Problem 35 - College Algebra with Modeling and Visualization

Question

Use transformations to explain how the graph of \(f\) can be found by using the graph of \(y=x^{2}\) \(y=\sqrt{x},\) or \(y=|x| .\) You do not need to graph \(y=f(x)\). \(f(x)=(x-3)^{2}+1\)

Step-by-Step Solution

Answer

Shift the graph of \(y = x^2\) 3 units to the right and 1 unit up.

Step 1: Identify the Basic Function

The basic function is identified from the equation, which is in the form of \(f(x) = (x-h)^2 + k\). The basic function is \(y = x^2\). This basic function involves squaring the input value.

Step 2: Determine Horizontal Transformations

The equation \((x-3)^2\) indicates a horizontal shift.The graph of \(y = x^2\) is shifted to the right by 3 units, as the transformation \(x-h\) inside the function implies a horizontal shift to the right if \(h ext{ is positive}\).

Step 3: Determine Vertical Transformations

The "+1" in the expression \((x-3)^2 + 1\) indicates a vertical shift.The entire graph moves upwards by 1 unit. This corresponds to the \(+k\) term in the general transformation equation \(y = (x-h)^2 + k\), indicating an upward shift of \|k|\ units.

Key Concepts

Quadratic Functions

Quadratic functions are fundamental in algebra, expressed in the standard form as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. At its simplest, the graph of a quadratic function is a parabola, a U-shaped curve that opens upward if \( a \) is positive and downward if \( a \) is negative. The most basic form of a quadratic function is \( y = x^2 \), which serves as the foundation for transformations. This simple parabola has its vertex at the origin, (0, 0), and is symmetric with respect to the y-axis.

Quadratic functions can be transformed in various ways to move and change the shape of the parabola. Understanding how each transformation affects the graph is crucial for mastering quadratic functions. Let's dive into these transformations, focusing on horizontal and vertical shifts.

Horizontal Shifts

A horizontal shift in a quadratic function involves moving the entire graph left or right. It happens when you modify the \( \left(x-h\right) \) part of the function. This shift is determined by the value of \( h \). For the function \( f(x) = (x-3)^2 \), the graph of \( y = x^2 \) shifts to the right by 3 units.

These shifts occur because when \( h \) is positive, you are effectively subtracting a larger amount from each \( x \)-value. As a result, to maintain the same output \( f(x) \), you have to increase \( x \), thus shifting the graph to the right. Conversely, if \( h \) were negative, you'd be adding to \( x \), which would push the graph to the left.

Horizontal shifts do not alter the shape of the parabola; they simply change where it's located on the x-axis. This concept can be applied to any basic function \( y = g(x) \), resulting in \( y = g(x-h) \). It is essential to absorb how these transformations work as they lay the groundwork for more complex manipulations in algebra.

Vertical Shifts

Vertical shifts involve moving the graph up or down, dictated by the \( +k \) in \( f(x) = (x-h)^2 + k \). This transformation affects the entire function, changing the y-values by adding or subtracting \( k \). In our exercise, the graph of \( y = (x-3)^2 + 1 \) shifts upward by 1 unit due to the \(+1\).

When \( k \) is positive, as it is here, the whole graph rises vertically by \( k \) units. The vertex of the parabola, which was at the origin in the basic form \( y = x^2 \), moves upwards. Conversely, a negative \( k \) would move the graph down.

This provides an excellent visual cue for understanding vertical shifts. They are straightforward as compared to horizontal shifts but are equally crucial in the transformation toolbox. Such shifts make quadratic functions versatile, allowing them to model a wide range of real-world situations. Mastering vertical shifts will enhance your ability to analyze and graph complex functions accurately.