Math Problem Statement

The graph of f(x) = x^2 is shown below. Sketch the graph of a function g(x) that has the same shape as f(x) but shifted up 3 spaces. Now write the equation for g(x).

Solution

The problem asks for the equation of the function g(x)g(x), which has the same shape as the given function f(x)=x2f(x) = x^2 but is shifted up by 3 units.

To find the equation of g(x)g(x), we start with the base function f(x)=x2f(x) = x^2. When a function is shifted vertically by 3 units, we add 3 to the function:

g(x)=f(x)+3g(x) = f(x) + 3

Since f(x)=x2f(x) = x^2, we substitute this into the equation:

g(x)=x2+3g(x) = x^2 + 3

Thus, the equation for g(x)g(x) is:

g(x)=x2+3g(x) = x^2 + 3

Would you like further details or have any questions?

Here are 5 related questions:

  1. What would happen if the function was shifted down by 3 units?
  2. How does shifting horizontally affect the function equation?
  3. What does the vertex of the parabola represent after the vertical shift?
  4. Can you describe how the graph of g(x)=x2+3g(x) = x^2 + 3 differs from f(x)=x2f(x) = x^2?
  5. How would you shift the function to the left or right?

Tip: When graphing, vertical shifts affect the yy-coordinates, while horizontal shifts modify the xx-coordinates.

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Math Problem Analysis

Mathematical Concepts

Graphing Functions
Transformations of Functions
Quadratic Functions

Formulas

g(x) = f(x) + k
g(x) = x^2 + 3

Theorems

Vertical Translation

Suitable Grade Level

Grades 7-9