Math Problem Statement

The graph of y = x^2 is the solid black graph below. Which function represents the dotted graph?

Solution

We are given the graph of y=x2y = x^2 (solid black curve) and asked to identify which function corresponds to the dotted orange curve from the provided options.

Observing the Dotted Graph:

  • The dotted graph represents a parabola that is similar to y=x2y = x^2, but it has been shifted horizontally and vertically.
  • The vertex of the dotted graph appears to be at (1,5)(1, 5), whereas the vertex of y=x2y = x^2 is at the origin (0,0)(0, 0).

Applying the Vertex Form of a Parabola:

The general form for a parabola is:

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

where:

  • (h,k)(h, k) is the vertex of the parabola,
  • aa determines the direction and width of the parabola.

In this case, the dotted parabola seems to be shifted horizontally by +1+1 (i.e., h=1h = 1) and vertically by +5+5 (i.e., k=5k = 5).

Thus, the equation of the dotted parabola is:

y=(x1)2+5y = (x - 1)^2 + 5

Answer:

The correct function representing the dotted graph is y=(x1)2+5y = (x - 1)^2 + 5.

Let me know if you'd like further details! Here are five related questions for practice:

  1. How would the equation change if the vertex were at (1,5)(-1, -5)?
  2. What is the effect of the coefficient aa on the width of the parabola?
  3. How would you graph y=(x+3)22y = (x + 3)^2 - 2?
  4. What is the vertex of the parabola y=(x4)2+1y = (x - 4)^2 + 1?
  5. How does shifting a parabola horizontally and vertically affect its equation?

Tip: The signs in (xh)2(x - h)^2 and kk directly tell you the horizontal and vertical shifts.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Transformations

Formulas

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

Theorems

Vertex Form of a Parabola

Suitable Grade Level

Grades 9-10