Problem 40 - Beginning and Intermediate Algebra

Question

Use the transformation techniques to graph each of the following functions. $$f(x)=(x+2)^{2}-3$$

Step-by-Step Solution

Answer

The function \(f(x) = (x + 2)^2 - 3\) is a quadratic function in vertex form. The base function is \(f(x) = x^2\). Apply the following transformations to graph the function: 1. Horizontal shift: 2 units to the left 2. Vertical shift: 3 units down 3. Vertical stretch/compression: None (a = 1) The final graph is a parabola opening upwards with a vertex at (-2, -3).

Step 1: Identify the base function and transformations

The given function, \(f(x) = (x + 2)^2 - 3\), is a quadratic function in vertex form, which can be written as \(f(x) = a(x - h)^2 + k\). In our case, \(a = 1\), \(h = -2\), and \(k = -3\). The base function is \(f(x) = x^2\). We have three transformations to apply: 1. Horizontal shift: h = -2 (shift the graph 2 units to the left) 2. Vertical shift: k = -3 (shift the graph 3 units down) 3. Vertical stretch/compression: a = 1 (no vertical stretch or compression)

Step 2: Graph the base function

Start by sketching the graph of the base function, \(f(x) = x^2\), which is a parabola opening upwards with its vertex at the origin (0,0).

Step 3: Apply horizontal shift

Apply the horizontal shift by moving the vertex of the base function 2 units to the left. The new vertex will be at (-2, 0).

Step 4: Apply vertical shift

Apply the vertical shift by moving the vertex of the shifted function 3 units down. The new vertex will be at (-2, -3).

Step 5: Apply vertical stretch/compression

Since the value of a = 1, there is no vertical stretch or compression. The shape of the parabola remains the same.

Step 6: Sketch the final graph

Sketch the final graph of the transformed function, \(f(x) = (x + 2)^2 - 3\), with a parabola opening upwards and vertex at (-2, -3). Now you have graphed the function \(f(x) = (x + 2)^2 - 3\) using transformation techniques.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that can be expressed in the standard form as \( f(x) = ax^2 + bx + c \), where \(a eq 0\). The graph of a quadratic function is a parabola, which is a symmetrical curve. Depending on the sign of \(a\), the parabola can open upwards (if \(a > 0\)) or downwards (if \(a < 0\)).

Key characteristics to remember are:

• The vertex, which is the highest or lowest point of the parabola depending on its direction.
• The axis of symmetry, a vertical line that goes through the vertex and divides the parabola into two mirrored halves.
• The y-intercept, the point where the parabola crosses the y-axis.

Quadratic functions can take several forms, including the standard form mentioned above and the vertex form, each offering different advantages for graphing and analysis.

Vertex Form

The vertex form of a quadratic function is incredibly useful for graphing because it reveals the vertex's position directly. It is written as \( f(x) = a(x-h)^2 + k \). Here, \( (h, k) \) is the vertex of the parabola.

Here's why vertex form is handy:

• The value \( h \) represents the horizontal shift from the origin, indicating how far left or right the vertex moves from the point \( (0,0) \).
• The value \( k \) signifies the vertical shift, showing how far up or down the parabola's vertex moves.
• The parameter \( a \) affects the parabola's width and direction. If \( a > 1 \), the parabola is narrower. If \( 0 < a < 1 \), it's wider. A negative \( a \) reflects the parabola downwards.

Using vertex form provides a straightforward path to sketching the graph by first plotting the vertex and then applying other transformations like horizontal or vertical shifts.

Horizontal Shift

In transformations, a horizontal shift slides the graph of a function left or right. In the vertex form of a quadratic function, \( f(x) = a(x - h)^2 + k \), the parameter \( h \) dictates this movement.

To correctly apply a horizontal shift, remember:

• If \( h \) is positive, the graph shifts to the right by \( h \) units.
• If \( h \) is negative, the graph shifts to the left, which is the case in \( f(x) = (x + 2)^2 - 3 \) because \( h = -2 \).

By determining the correct value of \( h \), you can accurately adjust the position of the parabola's vertex and thereby the entire graph horizontally across the coordinate plane.

Vertical Shift

A vertical shift moves the graph of a function up or down. In the vertex form for quadratic functions, \( f(x) = a(x-h)^2 + k \), the value \( k \) is responsible for this vertical movement.

To implement a vertical shift effectively:

• A positive \( k \) will push the graph upwards by \( k \) units.
• A negative \( k \) will move the graph downward, which occurs in \( f(x) = (x + 2)^2 - 3 \) as \( k = -3 \).

Vertical shifts do not affect the shape or the horizontal location of the graph. They simply elevate or lower the parabola without altering its curvature or width.