Question

Graph the function \(f\) by starting with the graph of \(y=x^{2}\) and using transformations. \(f(x)=(x+2)^{2}-2\)

Short answer

Shift the graph of \(y=x^2\) left by 2 units and down by 2 units.

Step-by-step solution

Understand the Base Function

The base function is given as \(y = x^2\). This is a standard parabola that opens upwards with its vertex at the origin (0,0).

Horizontal Shift

Identify the horizontal shift in the function. The term \(x+2\) within the squared term \((x+2)^2-2\) indicates a horizontal shift to the left by 2 units. Shift the entire graph of \(y = x^2\) 2 units left to get the new function \(y = (x+2)^2\).

Vertical Shift

Next, identify the vertical shift. The \(-2\) outside the squared term signifies a vertical shift downward by 2 units. Shift the graph of \y = (x+2)^2\ downward by 2 units to get the final transformed function \(f(x)=(x+2)^2 - 2\).

Plot the Transformed Function

Now that the transformations have been applied, plot the transformed function. The vertex of the new graph is at (-2, -2) and it opens upwards similar to the base function. The shape remains the same; only the vertex has changed.

Key concepts

Graphing Functions

Graphing functions is a fundamental skill in mathematics, helping you visualize how different functions behave. The base function provided in the exercise is a quadratic function, which graphically forms a parabola. To graph such a function, you typically start with its simplest form. For quadratic functions, this is often the parent function, such as \(y = x^2\). This forms a U-shaped curve called a parabola, opening upwards, with its vertex at the origin (0, 0). Knowing the vertex and the direction the parabola opens is crucial.
The main idea is to apply transformations to this base function to get the desired graph. Transformations can include shifts (vertical or horizontal) and stretches or compressions. Each of these transformations affects the graph in a specific way, and understanding them helps you plot the function accurately.

Vertical Shift

A vertical shift occurs when we add or subtract a constant outside a function. For instance, if we have \(y = f(x) + c\) or \(y = f(x) - c\), it means we are shifting the graph of the function up or down by c units, respectively.
In the exercise, the term \-2\ outside the squared term \((x+2)^2 - 2\) signals a vertical shift downward by 2 units. This means every point on the graph moves down by 2 units. If the vertex of the base function \(y = x^2\) is at (0, 0), after applying the vertical shift, the vertex will be at (0, -2). It's crucial to note that this doesn’t affect the shape of the graph, only its placement on the y-axis.

Horizontal Shift

A horizontal shift involves adding or subtracting a constant within the function's input (inside the brackets). For example, \(y = f(x + c)\) or \(y = f(x - c)\) means shifting the graph left by c units or right by c units, respectively.
In \((x+2)^2 - 2\), the \(x+2\) indicates a leftward shift of the parabola by 2 units. It may seem counterintuitive, but adding inside the function results in a left shift. Thus, the vertex of the parabola that was at (0, 0) in the base function \((y = x^2)\) moves to (-2, 0). This shift affects the x-coordinates of all points on the graph.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, with the general form \(y = ax^2 + bx + c\). Their graphs form parabolas, which can open upwards or downwards depending on the sign of the leading coefficient a. If \a > 0\, it opens upwards; if \a < 0\, it opens downwards.
The standard base form \(y = x^2\) is the simplest quadratic function, where the parabola opens upwards with its vertex at the origin (0, 0). When transformations like vertical or horizontal shifts are applied, the vertex moves, but the parabola's shape remains the same. Understanding these transformations allows you to manipulate and graph various quadratic functions easily.