Problem 33 - Algebra and Trigonometry

Question

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=(x-2)^{2} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = (x-2)^2 \) is a parabola shifted 2 units to the right.

Step 1: Identify the Base Function

The given function is \( f(x) = (x-2)^2 \). The base function, a standard function from which we will derive this graph, is \( g(x) = x^2 \), which is the parabola centered at the origin \((0,0)\) with a symmetric U-shape opening upwards.

Step 2: Analyze the Transformation

The transformation in the function \( f(x) = (x-2)^2 \) comes from the term \( (x-2) \). This represents a horizontal shift. Specifically, replacing \( x \) with \( x-2 \) shifts the graph of the base function \( g(x) = x^2 \) to the right by 2 units.

Step 3: Sketch the Transformed Graph

Start with the standard parabola \( g(x) = x^2 \). To apply the transformation: take each point on the parabola and move it 2 units to the right. The vertex of the parabola at \((0,0)\) will move to \((2,0)\), which reflects the horizontal shift. The shape of the parabola remains a symmetrical U-shape opening upwards.

Key Concepts

Quadratic Functions

Quadratic functions are one of the fundamental types of functions in mathematics. They are expressed in the form of a polynomial equation: \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The simplest quadratic function is \( f(x) = x^2 \), which forms a basic parabola.

• The graph of a quadratic function is called a parabola.
• Parabolas have a single point called the vertex, which can be a maximum or minimum, depending on the orientation of the parabola.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirroring halves.

Understanding quadratic functions is key to mastering various transformations, as every transformation alters the graph in predictable ways, allowing us to sketch graphs without plotting numerous points.

Graphing Functions

Graphing functions involves creating a visual representation of mathematical equations on a coordinate plane. For quadratic functions like \( f(x) = x^2 \), it manifests as a smooth U-shaped curve called a parabola.

• To graph a function, identify its standard form and then apply any transformations.
• Start by plotting the base function, such as \( g(x) = x^2 \), which centers at the origin.
• Shift, stretch, compress, or reflect the graph by implementing appropriate transformations to display the desired behavior.

Graphing isn't just plotting points; it's about recognizing patterns and utilizing transformations. This approach simplifies understanding and predicting how changes in equations affect their graphs.

Horizontal Shifts

The concept of horizontal shifts is crucial when working with function transformations, especially with quadratic functions. Horizontal shifts occur when the entire graph moves left or right along the x-axis.

• When you see \( f(x) = (x-h)^2 \), the graph of the function \( g(x) = x^2 \) shifts horizontally.
• If \( h \) is positive, the graph shifts to the right by \( h \) units.
• If \( h \) is negative, the graph shifts to the left by \( |h| \) units.

For example, \( f(x) = (x-2)^2 \) results in a horizontal shift of the base parabola to the right by 2 units. The entire graph moves, but its shape remains unchanged, maintaining its symmetric U-shape.