In Exercises 13-16, sketch the graph of each quadratic...

In Exercises $13-16,$ sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=(x-1)^{2}$
(b) $g(x)=(3 x)^{2}+1$
(c) $h(x)=\left(\frac{1}{3} x\right)^{2}-3$
(d) $k(x)=(x+3)^{2}$

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form f(x) = ax² + bx + c. Their graphs are parabolic curves that open upward if the coefficient a is positive and downward if a is negative. Understanding quadratic functions is essential as they serve as a basis for more complex algebraic and geometric analyses.

Parent Function (y = x²)

The parent function y = x² is the simplest quadratic function and serves as the reference for comparing other quadratic graphs. It has its vertex at (0, 0), is symmetric about the y-axis, and opens upward. Its graph provides a basis for understanding how translations, stretches, and compressions affect the shape and position of related quadratic functions.

Graph Transformations

Graph transformations include shifts, stretches, compressions, and reflections applied to the parent function. These operations allow us to modify the position and shape of the graph. For example, horizontal shifts occur when the input variable is adjusted (such as replacing x with x - h), while vertical shifts occur when a constant is added to the function output.

Horizontal Translation

A horizontal translation involves shifting the graph left or right. This is achieved by modifying the input of the function. For instance, replacing x with (x - h) moves the graph h units to the right if h is positive, and to the left if h is negative.

Vertical Translation

Vertical translation shifts the graph up or down. This transformation is carried out by adding or subtracting a constant to the entire function. For example, adding a constant k to f(x) moves the graph upward by k units, whereas subtracting k moves it downward.

Scaling (Stretching and Compression)

Scaling transformations alter the shape of the graph by stretching or compressing it along the vertical or horizontal axes. A vertical stretch or compression is applied by multiplying the function by a constant. If the factor is greater than one, the graph stretches vertically, making it narrower; if it is between zero and one, the graph compresses vertically, making it wider.

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