Given The Equation $y=(x+5)^2$, Provide The Following Information:- Axis Of Symmetry:- Vertex:- Domain:- Range:- Transformations:
Axis of Symmetry
The axis of symmetry is a line that passes through the vertex of a parabola and is equidistant from the two arms of the parabola. To find the axis of symmetry, we need to find the vertex of the parabola.
The given equation is in the form y = (x + h)^2, where h is the x-coordinate of the vertex. In this case, h = -5. Therefore, the axis of symmetry is the vertical line x = -5.
Vertex
The vertex of a parabola in the form y = (x + h)^2 is (h, k), where k is the y-coordinate of the vertex. In this case, h = -5 and k = 0 (since the equation is in the form y = (x + h)^2, the y-coordinate of the vertex is always 0).
Therefore, the vertex of the parabola is (-5, 0).
Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of a quadratic function in the form y = (x + h)^2, the domain is all real numbers, or (-∞, ∞).
Range
The range of a function is the set of all possible output values (y-values) for which the function is defined. In the case of a quadratic function in the form y = (x + h)^2, the range is all non-negative real numbers, or [0, ∞).
Transformations
The given equation y = (x + 5)^2 represents a parabola that has been shifted 5 units to the left. This is because the equation is in the form y = (x + h)^2, where h = -5.
To find the original equation of the parabola, we need to replace x with (x + 5) in the original equation. This gives us:
y = (x + 5)^2 y = (x + 5 + 5)^2 y = (x + 10)^2
Therefore, the original equation of the parabola is y = (x + 10)^2.
Graph of the Parabola
The graph of the parabola y = (x + 5)^2 is a U-shaped curve that opens upwards. The vertex of the parabola is at (-5, 0), and the axis of symmetry is the vertical line x = -5.
Key Features of the Parabola
- The parabola has a minimum point at (-5, 0).
- The parabola opens upwards.
- The axis of symmetry is the vertical line x = -5.
- The domain of the parabola is all real numbers, or (-∞, ∞).
- The range of the parabola is all non-negative real numbers, or [0, ∞).
Real-World Applications of the Parabola
The parabola y = (x + 5)^2 has many real-world applications, including:
- Projectile Motion: The parabola can be used to model the trajectory of a projectile, such as a thrown ball or a rocket.
- Optics: The parabola can be used to design optical systems, such as telescopes and microscopes.
- Engineering: The parabola can be used to design structures, such as bridges and buildings.
Conclusion
In conclusion, the equation y = (x + 5)^2 represents a parabola that has been shifted 5 units to the left. The axis of symmetry is the vertical line x = -5, and the vertex is at (-5, 0). The domain of the parabola is all real numbers, or (-∞, ∞), and the range is all non-negative real numbers, or [0, ∞). The parabola has many real-world applications, including projectile motion, optics, and engineering.
Frequently Asked Questions
Q: What is the axis of symmetry of the parabola y = (x + 5)^2?
A: The axis of symmetry is the vertical line x = -5.
Q: What is the vertex of the parabola y = (x + 5)^2?
A: The vertex of the parabola is (-5, 0).
Q: What is the domain of the parabola y = (x + 5)^2?
A: The domain of the parabola is all real numbers, or (-∞, ∞).
Q: What is the range of the parabola y = (x + 5)^2?
A: The range of the parabola is all non-negative real numbers, or [0, ∞).
Q: How do I graph the parabola y = (x + 5)^2?
A: To graph the parabola, start by plotting the vertex at (-5, 0). Then, use the axis of symmetry to draw the parabola. The parabola will open upwards.
Q: What are some real-world applications of the parabola y = (x + 5)^2?
A: Some real-world applications of the parabola include:
- Projectile Motion: The parabola can be used to model the trajectory of a projectile, such as a thrown ball or a rocket.
- Optics: The parabola can be used to design optical systems, such as telescopes and microscopes.
- Engineering: The parabola can be used to design structures, such as bridges and buildings.
Q: How do I find the equation of a parabola that has been shifted to the left?
A: To find the equation of a parabola that has been shifted to the left, replace x with (x + h) in the original equation, where h is the number of units the parabola has been shifted to the left.
Q: What is the difference between a parabola that opens upwards and a parabola that opens downwards?
A: A parabola that opens upwards has a minimum point, while a parabola that opens downwards has a maximum point.
Q: How do I determine whether a parabola opens upwards or downwards?
A: To determine whether a parabola opens upwards or downwards, look at the coefficient of the x^2 term. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.
Q: What is the significance of the axis of symmetry in a parabola?
A: The axis of symmetry is a line that passes through the vertex of a parabola and is equidistant from the two arms of the parabola. It is an important concept in graphing and analyzing parabolas.
Q: How do I find the equation of a parabola that has been reflected across the x-axis?
A: To find the equation of a parabola that has been reflected across the x-axis, multiply the y-coordinate of the vertex by -1.
Q: What is the difference between a parabola and a circle?
A: A parabola is a U-shaped curve that opens upwards or downwards, while a circle is a round shape that is centered at a point.
Q: How do I determine whether a graph is a parabola or a circle?
A: To determine whether a graph is a parabola or a circle, look at the equation of the graph. If the equation is in the form y = (x + h)^2, the graph is a parabola. If the equation is in the form x^2 + y^2 = r^2, the graph is a circle.
Q: What is the significance of the vertex in a parabola?
A: The vertex is the point on a parabola that is the lowest or highest point on the graph. It is an important concept in graphing and analyzing parabolas.
Q: How do I find the equation of a parabola that has been stretched or compressed?
A: To find the equation of a parabola that has been stretched or compressed, multiply the x-coordinate of the vertex by a constant factor.
Q: What is the difference between a parabola and an ellipse?
A: A parabola is a U-shaped curve that opens upwards or downwards, while an ellipse is a curved shape that is centered at a point and has two axes of symmetry.
Q: How do I determine whether a graph is a parabola or an ellipse?
A: To determine whether a graph is a parabola or an ellipse, look at the equation of the graph. If the equation is in the form y = (x + h)^2, the graph is a parabola. If the equation is in the form (x - h)2/a2 + (y - k)2/b2 = 1, the graph is an ellipse.