Question

Draw a graph that contains the points \((-2,5),(-1,3),\) and (0,2) and is symmetric with respect to the \(y\) -axis. Compare your graph with those of other students; comment on any similarities. Can a graph contain these points and be symmetric with respect to the \(x\) -axis? the origin? Why or why not?

Short answer

The graph can be symmetric about the \( y \)-axis but not the \( x \)-axis or the origin because neither corresponding points \((x, -y)\) nor \((-x, -y)\) exist.

Step-by-step solution

Plot the Given Points

Begin by plotting the points \((-2, 5)\), \((-1, 3)\), and \( (0, 2) \) on the coordinate plane.

Reflect Points Across the y-axis

To ensure the graph is symmetric with respect to the \( y \)-axis, reflect each point across the \( y \)-axis. This means converting \( x \) to \(-x\) while keeping \( y \) the same. The reflected points will be \( (2, 5) \) for \((-2, 5)\), \( (1, 3) \) for \((-1, 3)\), and \( (0, 2) \) remains the same.

Draw the Symmetric Graph

Next, plot the reflected points \( (2, 5) \) and \( (1, 3) \) along with the original points. Draw a smooth curve or line passing through all these points to complete the graph.

Compare With Other Graphs

Compare the graph with those of other students. Note that graphs symmetric with respect to the \( y \)-axis should look like mirror images across the \( y \)-axis.

Graph Symmetry Considerations

To determine if the graph can be symmetric with respect to the \( x \)-axis or the origin, recall: Symmetry with respect to the \( x \)-axis means that the point \( (x, y) \) would need a corresponding point \((x, -y) \). For these points, there are no symmetrical \((x, -y) \) points, so symmetry with the \( x \)-axis is not possible. Symmetry with respect to the origin means that a point \((x, y) \) would have a corresponding point at \((-x, -y)\). Similar reasoning applies, and none of the necessary corresponding points are plotted.

Key concepts

coordinate plane

The coordinate plane is a 2-dimensional surface where we can plot points, lines, and curves. It is divided by the x-axis (horizontal) and the y-axis (vertical) which intersect at the origin \((0, 0)\). Each point on this plane is represented by a pair of numbers \((x, y)\), where \x\ is the horizontal distance from the origin and \y\ is the vertical distance.
For example, the point \((-2, 5)\) means the point is 2 units to the left of the origin and 5 units up. On a graph, this helps us visualize relationships between different datasets.

• x-axis: The horizontal line on the plane.
• y-axis: The vertical line on the plane.
• Origin: Where the x-axis and y-axis intersect at (0, 0).

reflection

Reflection in mathematics refers to flipping a point or a shape across a line, creating a mirror image. When we reflect points on the coordinate plane, we use these principles:
Reflection across the y-axis means keeping the y-coordinate the same while changing the x-coordinate to its opposite. For instance, reflecting \((-2, 5)\) across the y-axis gives \(2, 5)\).
Reflection across the x-axis similarly keeps the x-coordinate the same but changes the y-coordinate to its opposite. For example, reflecting (0, 2) across the x-axis would result in (0, -2).

• y-axis reflection: (x, y) becomes (-x, y)
• x-axis reflection: (x, y) becomes (x, -y)

This is useful when determining symmetry for graphs and equations.

symmetry with respect to y-axis

A graph is symmetric with respect to the y-axis if one side of the graph is a mirror image of the other side about the y-axis. This means reflecting every point \((x, y)\) of the graph would have a corresponding point \((-x, y)\).
For example, in the given exercise, the points \((-2, 5), (-1, 3),\) and (0, 2) have corresponding points (2, 5) and (1, 3), respectively. This indicates that the graph is symmetric with respect to the y-axis.

• Original points: \((-2, 5)\)
• Reflected points: \(2, 5)\)

This concept helps us understand the balance and structure in various mathematical graphs and functions.

symmetry with respect to x-axis

Symmetry with respect to the x-axis is when every point \((x, y)\) on the graph has a corresponding point \((x, -y)\). This means the graph looks identical above and below the x-axis, upside-down mirror image.
In the exercise, points like \((-2, 5)\) do not have corresponding points \((-2, -5)\), so the given graph isn't symmetric with respect to the x-axis.
Key things to remember:

• Original point: \((x, y)\) should have \((x, -y)\)
• Example: (3, 4) should have (3, -4)

Understanding this concept helps us analyze and interpret different types of functions and their shapes on the coordinate plane.