Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of \(f(x)=(x-2)^{2}-4\) using translations.

Step-by-Step Solution

Answer

The graph of \(f(x) = (x-2)^2 - 4\) is a parabola opening upwards with vertex at (2, -4).

Step 1: Identify the Base Graph

The base graph for the function is \(y = x^2\), which is a parabola opening upwards with its vertex at the origin (0, 0).

Step 2: Translate Horizontally

The function \(f(x) = (x-2)^2 - 4\) has a horizontal translation. The expression \((x-2)\) indicates a shift of the vertex 2 units to the right, changing the vertex from (0, 0) to (2, 0).

Step 3: Translate Vertically

The \(-4\) in \((x-2)^2 - 4\) indicates a vertical translation. This translates the graph 4 units downwards, moving the vertex from (2, 0) to (2, -4).

Step 4: Sketch the Final Graph

Start with the base graph of \(y = x^2\) and apply the transformations. Shift the graph 2 units to the right and then 4 units down to obtain the graph with the vertex at (2, -4). Draw the parabola opening upwards with this new vertex.

Key Concepts

Parabola

A parabola is a U-shaped curve that is a graphical representation of a quadratic function. It can open upwards or downwards, depending on the function's coefficients.
In a standard form of a quadratic function, such as \(y = ax^2 + bx + c\), the parabola will open upwards if \(a > 0\) and downwards if \(a < 0\). The point where the parabola changes direction is known as the vertex. This is either the highest or lowest point on the graph.
Parabolas are symmetrical around a vertical line that passes through the vertex, called the axis of symmetry. This symmetry makes them interesting and useful in mathematics and physics, particularly in understanding trajectories and optimization problems.
Parabolas are foundational in studying quadratic functions and serve as the basis for transformations in many graphing exercises.

Vertex Form

The vertex form of a quadratic function is a way to write the equation of the parabola so that its vertex and transformations are evident. The vertex form is expressed as:
\[f(x) = a(x-h)^2 + k\]
Here, \((h, k)\) is the vertex of the parabola, making it easy to identify the location where the parabola changes direction. The value of \(a\) dictates the parabola's width and direction (upwards if \(a > 0\) and downwards if \(a < 0\)).
Vertex form is particularly useful because it directly shows the transformations applied to the basic parabola \(y = x^2\). It becomes straightforward to determine horizontal and vertical shifts just by looking at this form. For example, the expression \((x-2)^2 - 4\) reveals the vertex to be at \((2, -4)\).
Understanding vertex form helps in quickly sketching the graph and applying further transformations effectively.

Transformations

Transformations are changes applied to the basic graph of a function, which shift, stretch, or flip it. In our exercise, the base graph is \(y=x^2\), and we apply transformations to get the function \(f(x)=(x-2)^2-4\).
Transformations come in several types:

• Translation: Shifting the entire graph horizontally or vertically without altering its shape. In the given exercise, the graph shifts 2 units to the right (horizontal translation) and 4 units downwards (vertical translation).
  
• Reflection: Flipping the graph over a specific axis. Although this isn't used in our example, it can be applied with changes to the sign of \(a\).
• Dilation: Stretching or compressing the graph, which reflects changes in the width and steepness of the parabola.

By understanding these types of transformations, sketching complex graphs becomes a simpler task. Each transformation can be visually mapped on the basic graph to see the resulting changes.

Quadratic Functions

Quadratic functions are polynomial functions with a degree of two. They are expressed in the standard form \(y = ax^2 + bx + c\), and their graph is a parabola.
Quadratic functions have several important aspects:

• Coefficients: The values \(a\), \(b\), and \(c\) determine the parabola's form, position, and orientation.
• Vertex: The point where the parabola reaches its maximum or minimum, calculable directly in vertex form \((h, k)\) or derived from the standard form.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves, passing through the vertex.

Quadratic functions model many real-world phenomena such as projectile motion and parabolas of satellite dishes. Learning to manipulate and graph these functions is essential for students to grasp the broader concept of polynomial functions and their applications.