Graph the following on the same coordinate system.
(a) y = (x - 2)²
(b) y = (x + 1)²
(c) y = (x + 3)²
(d) How do these graphs differ from the graph of y = x²?

Verified step by step guidance

1. Identify the base graph, which is the parent function \(y = x^{2}\). This is a parabola with its vertex at the origin \((0,0)\) and opens upward.
2. For each given function, recognize that they are transformations of the parent function involving horizontal shifts. Specifically, \(y = (x - 2)^{2}\) shifts the graph of \(y = x^{2}\) to the right by 2 units.
3. Similarly, \(y = (x + 1)^{2}\) shifts the graph of \(y = x^{2}\) to the left by 1 unit, and \(y = (x + 3)^{2}\) shifts it to the left by 3 units.
4. To graph each function, plot the vertex at the shifted point: \((2,0)\) for \(y = (x - 2)^{2}\), \((-1,0)\) for \(y = (x + 1)^{2}\), and \((-3,0)\) for \(y = (x + 3)^{2}\). Then sketch the parabola opening upward from each vertex.
5. In summary, these graphs differ from \(y = x^{2}\) by horizontal translations. The shape and orientation remain the same, but the vertex moves left or right depending on the sign and value inside the parentheses.

Key Concepts

Graphing Quadratic Functions

Quadratic functions are represented by equations of the form y = ax² + bx + c. Their graphs are parabolas, which are U-shaped curves. Understanding how to plot points and identify the vertex helps in sketching these graphs accurately.

Horizontal Shifts of Parabolas

The expressions (x - h)² represent horizontal shifts of the basic parabola y = x². If h is positive, the graph shifts h units to the right; if h is negative, it shifts |h| units to the left. This shift changes the vertex position without altering the shape.

Vertex Form of a Quadratic Function

The vertex form y = (x - h)² shows the parabola's vertex at (h, 0). This form makes it easy to identify the vertex and understand transformations like shifts. Comparing graphs in vertex form reveals how each parabola moves relative to y = x².