Pregunta
QUESTION 3
The equation of a hyperbola is given by
.
Write down the equation of the new function that is forme
Shift two units to the left
Shift 3 units up
Shift 1 unit right and 2 units down
The equation of the new hyperbola has new asyr
QUESTION 4
Sketch on the same set of axes the graphs of
.
Clearly indicate all intercepts with the axes, turning point
The equation of a hyperbola is given by
Write down the equation of the new function that is forme
QUESTION 4
Sketch on the same set of axes the graphs of
Clearly indicate all intercepts with the axes, turning point
Ask by Burton Vargas. in South Africa
Jan 08,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
QUESTION 3
-
Shift Two Units to the Left:
-
Shift Three Units Up:
-
Shift One Unit Right and Two Units Down:
-
Asymptotes After Transformations:
- After 3.1: Vertical asymptote at
, horizontal asymptote at - After 3.2: Vertical asymptote at
, horizontal asymptote at - After 3.3: Vertical asymptote at
, horizontal asymptote at
- After 3.1: Vertical asymptote at
QUESTION 4
- Graph of
: - Vertex: (0, 0)
- Y-intercept: (0, 0)
- X-intercept: (0, 0)
- Symmetric about the y-axis
- Opens downward
- Additional Points: (1, -2) and (-1, -2)
By plotting these points and understanding the symmetry, the graph of
can be accurately sketched.
Solución
QUESTION 3
Given the hyperbola
, we’ll apply various transformations as specified.
3.1 Shift Two Units to the Left
Transformation Steps:
- Horizontal Shift: Shifting the graph two units to the left involves replacing
with .
New Function:
Final Equation:
3.2 Shift Three Units Up
Transformation Steps:
- Vertical Shift: Shifting the graph three units up involves adding 3 to the function.
New Function:
Final Equation:
3.3 Shift One Unit Right and Two Units Down
Transformation Steps:
- Horizontal Shift: Shifting the graph one unit to the right involves replacing
with . - Vertical Shift: Shifting the graph two units down involves subtracting 2 from the function.
New Function:
Final Equation:
3.4 New Asymptotes of the Transformed Hyperbola
For the transformed hyperbolas obtained in parts 3.1 to 3.3, the asymptotes change based on the transformations applied.
-
Original Function
: - Vertical Asymptote:
- Horizontal Asymptote:
- Vertical Asymptote:
-
After 3.1
: - Vertical Asymptote:
- Horizontal Asymptote:
- Vertical Asymptote:
-
After 3.2
: - Vertical Asymptote:
- Horizontal Asymptote:
- Vertical Asymptote:
-
After 3.3
: - Vertical Asymptote:
- Horizontal Asymptote:
- Vertical Asymptote:
QUESTION 4
Sketching the Graph of
:
Since a sketch cannot be drawn here, I’ll describe the key features of the graph:
-
Shape of the Graph:
- The equation
represents a downward-opening parabola because the coefficient of is negative.
- The equation
-
Vertex (Turning Point):
- The vertex of the parabola is at the origin.
-
Intercepts:
-
Y-intercept: Set
. The y-intercept is at the origin. -
X-intercepts: Set
. The only x-intercept is also at the origin.
-
-
Additional Points:
- For
: - For
: - These points help in sketching the symmetry of the parabola about the y-axis.
- For
-
Symmetry:
- The parabola is symmetric with respect to the y-axis.
-
Behavior:
- As
increases, decreases without bound due to the negative coefficient.
- As
Summary of Key Features:
- Vertex: (0, 0)
- Y-intercept: (0, 0)
- X-intercept: (0, 0)
- Opening Direction: Downward
- Symmetry: About the y-axis
By plotting these points and understanding the symmetry, one can sketch an accurate graph of the parabola
.
Respondido por UpStudy AI y revisado por un tutor profesional
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Beyond the Answer
To shift the hyperbola
two units to the left, you need to replace
with
. So, the new function becomes:
For a 3-unit upward shift, simply add 3 to the entire function, resulting in:
By shifting 1 unit to the right and 2 units down, replace
with
and subtract 2:
Thus, the equation becomes:
The new asymptotes will change as you perform these transformations. The original had vertical asymptote at
. After the shifts, the vertical asymptote for each transformation adjusts accordingly.
The graph of
is a downward-opening parabola with its vertex at the origin (0,0). The x-intercepts occur where
, giving
as the only intercept, while the y-intercept is also at (0,0). The parabola has no turning points other than the vertex since it continuously decreases. The sketch should illustrate the origin and show the arms of the parabola extending downwards.

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