Problem 25 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=(x-5)^{2} $$

Step-by-Step Solution

Answer

Shift the graph of \( y = x^2 \) 5 units right.

Step 1: Identifying the Standard Function

The given function is based on the standard quadratic function, which is \( f(x) = x^2 \). This is the simplest form of a parabola.

Step 2: Understanding the Transformation

The function \( f(x) = (x-5)^2 \) represents a transformation of the basic parabola \( y = x^2 \). This transformation does not alter the shape of the graph but shifts it horizontally.

Step 3: Determining the Horizontal Shift

The expression \( (x-5)^2 \) indicates a horizontal shift to the right. To find the direction and magnitude of the shift, notice that \( x \) is replaced with \( x-5 \). This results in a shift 5 units to the right.

Step 4: Sketching the Transformed Graph

Start with the graph of \( y = x^2 \), which is a parabola with its vertex at the origin (0,0). Apply the horizontal shift by moving the entire graph 5 units to the right. The vertex of the parabola is now at (5,0). The graph opens upwards, maintaining the same width and shape as \( y = x^2 \).

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental part of algebra, often represented by the equation \( f(x) = ax^2 + bx + c \). These functions graph as parabolas, which are symmetrical curves. In their simplest form, quadratic functions have the equation \( f(x) = x^2 \). Here, \( a \), \( b \), and \( c \) are constants which determine the shape and position of the parabola.

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.
• The value of \( c \) shifts the entire graph up or down depending on its value.

The vertex, which is the highest or lowest point of the parabola, plays a key role in understanding its positioning. In the simplest quadratic form \( y = x^2 \), the vertex is at the origin (0, 0). This base parabola is used for many graph transformations.

Horizontal Shift

A horizontal shift in a function involves moving the entire graph left or right along the x-axis. For quadratic functions, like \( y = (x-h)^2 \), this shift is determined by the value inside the parentheses. Example: The function \( f(x) = (x-5)^2 \) represents a horizontal shift of the basic parabola \( y = x^2 \).

• The subtraction of 5 inside the parentheses indicates a shift to the right by 5 units.
• If it were \( (x+5)^2 \), the graph would shift 5 units to the left.

This shift changes the location of the vertex from the origin to the new point. For \( f(x) = (x-5)^2 \), the vertex moves to (5, 0). Importantly, the shape and orientation (opening upwards or downwards) of the parabola remain unchanged by this transformation.

Parabola Shape

The shape of the parabola in a quadratic function is primarily affected by the coefficient \( a \) in the standard form \( ax^2 \). However, transformations like horizontal and vertical shifts can relocate the parabola on the graph without changing its fundamental form. In the function \( y = (x-5)^2 \), we observe that:

• The parabola maintains its symmetrical U-shape.
• Its width does not change because the transformation only involves a shift.
• The vertex shifts to (5, 0) as a result of the horizontal shift.

Remember, the parabola's shape remains the same even if it's moved around on the graph. In this example, despite shifting horizontally by 5 units, the parabola still opens upwards and looks identical to the basic \( y = x^2 \) graph, underscoring the symmetry and elegance of parabolic functions. This makes graph transformations crucial tools for understanding how various shifts affect graphs without altering their core shapes.