  4. What are some applications of quadratic functions like these in real-world problems?
  5. How would the graph change if we added a coefficient in front of the squared term, like y=2(x3)2y = 2(x - 3)^2?

Tip:

The vertex form of a quadratic function is useful for easily identifying the vertex and direction of the parabola, making graphing simpler.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Graphing
Parabolas

Formulas

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

Theorems

Vertex theorem for parabolas

Suitable Grade Level

Grades 9-12