Question

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?

Accepted Answer

Identify the base graph, which is the parent function $$y = x$$^{2}$$. This is a parabola with its vertex at the origin (0$$,0)$$ and opens upward. For each given function, recognize that they are transformations of the parent function involving horizontal shifts. Specifically, y = (x$$ - $$2)^{2}$$ shifts the graph of $$y = x$$^{2}$$ to the right by 2 units. Similarly, y = (x$$ + $$1)^{2}$$ 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. 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}$$. Then sketch the parabola opening upward from each vertex. In summary, these graphs differ from y$$ = $$x^{2} by $$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.

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?

Key Concepts

Graphing Quadratic Functions

Horizontal Shifts of Parabolas

Vertex Form of a Quadratic Function