Problem 41
Question
The graph of each equation is to be translated 3 units right and 5 units up. Write each new equation. \(\frac{(x-2)^{2}}{36}-\frac{(y-3)^{2}}{25}=1\)
Step-by-Step Solution
Verified Answer
The equation of the hyperbola after being translated 3 units to the right and 5 units upwards is \(\(\frac{(x-5)^{2}}{36}-\frac{(y-8)^{2}}{25}=1\)\).
1Step 1: Identify the center of the hyperbola
The given hyperbola's equation has coefficient of x as 2 and y as 3 which are opposite of the center's coordinates. Therefore, the center of the given hyperbola is at \((h, k) = (2, 3)\).
2Step 2: Apply shift changes
To translate or shift the graph to the right by 3 units and up by 5 units, we add these transformations to the center coordinates of the original hyperbola. Therefore, the new center of the hyperbola \((h', k') = (2+3, 3+5) = (5, 8)\).
3Step 3: Frame the new equation
Substitute the new center coordinates back into the original form of the hyperbola equation. Hence, the new equation after the transformation is \(\(\frac{(x-5)^{2}}{36}-\frac{(y-8)^{2}}{25}=1\)\).
Key Concepts
Translation of GraphsCoordinate SystemsEquation Transformation
Translation of Graphs
Translating graphs is a fundamental concept in studying hyperbolas and other conic sections. When we talk about translating a graph, we essentially mean moving it around in the plane without altering its shape or orientation.
Think of it as sliding a piece of paper across a desk - the paper stays the same, but its position changes. In the context of equations, this process involves adjusting the constants within the equation to reflect the new positions of the graph's essential points, such as the center in the case of a hyperbola.
For a hyperbola given by \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]the center is \((h, k)\). When translating the graph, we adjust \(h\) for horizontal shifts and \(k\) for vertical shifts.
In this exercise, by shifting the hyperbola 3 units to the right (adjust \(h\) by adding 3) and 5 units up (adjust \(k\) by adding 5), the center shifts from \((2, 3)\) to \((5, 8)\). This is a straightforward application of translation, helping to either move the graph into a more analyzed or desired position.
Think of it as sliding a piece of paper across a desk - the paper stays the same, but its position changes. In the context of equations, this process involves adjusting the constants within the equation to reflect the new positions of the graph's essential points, such as the center in the case of a hyperbola.
For a hyperbola given by \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]the center is \((h, k)\). When translating the graph, we adjust \(h\) for horizontal shifts and \(k\) for vertical shifts.
In this exercise, by shifting the hyperbola 3 units to the right (adjust \(h\) by adding 3) and 5 units up (adjust \(k\) by adding 5), the center shifts from \((2, 3)\) to \((5, 8)\). This is a straightforward application of translation, helping to either move the graph into a more analyzed or desired position.
Coordinate Systems
The coordinate system is a crucial part of understanding graphs like hyperbolas. The standard Cartesian coordinate system consists of an \(x\)-axis and a \(y\)-axis intersecting at a point called the origin \((0, 0)\).
Each point in this plane is described by a set of numerical coordinates \((x, y)\). These coordinates indicate the point’s distance from the \(y\)-axis (horizontally) and the \(x\)-axis (vertically).
Understanding how translations affect coordinates is vital. For instance, shifting a graph to the right by 3 units means increasing the \(x\)-coordinate of every point by 3. Similarly, moving up by 5 units increases the \(y\)-coordinate by 5. Each change in coordinates reflects a corresponding movement in the graph within the coordinate system.
In problems like the one provided, translating the coordinates affects the graph's position while maintaining its shape, helping visualize how alterations in equations map to physical shifts in appearance.
Each point in this plane is described by a set of numerical coordinates \((x, y)\). These coordinates indicate the point’s distance from the \(y\)-axis (horizontally) and the \(x\)-axis (vertically).
Understanding how translations affect coordinates is vital. For instance, shifting a graph to the right by 3 units means increasing the \(x\)-coordinate of every point by 3. Similarly, moving up by 5 units increases the \(y\)-coordinate by 5. Each change in coordinates reflects a corresponding movement in the graph within the coordinate system.
In problems like the one provided, translating the coordinates affects the graph's position while maintaining its shape, helping visualize how alterations in equations map to physical shifts in appearance.
Equation Transformation
Transforming equations is another essential aspect when working with graph translations. It involves rewriting the equation of a graph to reflect changes in position, such as those resulting from translation.
The initial equation of a hyperbola might be expressed as \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]where \((h, k)\) is the center. When the hyperbola is translated, these values \(h\) and \(k\) get adjusted, and thus, the equation itself must be transformed to maintain accurate representation.
In this exercise, evaluating the impact of moving 3 units right and 5 units up informs us to modify the \(x\) and \(y\) terms:
- Replacing \((x - h)\) with \((x - (h+3))\)- Replacing \((y - k)\) with \((y - (k+5))\)
Hence, the transformed equation becomes \[\frac{(x-5)^2}{36} - \frac{(y-8)^2}{25} = 1\]This new equation represents the hyperbola shifted in accordance to the given translations, showcasing how algebraic manipulation of equations can directly correspond to graphical changes.
The initial equation of a hyperbola might be expressed as \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]where \((h, k)\) is the center. When the hyperbola is translated, these values \(h\) and \(k\) get adjusted, and thus, the equation itself must be transformed to maintain accurate representation.
In this exercise, evaluating the impact of moving 3 units right and 5 units up informs us to modify the \(x\) and \(y\) terms:
- Replacing \((x - h)\) with \((x - (h+3))\)- Replacing \((y - k)\) with \((y - (k+5))\)
Hence, the transformed equation becomes \[\frac{(x-5)^2}{36} - \frac{(y-8)^2}{25} = 1\]This new equation represents the hyperbola shifted in accordance to the given translations, showcasing how algebraic manipulation of equations can directly correspond to graphical changes.
Other exercises in this chapter
Problem 41
The eccentricity of an ellipse is a measure of how nearly circular it is. Eccentricity is defined as \(\frac{s}{a},\) where \(c\) is the distance from the cente
View solution Problem 41
Find the vertices and the asymptotes of each hyperbola. $$ x^{2}-9 y^{2}=1 $$
View solution Problem 41
Mental Math Each given point is on the graph of the given equation. Use symmetry to find at least one more point on the graph. $$ (-\sqrt{2}, 1), x^{2}+y^{2}=3
View solution Problem 41
Write an equation of a parabola with a vertex at the origin. directrix \(y=2.8\)
View solution