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 vertex is at (2,-4); graph shifts right 2 units, down 4 units.

Step 1: Identify the Base Function

The base function for this transformation is the quadratic function \(f(x) = x^2\), which is a standard parabola that opens upwards with its vertex at the origin (0,0).

Step 2: Translate the Base Function Horizontally

The term \((x-2)^2\) indicates a horizontal shift to the right by 2 units. This means the parabola's vertex, which was originally at (0,0), moves to (2,0). This translation is due to replacing \(x\) with \(x-2\).

Step 3: Translate the Base Function Vertically

The subtraction of 4 in \((x-2)^2 - 4\) translates the graph of the function vertically downward by 4 units. So, the new vertex of the parabola becomes (2, -4).

Step 4: Plot Key Points and Sketch

To sketch the graph, plot the new vertex at (2, -4). Since it's a basic quadratic transformation, you can plot additional points by choosing x-values around the vertex point and calculate the corresponding y-values, e.g., (1,-3), (3,-3). Then draw a symmetric parabola opening upward.

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two. It is generally expressed in the form \(f(x) = ax^2 + bx + c\). The graph of a quadratic function is a smooth, curved shape known as a parabola. The parameter \(a\) determines the parabola's direction:

• If \(a > 0\), the parabola opens upwards, resembling a smiley face.
• If \(a < 0\), the parabola opens downwards, like a frown.

Quadratic functions are essential in understanding various mathematical and physical phenomena.
They appear in scenarios like projectile motion and optimization problems.

Horizontal Translation

Horizontal translation refers to moving a graph left or right along the x-axis. This kind of shift is controlled by altering the x-variable in the function. For example, in \(f(x) = (x-2)^2\), replacing \(x\) with \(x-2\) results in a horizontal translation.
It means moving the base graph of \(f(x) = x^2\) to the right by 2 units.

• If \(x\) is replaced with \(x-h\), shift right by \(h\) units.
• If \(x\) is replaced with \(x+h\), shift left by \(h\) units.

Understanding horizontal translations helps in graph sketching, allowing one to precisely locate features of the graph like vertices.

Vertical Translation

Vertical translation involves shifting the entire graph of a function up or down the y-axis. This is achieved by modifying the constant term in the function. In \(f(x) = (x-2)^2 - 4\), the \