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Chapter 9: Problem 84

The graph of \(x=-y^{2}\) is a parabola, but the equation does not define a function. Explain.

Short Answer

Expert verified
The equation \(x=-y^{2}\) does not define a function because it fails the 'vertical line test', meaning there are not unique outputs for each input.

Step by step solution

01

Plotting the function

Start by plotting the equation \(x=-y^{2}\) on a plane. The graph will be a parabola, but instead of opening upwards or downwards (like \(y=x^{2}\) or \(y=-x^{2}\)), the parabola will open toward the left because of the negative sign on the right-hand-side.
02

Testing the 'vertical line test'

The 'vertical line test' is a method used to verify if a curve is a graph of a function or not. This involves imagining or drawing a vertical line along the plane where the function is plotted. If the line crosses the curve at more than one point, then the curve does not represent a function. In this case, the vertical lines would cross our parabola in two places which signifies that for a given input 'x', there are two corresponding outputs 'y'.
03

Conclusion

Thus, even though our given equation \(x=-y^{2}\) can be plotted and result in a familiar shape (a parabola), it does not pass the vertical line test and therefore does not represent a function. This is because it violates the key rule of a function that each input should correspond to exactly one output.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph of Equation
When we refer to the "graph of an equation," it means plotting all possible pairs \(x, y\) that satisfy the equation on a coordinate plane. This visual representation helps us understand the relationship between the variables involved.
For example, in the equation \(x = -y^2\), instead of having \(y\) as the dependent variable and \(x\) as the independent one, it is reversed. Here, \(y\) values are chosen, and corresponding \(x\) values are calculated, resulting in a parabola that opens sideways.
  • The negative sign before \(y^2\) causes the parabola to open leftwards.
  • Each point on the graph corresponds to a valid pair that satisfies \(x = -y^2\).

Thus, the graph of this equation allows us to visualize its structure, shape, and the relation between its variables.
Function Definition
In mathematics, a function is defined as a relationship between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. This means for a given input \(x\), there must be only one possible \(y\) value.
To check if a graph represents a function, we often use what's known as the Vertical Line Test:
  • Imagine drawing vertical lines (parallel to the y-axis) across your graph.
  • If any vertical line crosses the graph at more than one point, the graph is not a function because it shows that a single \(x\) value leads to more than one \(y\) value.
  • If each line touches only once, it confirms each \(x\) is paired with exactly one \(y\).

With the graph of \(x = -y^2\), vertical lines will intersect the parabola in two places, indicating that it does not meet the function criteria.
Parabola
Parabolas are a specific kind of curve that can be defined by quadratic equations. Typically expressed as \(y = ax^2 + bx + c\), they are symmetrical and can open upwards, downwards, right, or left depending on orientation and signs within the equation.
For \(x = -y^2\),
  • The standard form \(y = ax^2\) is flipped to \(x = ay^2\) because \(y\) is the independent variable in this context.
  • The negative sign indicates the parabola opens leftwards, as opposed to the common up or down direction.

This distinct feature of orientation shows the flexibility of parabolas and how their graphs can be manipulated by changing the variable and sign arrangement. Understanding these properties is crucial when analyzing different quadratic equations and predicting their graphical nature.

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Most popular questions from this chapter

Simplify. Write each result in a + bi form. $$(4+3 i)-(-4-i)$$

Use the quadratic formula to find all real solutions of each equation. SEE EXAMPLE \(2 .\) (OBJECTIVE 1) $$7 x^{2}-21=14 x$$

Simplify. Write each result in a + bi form. $$(-2-\sqrt{-16})(1+\sqrt{-4})$$

Simplify. Write each result in a + bi form. $$(1+\sqrt{-5})(2-\sqrt{-3})$$

Use the quadratic formula to find all real solutions of each equation. SEE EXAMPLE \(2 .\) (OBJECTIVE 1) $$3 x^{2}=-7 x-4$$
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