Problem 13 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$S(x)=(x+3)^{2}-1$$

Step-by-Step Solution

Answer

The underlying basic function is \(y = x^2\). The transformed graph is obtained by shifting the basic function left by 3 units and down by 1 unit. The resultant graph is a parabola opening upwards with vertex (-3,-1).

Step 1: Identify the basic function

The basic function here is \(f(x) = x^2\), which is a parabola. This parabola opens upward and its vertex is at the origin (0,0).

Step 2: Identify the transformations

The full function \(S(x)=(x+3)^{2}-1\), compared to the basic function \(f(x) = x^2\), has been shifted left by 3 units and down by 1 unit. This occurs due to the +3 and -1 in the function.

Step 3: Apply the transformations

Shift the graph of \(y=x^2\) left by 3 units to get the graph of \(y = (x+3)^2\). Then, shift the result down by 1 unit, to get the graph of \(y = (x+3)^{2}-1\).

Step 4: Sketch the resultant graph

The graph has the shape of a vertically opened parabola with a vertex at the point (-3, -1). All other points on the graph will have their x-coordinates 3 less than the corresponding points on the graph of the basic function \(y=x^2\) and their y-coordinates 1 less. For instance, the point (0,0) on the graph of \(y=x^2\) will correspond to the point (-3,-1) on the graph of \(y = (x+3)^{2}-1\).

Key Concepts

Basic Functions

Understanding basic functions is like having a key to the world of mathematics. A basic function is a simple, foundational mathematical expression that serves as a building block for more complex functions. The most common basic functions include lines (y = mx + b), parabolas (y = x^2), and more.
Each basic function has a unique graph that represents every possible point (x, y) that satisfies the function's equation. For instance, the graph of a linear function is a straight line, while a quadratic function, such as y = x^2, is represented by a parabola.
Knowing these basic graphs sets the stage for understanding how more complicated functions behave and for grasping their visual representations.

Parabola Graphing

The graph of a parabola, a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix), is essential in algebra and appears frequently in different forms.
The simplest form of a parabola is y = x^2, which opens upwards with the vertex at the origin (0,0). Graphing a parabola involves plotting a set of points that follow the function's rule and connecting them to reveal the 'U' shape characteristic of the parabola.

Shape and Symmetry

• All parabolas are symmetric about a line called the axis of symmetry.
• The vertex is the point where the parabola changes direction and is also the maximum or minimum point of the function.

The vertex, direction, width, and position of the parabola can all change with different transformations applied to the basic quadratic function.

Function Shifts

Altering the graph of a basic function without changing its shape is known as function shifts or transformations. These shifts can move the graph horizontally, vertically, or both.
There are two main types of shifts:

• Horizontal shifts: These occur when you add or subtract a constant from the x variable.
• Vertical shifts: These occur when you add or subtract a constant from the entire function.

For instance, in the exercise function S(x) = (x+3)^2 - 1, there is a horizontal shift to the left by 3 units and a vertical shift downward by 1 unit. It is crucial to recognize these transformations to accurately graph the function without plotting numerous points.

Sketching Graphs

Being able to sketch graphs of functions is a valuable skill, especially for visualization and understanding complex behaviors of functions.
The process often starts with identifying the basic graph of the function — whether it's a line, parabola, hyperbola, etc. Then, you look at the transformations that have been applied, like shifts, stretches, or reflections, and adjust the graph accordingly.

Steps for Sketching

When sketching the graph of a function with transformations, consider following these methods:

• Identify the type of the basic function and knowledge of its shape and properties.
• Observe transformation indicators in the function, such as added or subtracted terms.
• Apply the transformations to the basic graph in the proper order.
• Plot the vertex (for parabolas) or intercepts (for lines), which serve as reference points.
• Draw the transformed graph while maintaining the original shape characteristics.

With practice, you’ll improve in visualizing and sketching graphs quickly and accurately.