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Chapter 5: Problem 9

Consider what happens when you rotate a linear graph \(180^{\circ} .\) a. Graph the line \(y=2 x+4\) b. On the same grid, draw the image of the line under a \(180^{\circ}\) rotation centered at the origin. c. What do you notice about the image line and the original line? d. Write an equation of the new line. e. Does your equation in Part d support your observation in Part c? Explain.

Short Answer

Expert verified
The new line equation is \(y = -2x - 4\). The lines are symmetric about the origin.

Step by step solution

01

Graph the original line

Plot the line given by the equation \(y = 2x + 4\). Start by determining the y-intercept, which is 4 (when x = 0, y = 4). Then find another point on the line using the slope 2 (rise over run), such as moving up 2 units and 1 unit to the right, arriving at the point (1, 6). Draw the line passing through these points.
02

Understand 180-degree rotation

Rotation by 180 degrees around the origin transforms each point (x, y) on the plane to (-x, -y). That is, the coordinates of each point are negated.
03

Rotate the original line

Take points on the original line and apply the 180-degree rotation. For example, the original points (0, 4) becomes (0, -4), and (1, 6) becomes (-1, -6). Plot these new points on the same grid.
04

Draw the rotated line

Connect the points (0, -4) and (-1, -6) to form the image of the line under the 180-degree rotation.
05

Observe the relationship between the lines

Notice that the rotated line appears to be a reflection of the original line in the opposite quadrants (symmetric with respect to the origin).
06

Derive the equation of the new line

To derive the new equation, recognize that rotating \(y = 2x + 4\) by 180 degrees, changes the signs of both x and y coordinates. Substitute \(x\) with \(-x\) and \(y\) with \(-y\): \(-y = 2(-x) + 4\) Simplify to get: \(y = -2x - 4\)
07

Analyze the new equation

Compare the new equation \(y = -2x - 4\) with the original one \(y = 2x + 4\). This confirms the reflection relationship as observed in Part c, where the signs of both the slope and y-intercept have reversed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

180-degree rotation
To understand a 180-degree rotation, imagine spinning an object halfway around a circle. For points on a graph, this means flipping them over the origin. If you have a point (x, y), after a 180-degree rotation, its coordinates become (-x, -y). This is because every point鈥檚 horizontal and vertical coordinates change sign. For example, if you rotate (2, 3) by 180 degrees, it lands at (-2, -3). It's like taking a point in one quadrant of the coordinate plane and moving it to the opposite quadrant.
Linear transformations
Linear transformations change the position of a graph while maintaining a straight line. A 180-degree rotation is one such transformation. During this rotation, each point on a line is flipped across the origin. This changes the equation of the line as well. For instance, when y = 2x + 4 rotates 180 degrees, every point (x, y) on the line changes to (-x, -y). This step results in the new line equation y = -2x - 4. Here, both the slope (2) and the y-intercept (4) reverse their signs due to the rotation.
Symmetry in algebra
Symmetry in algebra occurs when a shape or graph mirrors another. In the context of linear equations and rotations, a 180-degree rotation results in symmetric lines about the origin. If you look at the lines y = 2x + 4 and y = -2x - 4, they are symmetric. This symmetry means that if one line is in the first and third quadrants, the other will be in the second and fourth quadrants after rotation. This happens because each coordinate pair (x, y) moves to (-x, -y), maintaining the line's structure while flipping its position.
Slope-intercept form
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. Understanding this form is crucial for graphing lines and predicting their changes under transformations like rotations. For example, y = 2x + 4 has a slope (m) of 2, indicating the line rises 2 units for every 1 unit it runs right. The y-intercept (b) is 4, meaning the line crosses the y-axis at y = 4. When you rotate this line 180 degrees, the new line, y = -2x - 4, has a slope of -2 and a y-intercept of -4, showing how the transformation changes both the slope and the intercept.

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Most popular questions from this chapter

Perspective drawings look three-dimensional. The projection method for making scale drawings is related to a method for making perspective drawings. On your own paper, follow the steps below to make a perspective drawing of a box. Use a pencil. a. Start by drawing a rectangle. This will be the front of your box. b. Choose a point outside your rectangle. This point is called the vanishing point for your drawing. Connect each vertex to that point, and then find the midpoint of each connecting segment. c. Connect the four midpoints you found in Part b to each other, in order. This gives you the back of the box. Then erase the lines connecting them to the vanishing point. d. To make the box clearer, erase the lines that should be hidden on the back of the box, or make them dashed. e. Follow the same steps to make a perspective drawing of a triangular prism. That is, start with a triangle (instead of a rectangle) and follow Part a鈥揹. f. In this method of three-dimensional drawing, at what step do you create a pair of similar figures? Explain.

Find the value of \(t\) in each equation. \(3^{t}=729\)

Write an equation of a line that is parallel to the given line. $$ 2 x=4 y $$

Complete Parts a鈥揺. a. Explain the rule in words. b. Find the coordinates of each vertex, and copy the figure onto graph paper. c. Perform the given rule on the coordinates of each vertex to find the image vertices. d. On the same set of axes, plot each image point. Connect them in order. e. Compare the image to the original: is it a reflection, a rotation, a translation, or some other transformation? Rule: \((x, y) \rightarrow(-x, y)\)

Suppose you translate a figure using this rule: \((x, y) \rightarrow(x+2, y-3)\) You then translate the image using this rule: \((x, y) \rightarrow(x-1, y-1)\) Where is the final image in relation to the original figure?
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