Problem 23 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

Reflect \( y = x^2 \) across the x-axis to get \( y = -x^2 \).

Step 1: Sketch the graph of \( y = x^2 \)

The function \( y = x^2 \) is a basic quadratic function. It forms a parabola that opens upwards with its vertex at the origin (0,0). As \( x \) increases or decreases from 0, \( y \) increases at an accelerating rate, forming a symmetric U-shaped curve.

Step 2: Sketch the graph of \( y = -x^2 \)

The function \( y = -x^2 \) is similar to \( y = x^2 \) but with a negative coefficient for \( x^2 \). This forms a parabola that opens downwards with its vertex at the origin (0,0). As \( x \) increases or decreases from 0, \( y \) decreases at an accelerating rate, forming an inverted U-shaped curve.

Step 3: Compare the graphs of \( y = x^2 \) and \( y = -x^2 \)

The graph of \( y = -x^2 \) is a vertical reflection of \( y = x^2 \). Both graphs are parabolas with their vertices at the origin, but one opens upwards while the other opens downwards. This reflection occurs across the x-axis.

Step 4: Identify the transformation

The transformation that maps \( y = x^2 \) to \( y = -x^2 \) is a reflection across the x-axis. This means every point \( (x, y) \) on the graph of \( y = x^2 \) is transformed to \( (x, -y) \) on the graph of \( y = -x^2 \).

Key Concepts

Understanding Parabolas

A parabola is a symmetrical curve that appears in the graph of a quadratic function. The simplest form of a quadratic function is given by the equation \( y = x^2 \). This forms the basic parabola. A few key features define parabolas in general:

• The **vertex**, which is the point where the parabola changes direction, is at the origin, (0, 0), in the case of \( y = x^2 \).
• **Symmetry** is a hallmark characteristic of parabolas. The product of this symmetry is a U-shaped curve.
• As you move away from the vertex, both left and right along the x-axis, the y-values increase, creating the distinct curve.

Understanding these features helps in predicting how changes to the standard equation \( y = x^2 \) affect the shape and orientation of its graph. This basis is essential when exploring graph transformations and reflections.

Exploring Graph Transformations

Graph transformations are alterations made to the graph of a function that changes its position or shape. In the given exercises, we explore basic transformations that affect parabolas:

When viewing transformations, consider these effects:

• **Translation** shifts the graph along the x or y-axis. This occurs when you add or subtract constants to the function.
• **Dilation**, or stretching/compressing, alters the steepness. Multiplying by a coefficient greater than 1 stretches the graph, while a coefficient between 0 and 1 compresses it.
• **Reflection**, which we'll explore in detail in the next section, flips the graph across an axis.

By applying these transformations to the quadratic equation \( y = x^2 \), you can shift, stretch, compress, or even flip the parabola to explore countless variations. Understanding each will deepen your insight into how they transform the basic parabolic shape.

Reflection Across the X-axis

Reflecting a graph means flipping it over a designated axis. For parabolas, reflection across the x-axis is a fundamental transformation. Specifically, this transformation turns a parabola that opens upwards into one that opens downwards.

To achieve this, the function \( y = x^2 \) is multiplied by -1, becoming \( y = -x^2 \). This process has several clear outcomes:

• The **vertex** of the parabola remains at the origin, (0, 0).
• The **direction** of the parabola changes; instead of opening upwards, it now opens downwards.
• For each point on the original parabola \( (x, y) \), its counterpart on the reflected graph will be \( (x, -y) \).

Through understanding reflections across the x-axis, you can predict the behavior of the parabola when the signs are changed in its equation. This reflection happens symmetrically, maintaining the general shape of the parabola while modifying its direction.