Question

Graph \(y=2(x+1)^{2}-3\) using transformations.

Short answer

Graph shifts left 1 unit, stretches by 2, and shifts down 3 units.

Step-by-step solution

Identify the Parent Function

The given function is a transformation of the parent function for a parabola, which is \(y = x^2\).

Apply Horizontal Shift

The equation has \(x + 1\) inside the square term indicating a horizontal shift. The function shifts to the left by 1 unit.

Apply Vertical Stretch

The coefficient 2 in front of \( (x + 1)^2 \) indicates a vertical stretch by a factor of 2. This means that the graph will be narrower compared to the parent function.

Apply Vertical Shift

The term -3 at the end of the equation indicates a vertical shift down by 3 units.

Sketch the Graph

Start with the point \((-1, -3)\) as this is the vertex of the transformed function. Then, apply the stretches and shifts identified in the previous steps to sketch the graph.

Key concepts

Horizontal Shift

A horizontal shift occurs when the graph of a function moves left or right along the x-axis. This can be easily identified by examining the inside of the function's squared term. In our exercise, we see the term \( x + 1 \).

Vertical Stretch

A vertical stretch or compression modifies how

Vertical Shift

Vertical shifts involve moving the graph up or down along the y-axis. This is shown by a constant added to or subtracted from the function. In our example, the -3 at the end of \( y = 2(x + 1)^2 - 3\)

Parent Function of a Parabola

The parent function of a parabola is the simplest form of a quadratic equation, \( y = x^2 \). It provides the basic shape of a parabola,