Problem 33 - Intermediate Algebra

Question

Use the DRAW feature of your graphing calculator to draw each of the following. (a) A line segment between \((-2,-4)\) and \((-2,5)\) (b) A line segment between \((2,2)\) and \((5,2)\) (c) A line segment between \((2,3)\) and \((5,7)\) (d) A triangle with vertices at \((1,-2),(3,4)\), and \((-3,6)\)

Step-by-Step Solution

Answer

Use DRAW function to connect specified points for line segments and a triangle.

Step 1: Plot Line Segment (a)

Use your graphing calculator to plot a vertical line segment. Input the points \((-2, -4)\) and \((-2, 5)\) into the calculator's DRAW function. The line will be vertical because both endpoints have the same x-coordinate (-2).

Step 2: Plot Line Segment (b)

Input the points \((2, 2)\) and \((5, 2)\) into the DRAW function. This will plot a horizontal line segment because the y-coordinates are the same (2) for both points.

Step 3: Plot Line Segment (c)

Input the points \((2, 3)\) and \((5, 7)\). This line segment is neither purely vertical nor purely horizontal. It is diagonal, and the slope can be calculated as \(m = \frac{7 - 3}{5 - 2} = \frac{4}{3}\). This gradient indicates a rise of 4 units for every 3 units moved along the x-axis.

Step 4: Plot Triangle (d)

Input the vertices \((1, -2)\), \((3, 4)\), and \((-3, 6)\) into the DRAW function. Use the calculator's feature to connect these points with line segments, forming a triangle. Check that all lines are visible and connect between each pair of points.

Key Concepts

Line Segments

A line segment is a part of a line bounded by two distinct endpoints. When plotting a line segment on a graphing calculator, you simply connect these points with a straight line. Let's consider some examples to make this concept clearer:

• In exercise (a), the line segment connects (-2, -4) to (-2, 5). This segment is vertical because both points share the same x-coordinate. Vertical segments are easy to recognize, as they run up and down.
• For exercise (b), we connect (2, 2) to (5, 2). With identical y-coordinates, this line is horizontal, stretching left to right.
• Exercise (c) involves a diagonal segment between (2, 3) and (5, 7). Here, the line neither runs parallel to the x-axis nor the y-axis. Diagonal segments have a slope, which tells us how steep the line is.

Understanding line segments helps us visualize portions of data and various geometric shapes, such as triangles.

Plotting Points

Plotting points is an essential skill in coordinate geometry. It involves marking specific locations on a graph using coordinate pairs, such as (x, y). Each pair corresponds to a location in space defined by an x-coordinate (horizontal distance) and a y-coordinate (vertical distance).

When you enter coordinates into a graphing calculator, you should:

• Identify the x-value, move along the x-axis to reach it.
• Determine the y-value, move up or down from the x-value to plot the point accurately.

In our examples, once points are plotted, you can start creating other shapes. Whether plotting a line or a more complex geometric shape like a triangle, precise point location is key.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses a coordinate system to investigate geometric shapes. In this method, every point in the plane is specified by a pair of numerical coordinates. This is crucial when working with graphs and understanding spatial relationships between points.

With graphing calculators, coordinate geometry allows us to:

• Visualize how points relate spatially, forming line segments and shapes.
• Determine slopes of lines by calculating the rise and run between points, as seen in exercise (c) with a slope of \( \frac{4}{3} \).
• Analyze geometric properties and relationships using coordinates to gain a better understanding of your data.

Coordinate geometry bridges algebra and geometry, as it involves equations to represent a geometrical figure.

Triangles

A triangle is a polygon with three edges and three vertices. In coordinate geometry, you define a triangle with three pairs of coordinates, which are its vertices. Let's visualize this with our exercise

In exercise (d), the vertices are

• (1, -2)
• (3, 4)
• (-3, 6)

By marking these points and connecting them in sequence with line segments, you form a triangle. Graphing calculators simplify this process, enabling visual analysis of the shape.

Triangles have unique characteristics, such as the sum of internal angles equal to 180 degrees. Recognizing how triangles form and behave aids understanding in fields ranging from basic geometry to more advanced applications in physics and engineering.