Problem 21 - Algebra 2 and Trigonometry

Question

a. Sketch the graph of \(y=x^{2} .\) b. Sketch the graph of \(y=x^{2}+2\) c. Sketch the graph of \(y=x^{2}-3\) d. Describe the graph of \(y=x^{2}+a\) in terms of the graph of \(y=x^{2}\) . e. What transformation maps \(y=x^{2}\) to \(y=x^{2}+a ?\)

Step-by-Step Solution

Answer

Each graph is a vertical shift of \(y = x^2\); \(y = x^2 + a\) shifts by \(a\) units.

Step 1: Understanding the Base Function

The base function given is \(y = x^2\), which is a standard parabola that opens upwards. It has its vertex at the origin (0,0) and is symmetric about the y-axis. This is the graph we will start with for transformations in the following parts.

Step 2: Sketching the Function y = x^2

The graph of \(y = x^2\) is drawn by plotting several points: for example (0,0), (1,1), (-1,1), (2,4), and (-2,4). These points show the curve of the parabola, which gets wider as \(|x|\) increases. Connect these points smoothly to sketch the entire parabola.

Step 3: Sketching y = x^2 + 2

To graph \(y = x^2 + 2\), take the basic \(y = x^2\) graph and shift it up by 2 units on the y-axis. Every point on the graph of \(y = x^2\), such as (0,0) or (1,1), now becomes (0,2), (1,3), etc., maintaining the same shape but vertically translated.

Step 4: Sketching y = x^2 - 3

Start with the graph of \(y = x^2\) and shift every point down by 3 units. This means (0,0) becomes (0,-3), (1,1) becomes (1,-2), and so on. The shape and width of the parabola remain unchanged, but its vertex moves to (0,-3).

Step 5: Analyzing the Graph y = x^2 + a

The function \(y = x^2 + a\) represents a vertical shift of the graph \(y = x^2\). If \(a > 0\), the graph shifts up by \(a\) units; if \(a < 0\), it shifts down by \(|a|\) units. The parabola's shape does not change, only its vertical position does.

Step 6: Describing the Transformation

The transformation that maps \(y = x^2\) to \(y = x^2 + a\) is a vertical translation. The graph is moved up if \(a\) is positive and down if \(a\) is negative. This transformation does not affect the parabola's shape or the symmetry with respect to the y-axis.

Key Concepts

Vertical Shift

A vertical shift in a graph occurs when each point on the graph is moved either up or down along the y-axis. In the equation of a function, this is represented by adding or subtracting a constant value. For quadratic functions of the form \(y = x^2\), a vertical shift is indicated by an additional constant \(a\), resulting in the function \(y = x^2 + a\).

This transformation doesn't alter the shape of the graph; it simply affects the graph's vertical position.

• If \(a > 0\), the entire graph moves up \(a\) units.
• If \(a < 0\), the graph shifts down by \(|a|\) units.

As a result, the vertex of the parabola, initially at the origin \((0,0)\), moves to \((0,a)\). Thus, by analyzing the value of \(a\), we can easily determine the extent and direction of the vertical shift.

Graph Sketching

Graph sketching involves plotting points on a coordinate plane to visualize the graph of a function. Starting with a basic understanding of key points helps sketch accurate graphs, especially for quadratic functions like \(y = x^2\).

To begin sketching a standard parabola, it's useful to choose points that give a clear picture of the curve's shape. Typical points for \(y = x^2\) might include:

• \((0,0)\)
• \((1,1)\) and \((-1,1)\)
• \((2,4)\) and \((-2,4)\)

By plotting these and other strategic points, you'll see the curve of the parabola. This basic shape is symmetric around the y-axis and becomes wider as \(x\) moves away from zero in either direction.

Standard Parabola

The standard parabola refers to the graph of the function \(y = x^2\). This is a simple yet important function in algebra, characterized by its distinct U-shape. The parabola opens upward, with its narrowest point, known as the vertex, located at the origin \((0,0)\).

Key features include:

• Symmetry: The parabola is symmetric with respect to the y-axis, meaning it looks the same on both sides of this axis.
• Vertex: The point \((0,0)\) is the vertex, serving as both the lowest point on the graph and the point of symmetry.
• Width: As you move away from the vertex, the parabola widens. Points equidistant from the y-axis (like \((1,1)\) and \((-1,1)\)) have the same y-value.

The standard parabola provides the groundwork for understanding more complex transformations and quadratic functions.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, and they form the basis for understanding parabolas. They are usually expressed in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

For the simplest quadratic function, \(y = x^2\), the coefficients are \(a = 1\), \(b = 0\), and \(c = 0\), resulting in the standard parabola. However, changing these coefficients leads to various transformations, such as:

• Vertical shifts: Adding a constant \(c\) shifts the graph vertically.
• Horizontal shifts: The presence of \(b\) suggests a horizontal shift, though it's often combined with other terms to determine the specific transformation.
• Stretches and compressions: The coefficient \(a\) affects the width and direction of opening, with \(a > 1\) meaning a narrower parabola and \(0 < a < 1\) meaning a wider one.

Understanding these components helps in graphing and interpreting quadratic functions in various forms.