Chapter 2.5

Q1. Graph the function f(x) = (x + 1)^2 - 2 by starting with the basic function and using transformations.

Background

Topic: Transformations of Quadratic Functions

This question tests your understanding of how to graph quadratic functions by applying horizontal and vertical shifts to the basic parabola y = x^2.

Key Terms and Formulas

• Quadratic function: $y = x^2$

• Transformation: Shifting, stretching, compressing, and reflecting graphs

• General form: $y = (x + h)^2 + k$ where h shifts horizontally and k shifts vertically

Step-by-Step Guidance

1. Start with the basic graph of $y = x^2$, which is a parabola opening upwards with its vertex at (0, 0).

2. Apply the horizontal shift: The term $(x + 1)$ means the graph is shifted 1 unit to the left. So, the new vertex will be at (-1, 0).

3. Apply the vertical shift: The term $-2$ means the graph is shifted 2 units downward. So, the new vertex will now be at (-1, -2).

4. Sketch the transformed parabola, making sure the shape remains the same (opens upward), but the vertex is at the new location.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $f(x) = (x + 1)^2 - 2$ is a parabola opening upward with its vertex at (-1, -2). The graph is the basic parabola $y = x^2$ shifted 1 unit left and 2 units down.