How do i apply the vertical test to determine if a relation is a function?

Answer 1

The vertical line test is a visual way to determine whether a relation is a function. According to the definition of a function, each input (or x-value) must correspond to exactly one output (or y-value).

To apply the vertical line test, follow these steps:

1. Graph the relation: Plot the points of the relation on a coordinate system or draw the curve if it's defined by an equation.

2. Draw vertical lines: Imagine drawing vertical lines (or use a ruler or a straight edge) that can move across the graph from left to right.

3. Check for intersections: If any vertical line intersects the graph at more than one point, then the relation is not a function. This means that for the x-value corresponding to the intersection points, there are multiple y-values (outputs).

4. If no vertical line intersects more than once: If every vertical line intersects the graph at most once, then the relation is a function. This indicates that each x-value has a unique y-value.

Example

• Example of a function: The graph of \( y = x^2 \) passes the vertical line test, as any vertical line will intersect the graph at most once.

• Example of not a function: The graph of a circle, like \( x^2 + y^2 = r^2 \), fails the vertical line test because there are vertical lines that intersect the circle at two points, indicating that some x-values map to multiple y-values.

Using the vertical line test is a straightforward way to visually determine whether a relation is a function.

Answer 2

So if a graph shows a curve with y plot points at -6 and 6 and an x plot point at 6, its a function?

Answer 3

To determine if the described graph represents a function, we can apply the vertical line test.

1. Understanding the points: You mentioned that for an x-coordinate of 6, there are two different y-values: -6 and 6. This means that at \( x = 6 \), the relation has the points (6, -6) and (6, 6).

2. Applying the vertical line test: If you draw a vertical line at \( x = 6 \), it will intersect the graph at both points (6, -6) and (6, 6).

Since a vertical line intersects the graph at two points for that specific x-value, this indicates that there are multiple y-values (outputs) for the same x-value (input).

Therefore, according to the vertical line test, the relation is not a function.

In summary, a relation that assigns more than one output (y-value) for a single input (x-value) fails the definition of a function. In this case, it is not a function.

Answer 4

How can you tell the ordered pair for an equation like y = [frac/] -2/3x + 6?

Answer 5

It looks like you're trying to describe a linear equation in slope-intercept form, which is typically written as:

\[ y = mx + b \]

where \( m \) is the slope and \( b \) is the y-intercept.

In your example, the equation appears to have a formatting issue, but it seems like it may be written as:

\[ y = -\frac{2}{3}x + 6 \]

This indicates:

• The slope \( m = -\frac{2}{3} \)
• The y-intercept \( b = 6 \) (which means the graph intersects the y-axis at the point (0, 6)).

From this equation, you can derive various ordered pairs (points) that satisfy the equation by substituting different x-values and calculating the corresponding y-values. Here’s how to find some ordered pairs:

1. Choose x-values: Pick any values for \( x \). Common choices include integers like -3, -2, -1, 0, 1, 2, 3, etc.

2. Calculate y-values: Plug these x-values into the equation and solve for \( y \).

Example Ordered Pairs

1. For \( x = 0 \): \[ y = -\frac{2}{3}(0) + 6 = 6. \] Ordered pair: (0, 6).

2. For \( x = 3 \): \[ y = -\frac{2}{3}(3) + 6 = -2 + 6 = 4. \] Ordered pair: (3, 4).

3. For \( x = -3 \): \[ y = -\frac{2}{3}(-3) + 6 = 2 + 6 = 8. \] Ordered pair: (-3, 8).

4. For \( x = 6 \): \[ y = -\frac{2}{3}(6) + 6 = -4 + 6 = 2. \] Ordered pair: (6, 2).

5. For \( x = -6 \): \[ y = -\frac{2}{3}(-6) + 6 = 4 + 6 = 10. \] Ordered pair: (-6, 10).

Summary

You can generate an infinite number of ordered pairs \((x, y)\) that satisfy the equation by selecting different values for \( x \) and calculating the corresponding \( y \) values using the equation \( y = -\frac{2}{3}x + 6\).

Answer 6

How can you tell if a function is proporsional

Answer 7

A function is proportional if it can be expressed in the form

\[ y = kx \]

where \( k \) is a non-zero constant. This means that the relationship between \( y \) and \( x \) is linear, passing through the origin (0,0), with a constant ratio of \( y \) to \( x \) for any pair of corresponding values.

Here are a few criteria to determine if a function is proportional:

1. Linear Equation Through the Origin

• The equation of a proportional relationship is linear and has no constant term (intercept). It must pass through the origin (0, 0), which means when \( x = 0 \), \( y \) should also be \( 0 \).

2. Constant Ratio

• For any two points \((x_1, y_1)\) and \((x_2, y_2)\) on the graph of the function, the ratio \( \frac{y_1}{x_1} \) should be equal to \( \frac{y_2}{x_2} \). In other words, the output \( y \) is always some constant multiple (the constant of proportionality \( k \)) of the input \( x \).
• You can check the ratio using any data points: \[ k = \frac{y}{x} \]

3. Graph Characteristics

• If you plot the function on a graph, a proportional function will appear as a straight line that passes through the origin. Any deviation from this will indicate that the function is not proportional.

4. Equation Check

• If you have a function in the form \( y = mx + b \) (where \( m \) is the slope and \( b \) is the y-intercept):
  • If \( b = 0 \), then the function is proportional.
  • If \( b \neq 0 \), then the function is not proportional.

Example

1. Proportional Function:

   • \( y = 3x \) — passes through the origin and has a constant ratio (\( \frac{y}{x} = 3 \)).

2. Not a Proportional Function:

   • \( y = 2x + 1 \) — does not pass through the origin (the y-intercept is 1) and the ratio \( \frac{y}{x} \) changes at different points.

In summary, to determine if a function is proportional, check for a linear relationship that passes through the origin, maintain a constant ratio of \( y \) to \( x \), and ensure no constant term is present in the equation.