Question

if a vertical line intersects a graph at more than one point, the graph is ____. the graph of ( y = |x| ) is a __ relationship. the graph of a circle, such as ( x^2 + y^2 = 9 ), is a __ relationship. each input in a ____ relationship has exactly one output.

Explanation:

First Question:

Step1: Recall vertical line test

The vertical line test determines if a graph is a function. If a vertical line intersects a graph at more than one point, it fails the vertical line test, so it's not a function (it's a relation that's not a function).

Step2: Determine the term

So the graph is a "non - function" or more precisely, just a "relation" (but in the context of the vertical line test, the key term here is that it's not a function, but the standard term for a graph that fails the vertical line test is that it's not a function, so we can say "not a function" or "a relation that is not a function". But the common fill - in here is "not a function" or "a relation (that is not a function)". However, the most precise term based on the vertical line test is that it's not a function.

Second Question:

Step1: Analyze \(y = |x|\)

For the function \(y=|x|\), we can perform the vertical line test. For any value of \(x\), there is exactly one value of \(y\) (since \(y=

$$\begin{cases}x, & x\geq0\\-x, & x < 0\end{cases}$$

\)). So it passes the vertical line test.

Step2: Determine the relationship type

A relationship where each input has exactly one output is a function. So the graph of \(y = |x|\) is a "function" relationship.

Third Question:

Step1: Analyze the circle equation \(x^{2}+y^{2}=9\)

If we solve for \(y\), we get \(y=\pm\sqrt{9 - x^{2}}\). For a given \(x\) (such as \(x = 0\), \(y=\pm3\)), there are two values of \(y\). So if we draw a vertical line (for example, \(x = 0\)) through the graph of the circle, it will intersect the circle at two points \((0,3)\) and \((0, - 3)\).

Step2: Determine the relationship type

Since it fails the vertical line test, the graph of the circle is a "non - function" (or "a relation that is not a function") relationship.

Fourth Question:

Answer:

s:

1. not a function (or a relation that is not a function)
2. function
3. non - function (or a relation that is not a function)
4. function