Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=x^{2}\\\&g(x)=x^{2}+2\\\&h(x)=(x-2)^{2}\end{aligned}$$.

Step-by-Step Solution

Answer

The function \(f(x) = x^2\) is a basic parabola. The function \(g(x) = x^2 + 2\) is the same parabola shifted upwards by 2 units, and the function \(h(x) = (x-2)^2\) is the parabola shifted right by 2 units.

Step 1: Sketch the basic parabola \(f(x) = x^2\)

Plot the points of \(f(x) = x^2\) in the grid. Remember a quadratic function forms a parabola. The points \((-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)\) can be used to sketch the basic parabolic curve \(f(x) = x^2\).

Step 2: Sketch the square function shifted upwards \(g(x) = x^2 + 2\)

The function \(g(x) = x^2 + 2\) represents a vertical shift of the basic parabolic curve upwards by 2 units. Plot the points, remembering that each 'y' coordinate of the original \(f(x) = x^2\) graph will be increased by 2. The shifted graph should look identical to the original, but moved higher.

Step 3: Sketch the squared function shifted to the right \(h(x) = (x-2)^2\)

The function \(h(x) = (x-2)^2\) represents a horizontal shift of the basic parabolic curve to the right by 2 units. Each 'x' coordinate of the original \(f(x) = x^2\) graph will be increased by 2. The shifted graph should appear identical to the original, but moved to the right.

Step 4: Verification with a graphing utility

Verify these graphs using a graphing utility. All the three equations should precisely overlay onto the graphs plotted by the utility. Verification is essential to ensure that the hand-drawn plots are correct.

Key Concepts

Parabolas

A parabola is a symmetrical, curve-shaped graph that is formed by the equation of a quadratic function. When a quadratic function is written in the general form, such as \( f(x) = x^2 \), its graph creates a U-shaped curve that opens upwards (for positive quadratic coefficients) or downwards (for negative quadratic coefficients).
A classic property of the parabola is its "vertex," which represents the curve's highest or lowest point, depending on the opening direction. This feature makes understanding their graph essential in studying quadratic functions. To sketch the parabola for \( f(x) = x^2 \), you plot several key points, such as \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), and \((2, 4)\). These points form the basic shape of the parabola in the coordinate system.

Vertex Form

The vertex form of a quadratic function provides a simplified way to understand the positioning of the parabola's vertex. This form is expressed as \( a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
In vertex form, you can directly see shifts of the parabolic graph from the origin (0,0). The form \( g(x) = x^2 + 2 \) indicates a vertical shift upward of the basic parabola\( f(x) = x^2 \) by 2 units. Here, the vertex is at \((0, 2)\), which is simply moved up from \( f(x) \)'s \((0, 0)\).
For \( h(x) = (x-2)^2 \), the vertex shifts to the right by 2 units to \((2, 0)\). Such transformations, visible in the vertex form, make it easier to sketch and understand the graph's behavior without manually calculating multiple points.

Transformations of Functions

Transformations of functions are modifications of the basic function graph, such as shifting and stretching, to achieve a desired position or shape. Understanding transformations is crucial when graphing quadratic functions as they help predict the graph's behavior without excessive calculations.

• **Vertical Shifts**: Moving the graph up or down by adding or subtracting a constant to/from the function. For example, adding 2 in \( g(x) = x^2 + 2 \) lifts the entire parabola up by 2 units.
• **Horizontal Shifts**: Shifts the graph left or right. The expression \( h(x) = (x-2)^2 \) moves the original parabolic curve of \( f(x) = x^2 \) to the right by 2 units.
• **Reflection and Stretching**: Other transformations involve flipping the graph over an axis or altering the graph's width by multiplying the function by a coefficient. These are less relevant for this exercise but critical in broader parabolic transformations.

Understanding these shifts is one of the first steps in mastering function transformations in mathematics. They help illustrate how simple modifications to an equation can lead to entirely different graph behaviors.