Problem 20
Question
Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ x^{2}+y^{2}=25 \quad \text { Up } 3, \text { left } 4 $$
Step-by-Step Solution
Verified Answer
The shifted equation is \\( (x+4)^2 + (y-3)^2 = 25 \\).
1Step 1: Identify Original Equation
The given equation is a circle. The equation of a circle in the standard form is \(x^2 + y^2 = r^2\) where \((h, k)\) is the center of the circle and \(r\) is the radius. Here, the original circle is \(x^2 + y^2 = 25\) with center at (0,0) and radius 5, since \(r^2 = 25\).
2Step 2: Determine the Shift
The problem states to shift the graph 'up 3' and 'left 4'. This implies a vertical translation of \(+3\) (up) and a horizontal translation of \(-4\) (left). These transformations affect the center of the circle.
3Step 3: Apply the Translation
The original circle has center (0,0). Apply the horizontal shift of \(-4\) to the x-coordinate to get \(-4 + 0 = -4\). Apply the vertical shift of \(+3\) to the y-coordinate to get \(0 + 3 = 3\). The new center is at (-4, 3).
4Step 4: Write the New Equation
The shifted equation of the circle is \((x + 4)^2 + (y - 3)^2 = 25\). This comes from substituting the shifted center \((-4, 3)\) into the circle equation form \((x - h)^2 + (y - k)^2 = r^2\).
5Step 5: Sketch the Graphs
Draw the original circle centered at (0,0) with radius 5. Then, draw the shifted circle centered at (-4, 3), ensuring both circles have the same radius.
Key Concepts
Circle EquationsCoordinate GeometryTranslation Transformations
Circle Equations
In coordinate geometry, a circle's equation gives us vital information about its size and position. The most common form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). This equation shows that a circle is centered at the point \((h,k)\), and \(r\) represents the radius.
The original equation provided is \(x^2 + y^2 = 25\). This means the circle is centered at the origin, \((0,0)\). The radius \(r\) can be found using the square root of 25, which is 5. Drawing this on a graph will give us a circle with all points precisely 5 units away from the origin in every direction.
Knowing circle equations helps us understand how to adjust circles with transformations, with insights into their basic geometric properties. You can quickly locate where it stands or how far it spreads with just a glance at the equation.
The original equation provided is \(x^2 + y^2 = 25\). This means the circle is centered at the origin, \((0,0)\). The radius \(r\) can be found using the square root of 25, which is 5. Drawing this on a graph will give us a circle with all points precisely 5 units away from the origin in every direction.
Knowing circle equations helps us understand how to adjust circles with transformations, with insights into their basic geometric properties. You can quickly locate where it stands or how far it spreads with just a glance at the equation.
Coordinate Geometry
Coordinate geometry is the art of using an x-y coordinate plane to visualize and solve geometric problems. It's incredibly handy for mapping patterns or shapes like lines, circles, and curves onto a graphical interface.
In our exercise, the circle's placement is described using its equation. Initially, it is positioned at the origin \((0,0)\) with a specific radius. This places our circle centrally on the graph, evenly distributed around this central point. Understanding this symmetry is key to grasping how transformations will impact the circle.
Coordinate geometry equips us with the tools needed for plotting points and shapes. It helps visualize mathematical concepts, making them tangible. Using coordinates \((x, y)\), along with circle formulas, brings clarity to how different transformations like shifts, stretches, and rotations wildly affect geometric figures.
In our exercise, the circle's placement is described using its equation. Initially, it is positioned at the origin \((0,0)\) with a specific radius. This places our circle centrally on the graph, evenly distributed around this central point. Understanding this symmetry is key to grasping how transformations will impact the circle.
Coordinate geometry equips us with the tools needed for plotting points and shapes. It helps visualize mathematical concepts, making them tangible. Using coordinates \((x, y)\), along with circle formulas, brings clarity to how different transformations like shifts, stretches, and rotations wildly affect geometric figures.
Translation Transformations
Translation transformations involve shifting shapes on the graph without altering their size or orientation. It’s like moving the entire shape to a new location while keeping its appearance intact.
For the circle in our example, we are instructed to move it up by 3 units and to the left by 4 units. These movements are known as translations. When moving "up," you increase the \(y\)-value of the center of the circle. Conversely, moving "left" decreases the \(x\)-value. Both shifts together adjust the center from \((0, 0)\) to \((-4, 3)\).
In translation, every point on the shape shifts equally, allowing it to glide across the plane smoothly. This avoids any change in shape or size. Whenever translating shapes, remember:
For the circle in our example, we are instructed to move it up by 3 units and to the left by 4 units. These movements are known as translations. When moving "up," you increase the \(y\)-value of the center of the circle. Conversely, moving "left" decreases the \(x\)-value. Both shifts together adjust the center from \((0, 0)\) to \((-4, 3)\).
In translation, every point on the shape shifts equally, allowing it to glide across the plane smoothly. This avoids any change in shape or size. Whenever translating shapes, remember:
- Adjust the \(x\) component with horizontal shifts (left or right).
- Modify the \(y\) component with vertical shifts (up or down).
Other exercises in this chapter
Problem 19
Solve the inequalities in Exercises \(19-34,\) expressing the solution sets as intervals or unions of intervals. Also, show each solution set on the real line.
View solution Problem 19
In Exercises 17–30, write an equation for each line described. Passes through \((3,4)\) and \((-2,5)\)
View solution Problem 20
In Exercises \(19-30,\) say whether the function is even, odd, or neither. Give reasons for your answer. $$ f(x)=x^{-5} $$
View solution Problem 20
Solve the inequalities in Exercises \(19-34,\) expressing the solution sets as intervals or unions of intervals. Also, show each solution set on the real line.
View solution