Which Parabola Will Have One Real Solution With The Line $y = X - 5$?A. Y = X 2 + X − 4 Y = X^2 + X - 4 Y = X 2 + X − 4 B. Y = − V 2 + 2 V − 1 Y = -v^2 + 2v - 1 Y = − V 2 + 2 V − 1
Introduction
When dealing with quadratic equations, it's essential to understand the relationship between the parabola and the line. In this case, we're given a line equation, y = x - 5, and we need to determine which parabola will have one real solution with this line. To solve this problem, we'll need to analyze the intersection points of the parabola and the line.
Understanding the Line Equation
The line equation y = x - 5 is a linear equation in the slope-intercept form, where the slope is 1 and the y-intercept is -5. This means that the line has a constant slope and a fixed y-intercept.
Understanding the Parabola Equations
We have two parabola equations to consider:
A. y = x^2 + x - 4 B. y = -v^2 + 2v - 1
To determine which parabola will have one real solution with the line y = x - 5, we need to analyze the intersection points of the parabola and the line.
Analyzing the Intersection Points
To find the intersection points, we need to set the two equations equal to each other and solve for x.
For parabola A, we have:
x^2 + x - 4 = x - 5
Simplifying the equation, we get:
x^2 - 4 = 0
This is a quadratic equation, and we can solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 0, and c = -4. Plugging these values into the formula, we get:
x = (0 ± √(0^2 - 4(1)(-4))) / 2(1) x = (0 ± √(16)) / 2 x = (0 ± 4) / 2
This gives us two possible values for x:
x = 2 or x = -2
For parabola B, we have:
-v^2 + 2v - 1 = x - 5
Simplifying the equation, we get:
-v^2 + 2v - 6 = 0
This is also a quadratic equation, and we can solve it using the quadratic formula:
v = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -1, b = 2, and c = -6. Plugging these values into the formula, we get:
v = (-2 ± √(2^2 - 4(-1)(-6))) / 2(-1) v = (-2 ± √(4 - 24)) / -2 v = (-2 ± √(-20)) / -2
This gives us two complex values for v:
v = (-2 ± 2i√5) / -2
Determining the Number of Real Solutions
To determine which parabola will have one real solution with the line y = x - 5, we need to analyze the number of real solutions for each parabola.
For parabola A, we have two real solutions: x = 2 and x = -2.
For parabola B, we have two complex solutions: v = (-2 ± 2i√5) / -2.
Since parabola A has two real solutions, it will have two intersection points with the line y = x - 5.
Conclusion
In conclusion, parabola A, y = x^2 + x - 4, will have two real solutions with the line y = x - 5. Parabola B, y = -v^2 + 2v - 1, will have two complex solutions with the line y = x - 5.
Recommendations
When dealing with quadratic equations, it's essential to understand the relationship between the parabola and the line. By analyzing the intersection points of the parabola and the line, we can determine the number of real solutions for each parabola.
Future Work
In the future, we can explore other quadratic equations and analyze their intersection points with the line y = x - 5. This will help us better understand the relationship between the parabola and the line.
Limitations
One limitation of this analysis is that we assumed the line equation y = x - 5 is a linear equation. In reality, the line equation may be a quadratic or cubic equation, which would require a more complex analysis.
Conclusion
In conclusion, parabola A, y = x^2 + x - 4, will have two real solutions with the line y = x - 5. Parabola B, y = -v^2 + 2v - 1, will have two complex solutions with the line y = x - 5.
Q: What is a parabola?
A: A parabola is a type of quadratic equation that has a U-shaped graph. It is defined by the equation y = ax^2 + bx + c, where a, b, and c are constants.
Q: What is a line?
A: A line is a type of linear equation that has a straight graph. It is defined by the equation y = mx + b, where m is the slope and b is the y-intercept.
Q: How do parabolas and lines intersect?
A: Parabolas and lines intersect when they have a common point. This can be found by setting the two equations equal to each other and solving for x.
Q: What is the relationship between the parabola and the line?
A: The parabola and the line have a relationship where the line can be tangent to the parabola at a single point, or the line can intersect the parabola at two points.
Q: How do you determine the number of real solutions for a parabola?
A: To determine the number of real solutions for a parabola, you need to analyze the discriminant of the quadratic equation. If the discriminant is positive, the parabola has two real solutions. If the discriminant is zero, the parabola has one real solution. If the discriminant is negative, the parabola has no real solutions.
Q: What is the significance of the line y = x - 5?
A: The line y = x - 5 is a specific line that is used to determine the number of real solutions for a parabola. It is a linear equation with a slope of 1 and a y-intercept of -5.
Q: Can a parabola have more than two real solutions?
A: No, a parabola cannot have more than two real solutions. This is because the quadratic equation has a maximum of two real roots.
Q: Can a line have more than one intersection point with a parabola?
A: Yes, a line can have more than one intersection point with a parabola. This occurs when the line is tangent to the parabola at a single point, or when the line intersects the parabola at two points.
Q: How do you graph a parabola?
A: To graph a parabola, you need to find the vertex of the parabola and then plot two points on either side of the vertex. The parabola is then drawn by connecting the two points with a smooth curve.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.
Q: How do you find the vertex of a parabola?
A: To find the vertex of a parabola, you need to use the formula x = -b / 2a, where a and b are the coefficients of the quadratic equation.
Q: What is the significance of the vertex of a parabola?
A: The vertex of a parabola is significant because it represents the minimum or maximum point of the parabola. It is also the point where the parabola changes direction.
Q: Can a parabola have a vertex that is not on the x-axis?
A: Yes, a parabola can have a vertex that is not on the x-axis. This occurs when the parabola is rotated or translated.
Q: How do you rotate a parabola?
A: To rotate a parabola, you need to multiply the x and y coordinates by a rotation matrix.
Q: How do you translate a parabola?
A: To translate a parabola, you need to add a constant value to the x and y coordinates.
Q: What is the significance of the rotation and translation of a parabola?
A: The rotation and translation of a parabola are significant because they allow you to change the orientation and position of the parabola in the coordinate plane.
Q: Can a parabola be rotated or translated in three dimensions?
A: Yes, a parabola can be rotated or translated in three dimensions. This is done by using a 3D rotation matrix and a 3D translation vector.
Q: What is the significance of the 3D rotation and translation of a parabola?
A: The 3D rotation and translation of a parabola are significant because they allow you to change the orientation and position of the parabola in three-dimensional space.
Q: Can a parabola be used to model real-world phenomena?
A: Yes, a parabola can be used to model real-world phenomena such as the trajectory of a projectile, the motion of a pendulum, and the shape of a satellite dish.
Q: What are some examples of parabolas in real-world applications?
A: Some examples of parabolas in real-world applications include:
- The trajectory of a baseball or a golf ball
- The motion of a pendulum
- The shape of a satellite dish
- The design of a parabolic mirror
- The shape of a parabolic antenna
Q: Can a parabola be used to solve real-world problems?
A: Yes, a parabola can be used to solve real-world problems such as optimizing the design of a satellite dish, determining the trajectory of a projectile, and designing a parabolic mirror.
Q: What are some challenges of working with parabolas?
A: Some challenges of working with parabolas include:
- Finding the vertex of the parabola
- Determining the number of real solutions for the parabola
- Rotating and translating the parabola
- Solving real-world problems using parabolas
Q: What are some tips for working with parabolas?
A: Some tips for working with parabolas include:
- Using the quadratic formula to find the roots of the parabola
- Graphing the parabola to visualize its shape
- Rotating and translating the parabola to change its orientation and position
- Solving real-world problems using parabolas to optimize design and performance.