Problem 111 - Introductory Algebra for College Students

Question

a. Graph each of the following points: $$ \left(1, \frac{1}{2}\right),(2,1),\left(3, \frac{3}{2}\right),(4,2) $$ Parts (b)-(d) can be answered by changing the sign of one or both coordinates of the points in part (a). b. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the \(y\) -axis of your graph in part (a)? c. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the \(x\) -axis of your graph in part (a)? d. What must be done to the coordinates so that the resulting graph is a straight-line extension of your graph in part (a)?

Step-by-Step Solution

Answer

For reflection about the y-axis, change the sign of x-coordinates. For reflection about the x-axis, change the sign of y-coordinates. For a straight-line extension, keep increasing x and y coordinates with the same pattern.

Step 1: Graphing points

Plot the points \( (1, \frac{1}{2}), (2,1), (3, \frac{3}{2}), (4,2) \) on your graph.

Step 2: Reflecting points about the y-axis

To transform the points such that the resulting graph is a mirror-image reflection about the y-axis of the original graph in part (a), the transformation rule is \( (x,y) \to (-x,y) \) . So, the new coordinates are \( (-1, \frac{1}{2}), (-2,1), (-3, \frac{3}{2}), (-4,2) \)

Step 3: Reflecting points about the x-axis

For transforming the points into a mirror-image reflection about the x-axis, the rule is \( (x,y) \to (x,-y) \). Using this rule, the new coordinates are \( (1, -\frac{1}{2}), (2,-1), (3, -\frac{3}{2}), (4,-2) \)

Step 4: Straight-line extension

The original graph forms a straight line with a slope of 1/2. To extend this line, continue the pattern in the coordinates. For instance, \( (5, \frac{5}{2}), (6,3) \) and so on.

Key Concepts

Graphing Points

Graphing points is a fundamental concept in coordinate geometry. It involves plotting points using ordered pairs, which are written in the form \(x, y\).
Each pair represents a location on a two-dimensional plane.
For example, the point \( (1, \frac{1}{2}) \) means you move 1 unit along the x-axis and \( \frac{1}{2} \) unit along the y-axis. To graph points:

• Identify the x-coordinate: It tells you how far to move left or right from the origin (0,0).
• Identify the y-coordinate: It guides you on how far to move up or down.
• Mark the spot where these positions intersect on the grid.

For instance, the points \( (2, 1) \) or \( (3, \frac{3}{2}) \) are plotted similarly.
Make sure each point is carefully marked to accurately represent the data.

Reflections

Reflections in coordinate geometry involve flipping a point over an axis. This concept is essential for understanding symmetry and transformations in a graph.
Two common types of reflections are about the x-axis and y-axis.When reflecting over the **y-axis**:

• The rule is \( (x, y) \to (-x, y) \).
• Here, the x-coordinate changes sign, while the y-coordinate remains unchanged.

For example, reflecting the point \( (2, 1) \) would result in \( (-2, 1) \).For a reflection over the **x-axis**:

• The rule is \( (x, y) \to (x, -y) \).
• The y-coordinate changes sign, while the x-coordinate stays the same.

So, reflecting \( (3, \frac{3}{2}) \) over the x-axis gives \( (3, -\frac{3}{2}) \).Using reflections, we can create mirror-like transformations of figures across the coordinate axes.

Transformations

Transformations are operations that alter the position of points or shapes in space. In coordinate geometry, transformations can include reflections, rotations, translations, and dilations.
We've already discussed reflections, and now we'll briefly touch on other aspects too. **Translations** involve shifting a shape horizontally or vertically without changing its orientation.
For instance, adding 2 to an x-coordinate moves the point 2 units to the right. **Rotations** turn a shape around a point, usually the origin, by a certain angle.
These are a bit more complex than reflections and translations, as they often require trigonometric understanding. Transformations are vital for visualizing movements and changes in geometric figures.
Understanding this helps in manually adjusting graphs and applying mathematical rules effectively.

Slope and Lines

The slope is a measure of how steep a line is and is calculated as the change in y divided by the change in x, or \( m = \frac{{\Delta y}}{{\Delta x}} \). This is crucial for understanding linear relationships and behaviors in graphs.
In the example, the line created by the points \( (1, \frac{1}{2}), (2,1), (3, \frac{3}{2}), (4,2) \) has a slope of \( \frac{1}{2} \).
This demonstrates that for every 2 units moved horizontally (along the x-axis), the line moves 1 unit vertically (along the y-axis).To extend a straight line:

• Continue using the slope to find additional points.
• For example, after \( (4, 2) \), adding another step gives \( (5, \frac{5}{2}) \).

Straight lines and slopes are foundational for graphing equations and understanding functions in algebra.
They allow for predicting future points and analyzing trends.