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Chapter 8: Problem 6

This table lists the vertices of a triangle. Name the vertex or vertices that will not be affected by doing a vertical stretch. (h)

Short Answer

Expert verified
Vertices with a y-coordinate of zero will not be affected by a vertical stretch.

Step by step solution

01

Understand Vertical Stretch

A vertical stretch is a transformation that stretches the figure vertically by multiplying the y-coordinates by a factor greater than 1, while the x-coordinates remain unchanged. For a point \((x,y) \,\rightarrow\, (x,k\cdot y)\)where \(k > 1\). This implies that only the y-coordinates are affected.
02

Identify Vertices Coordinates

Examine the given vertices of the triangle. Let鈥檚 assume the vertices of the triangle are:\(A(x_1, y_1), \ B(x_2, y_2), \ C(x_3, y_3)\)Identify the specific coordinates for each vertex to analyze the change.
03

Analyze Effect on Each Vertex

Determine the ideal vertices that will remain unaffected by the vertical stretch. Since a vertical stretch affects the y-coordinate, vertices with a y-coordinate of zero will remain unchanged. This is because:\((x,0) \,\rightarrow\, (x,0)\)Therefore, any vertex with a y-coordinate of zero will stay unaffected.
04

Determine the Vertices Unaffected

If there exists any vertex amongst \(A, B, \text{or} C\) whose y-coordinate is zero, that vertex will not be affected by the vertical stretch. Identify such vertices.For example, if vertex \(A\) has coordinates \(A(x_1, 0)\), it will not change.

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Key Concepts

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

Transformation
A transformation in mathematics refers to altering a shape or figure on a plane. Many transformations can be done, such as translation, rotation, reflection, and scaling (including stretching and compressing). Each type moves or changes the shape in specific ways.
A vertical stretch specifically changes the dimension of a shape along the y-axis.
This means that when applying a vertical stretch, only the height of the object is affected, while its width remains the same. This is done by multiplying the y-coordinate of each point by a scale factor greater than 1.
Coordinates
Coordinates are numerical values that define the position of a point or a vertex in a plane, typically denoted as \((x, y)\). The x-coordinate indicates the horizontal position, while the y-coordinate shows the vertical position.
When performing transformations such as a vertical stretch, the coordinates of the points are altered.
In the case of a vertical stretch, the transformation only affects the y-coordinate, leaving the x-coordinate unchanged.
Triangle Vertices
Triangles are polygons with three edges and three vertices. Each vertex of a triangle can be represented using its coordinates, such as \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\). These coordinates are helpful in analyzing transformations like a vertical stretch.
When stretching vertically, examine each vertex individually to determine the changes.
If a vertex has a y-coordinate of zero, it remains unaffected by the transformation, making it easier to predict and analyze outcomes.
Mathematical Analysis
Mathematical analysis involves examining numeric data and transformations to understand their impact. It's a detailed look into how changes affect the structure and properties of geometric figures.
For vertical stretch, analysis determines which vertices are modified by transforming the y-coordinates.
By reviewing the triangle's vertex coordinates, one can conclude that any vertex with a y-coordinate of zero stays the same after applying the transformation. This small detail highlights the importance of mathematical analysis in predicting outcomes.

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Most popular questions from this chapter

APPLICATION The equation \(y=a \cdot b^{x}\) models the decreasing voltage of a charged capacitor when connected to a load. Measurements for a particular 9 -volt battery are recorded in the table. \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline Time (s) & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline Voltage (volts) & \(6.579\) & \(6.285\) & \(5.992\) & \(5.738\) & \(5.484\) & \(5.230\) & \(4.995\) \\ \hline \end{tabular} [Data sets: CAPTM, CAPVT] a. The voltages are given beginning at time \(t=10 \mathrm{~s}\) rather than \(t=0 \mathrm{~s}\). How can the equation \(y=a \cdot b^{x}\) be changed to account for this? (a) b. To model the data, you need values of \(a\) and \(b\). For the value of \(b\), what is the average ratio between consecutive voltages? c. Find the value of \(a\) and write the equation that models these data. (h) d. Use your equation to predict the voltage at time 0 . e. Use your equation to predict when the voltage is less than \(1 .\)

APPLICATION Deshawn is designing a computer animation program. She has a set of coordinates for the tree shown on the right side. She wants to use 13 frames to move the tree from the right to the left. In each frame, she wants the tree height to shrink by \(80 \%\). How should she define the coordinates of each image using the coordinates from the previous frame? (a)

Jack lives in a cabin at the bottom of a hill. At the top of the hill, directly behind his cabin, is an observation tower. There is a creek at the bottom of the hill on the side opposite Jack's cabin. Match each description in a-c to one of the graphs below. Then answer part d. The horizontal axis in each graph shows time, and the vertical axis shows distance from the top of the hill. a. Jack walks steadily from the cabin to the observation tower. b. Jack walks steadily from the observation tower to the creek. c. Jack walks steadily from the cabin to the creek. d. Create a walking context and story for the unmatched graph.

Describe how the graph of \(y=x^{2}\) will be transformed if you replace \(x\) with \((x-3)\) b. \(x\) with \((x+2)\) (a) c. \(y\) with \((y+2)\) (a) d. \(y\) with \((y-3)\)

Perform the indicated operation and reduce the result to lowest terms. State any restrictions on the variable. Use your calculator to verify your answers. a. \(\frac{4 x^{3}}{24 x^{6}}+\frac{12 x^{4}}{15 x}\) b. \(\frac{3(x-6)}{18} \cdot \frac{4(x+6)}{8(x-6)}\) c. \(\frac{4 x y^{3}}{(2 x)^{3}} \div \frac{2 y^{2}}{1}\) d. \(\frac{3(x+4)}{5 x} \cdot \frac{20 x^{2}}{6 x^{2}+24 x}\)
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