Math Problem Statement

The point A(-3, 2) lies on the graph with equation y = f(x). If the graph of the related function y = -f(x) + 3 is drawn, what will the image of A be?

Solution

The problem gives us the point A(3,2)A(-3, 2) on the graph of y=f(x)y = f(x) and asks for the image of AA when the graph is transformed to y=f(x)+3y = -f(x) + 3.

Step-by-step approach:

  1. Given Information:

    • A(3,2)A(-3, 2) lies on the graph of y=f(x)y = f(x), meaning f(3)=2f(-3) = 2.
  2. Transformation: The transformation is y=f(x)+3y = -f(x) + 3.

    • The term f(x)-f(x) reflects the graph over the x-axis.
    • The term +3+3 shifts the graph vertically by 3 units upwards.
  3. Apply the transformation to the point:

    • Start with f(3)=2f(-3) = 2.
    • For the new function, y=f(x)+3y = -f(x) + 3: y=f(3)+3=(2)+3=2+3=1y = -f(-3) + 3 = -(2) + 3 = -2 + 3 = 1

Thus, the image of point A(3,2)A(-3, 2) under the transformation is (3,1)(-3, 1).

Conclusion:

The image of AA will be (3,1)(-3, 1).

Let me know if you'd like further details or have any questions!

Related questions:

  1. What is the effect of the transformation y=f(x)y = -f(x) on the graph?
  2. How does a vertical shift affect the coordinates of a point on a graph?
  3. What happens to the graph when a negative sign is applied to a function?
  4. How can we generalize transformations for any point (x1,y1)(x_1, y_1)?
  5. Can you explain what a horizontal reflection would look like?

Tip:

Whenever dealing with transformations, always consider how the reflection (negative sign) and shifts affect each point individually.

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Math Problem Analysis

Mathematical Concepts

Function Transformation
Coordinate Geometry

Formulas

y = -f(x) + 3

Theorems

Function Reflection and Vertical Shift

Suitable Grade Level

Grades 10-12