Problem 18 - Algebra 2

Question

Graph each relation and its inverse. $$ y=-x^{2} $$

Step-by-Step Solution

Answer

The graph for the function \(y=-x^{2}\) is a downward parabola. Its inverse relation \(x=-y^{2}\) doesn't have a function graph because it does not pass the vertical-line test. Thus, \(y=-x^{2}\) does not have an inverse function.

Step 1: Graph the given function

Let's first plot the function \(y=-x^{2}\). This is a downward opening parabola, with the vertex at the origin (0,0). As \(x\) increases or decreases, \(y\) decreases, which is characteristic of negative quadratic functions.

Step 2: Find the inverse of the given function

An inverse of a function is found by swapping 'x' and 'y'. So the inverse relation starts as \(x=-y^{2}\). However, to express this as a function of 'x', we must solve for 'y', which is a multi-valued solution because for a given 'x', there are two possible 'y' values corresponding to a positive and negative square root. As the existence of multiple values for 'y' contradicts the definition of a function, we conclude that the function \(y=-x^{2}\) does not have an inverse function.

Step 3: Graphical representation of the inverse relation

The graph of the inverse relation \(x=-y^{2}\) would be a parabola opening to the left, with all x-values being negative or zero. Plotting the inverse relation, it's obvious that it does not represent a function, as it does not pass the vertical line test. A vertical line intersects the graph in more than one place, meaning there are multiple 'y' values for a given 'x' value.

Key Concepts

Quadratic Functions

A quadratic function is an important concept in algebra and pre-calculus. It is generally expressed in the form \( y = ax^2 + bx + c \). In this case, the given function, \( y = -x^2 \), is a simple quadratic function where \( a = -1 \) and \( b = c = 0 \). The graph of such a function is a curve known as a "parabola".
This particular form, where \( b \) and \( c \) are zero, makes the parabola symmetric about the y-axis and simplifies graphing processes. Quadratic functions are useful in modeling numerous real-world scenarios, especially where there are acceleration and deceleration patterns, like projectile motion.

Graphing Parabolas

Graphing a parabola involves plotting points that satisfy the quadratic equation on a coordinate plane. For the function \( y = -x^2 \), the graph is a downward-opening parabola.
Here's how to graph it:

• Identify the vertex. For \( y = -x^2 \), the vertex is at the origin \((0,0)\).
• Determine the direction of the opening. The negative sign indicates the parabola opens downwards.
• Plot additional points to define the curve. For instance, at \( x = 1 \) and \( x = -1 \), \( y = -1 \).

Remember, for quadratic functions, the parabolas are symmetric about the vertical line passing through the vertex.

Function and Relation

In mathematics, a function is a special type of relation where each input (or 'x' value) corresponds to exactly one output (or 'y' value). The function \( y = -x^2 \) is indeed a function as it satisfies this criterion.
However, when finding its inverse, we swap 'x' and 'y', converting it to \( x = -y^2 \). Solving this for 'y' involves taking a square root, which results in two possible values (\( y = \pm \sqrt{-x} \)) for every 'x'.
This output is not a function since it does not have a unique 'y' for each 'x'. Instead, it remains a relation, illustrating how inverses can lose the function property.

Vertical Line Test

The vertical line test is a useful graphical method to determine if a curve represents a function. The rule is simple: if any vertical line intersects the curve more than once, then the curve is not a function.
For a parabola such as \( y = -x^2 \), any vertical line crosses the curve at exactly one point, satisfying the vertical line test, confirming it is a function.
In contrast, the inverse relation \( x = -y^2 \), plotted as a parabola opening leftward, fails this test. Here, a vertical line can intersect the curve at two points simultaneously, revealing it as not a function. This illustrates the importance of the vertical line test in understanding graphs.