Find The \[$x\$\]-intercept(s) And The Coordinates Of The Vertex For The Parabola \[$y = X^2 + 6x - 7\$\].If There Is More Than One \[$x\$\]-intercept, Separate Them With Commas.
Introduction
In mathematics, a parabola is a type of quadratic function that can be represented in the form y = ax^2 + bx + c, where a, b, and c are constants. The x-intercept(s) of a parabola are the points where the parabola intersects the x-axis, and the vertex is the point on the parabola that is the lowest or highest point. In this article, we will discuss how to find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 6x - 7.
Finding the x-Intercept(s)
To find the x-intercept(s) of a parabola, we need to set y = 0 and solve for x. This is because the x-intercept(s) are the points where the parabola intersects the x-axis, and at these points, the value of y is always 0.
For the parabola y = x^2 + 6x - 7, we can set y = 0 and solve for x as follows:
0 = x^2 + 6x - 7
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 6, and c = -7. Plugging these values into the quadratic formula, we get:
x = (-(6) ± √((6)^2 - 4(1)(-7))) / 2(1) x = (-6 ± √(36 + 28)) / 2 x = (-6 ± √64) / 2 x = (-6 ± 8) / 2
Simplifying this expression, we get two possible values for x:
x = (-6 + 8) / 2 = 1 x = (-6 - 8) / 2 = -7
Therefore, the x-intercept(s) of the parabola y = x^2 + 6x - 7 are x = 1 and x = -7.
Finding the Coordinates of the Vertex
The vertex of a parabola is the point on the parabola that is the lowest or highest point. To find the coordinates of the vertex, we need to use the formula:
x = -b / 2a
In this case, a = 1 and b = 6. Plugging these values into the formula, we get:
x = -6 / 2(1) x = -6 / 2 x = -3
To find the y-coordinate of the vertex, we need to plug the value of x into the equation of the parabola:
y = x^2 + 6x - 7 y = (-3)^2 + 6(-3) - 7 y = 9 - 18 - 7 y = -16
Therefore, the coordinates of the vertex of the parabola y = x^2 + 6x - 7 are (-3, -16).
Conclusion
In this article, we discussed how to find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 6x - 7. We used the quadratic formula to find the x-intercept(s) and the formula x = -b / 2a to find the x-coordinate of the vertex. We then plugged the value of x into the equation of the parabola to find the y-coordinate of the vertex. The x-intercept(s) of the parabola are x = 1 and x = -7, and the coordinates of the vertex are (-3, -16).
Example Problems
Problem 1
Find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 4x - 5.
Solution
To find the x-intercept(s), we need to set y = 0 and solve for x:
0 = x^2 + 4x - 5
Using the quadratic formula, we get:
x = (-(4) ± √((4)^2 - 4(1)(-5))) / 2(1) x = (-4 ± √(16 + 20)) / 2 x = (-4 ± √36) / 2 x = (-4 ± 6) / 2
Simplifying this expression, we get two possible values for x:
x = (-4 + 6) / 2 = 1 x = (-4 - 6) / 2 = -5
Therefore, the x-intercept(s) of the parabola y = x^2 + 4x - 5 are x = 1 and x = -5.
To find the coordinates of the vertex, we need to use the formula:
x = -b / 2a
In this case, a = 1 and b = 4. Plugging these values into the formula, we get:
x = -4 / 2(1) x = -4 / 2 x = -2
To find the y-coordinate of the vertex, we need to plug the value of x into the equation of the parabola:
y = x^2 + 4x - 5 y = (-2)^2 + 4(-2) - 5 y = 4 - 8 - 5 y = -9
Therefore, the coordinates of the vertex of the parabola y = x^2 + 4x - 5 are (-2, -9).
Problem 2
Find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 - 2x - 3.
Solution
To find the x-intercept(s), we need to set y = 0 and solve for x:
0 = x^2 - 2x - 3
Using the quadratic formula, we get:
x = (2 ± √((-2)^2 - 4(1)(-3))) / 2(1) x = (2 ± √(4 + 12)) / 2 x = (2 ± √16) / 2 x = (2 ± 4) / 2
Simplifying this expression, we get two possible values for x:
x = (2 + 4) / 2 = 3 x = (2 - 4) / 2 = -1
Therefore, the x-intercept(s) of the parabola y = x^2 - 2x - 3 are x = 3 and x = -1.
To find the coordinates of the vertex, we need to use the formula:
x = -b / 2a
In this case, a = 1 and b = -2. Plugging these values into the formula, we get:
x = -(-2) / 2(1) x = 2 / 2 x = 1
To find the y-coordinate of the vertex, we need to plug the value of x into the equation of the parabola:
y = x^2 - 2x - 3 y = (1)^2 - 2(1) - 3 y = 1 - 2 - 3 y = -4
Therefore, the coordinates of the vertex of the parabola y = x^2 - 2x - 3 are (1, -4).
Final Thoughts
In this article, we discussed how to find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 6x - 7. We used the quadratic formula to find the x-intercept(s) and the formula x = -b / 2a to find the x-coordinate of the vertex. We then plugged the value of x into the equation of the parabola to find the y-coordinate of the vertex. The x-intercept(s) of the parabola are x = 1 and x = -7, and the coordinates of the vertex are (-3, -16). We also provided example problems to help illustrate the concepts discussed in this article.
Introduction
In our previous article, we discussed how to find the x-intercept(s) and the coordinates of the vertex for a parabola. In this article, we will answer some frequently asked questions about finding the x-intercept(s) and the coordinates of the vertex for a parabola.
Q&A
Q: What is the x-intercept(s) of a parabola?
A: The x-intercept(s) of a parabola are the points where the parabola intersects the x-axis. At these points, the value of y is always 0.
Q: How do I find the x-intercept(s) of a parabola?
A: To find the x-intercept(s) of a parabola, you need to set y = 0 and solve for x. This can be done using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point on the parabola that is the lowest or highest point. It is the point where the parabola changes direction.
Q: How do I find the coordinates of the vertex of a parabola?
A: To find the coordinates of the vertex of a parabola, you need to use the formula: x = -b / 2a. This will give you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you need to plug the value of x into the equation of the parabola.
Q: What is the difference between the x-intercept(s) and the vertex of a parabola?
A: The x-intercept(s) of a parabola are the points where the parabola intersects the x-axis, while the vertex is the point on the parabola that is the lowest or highest point.
Q: Can a parabola have more than one x-intercept?
A: Yes, a parabola can have more than one x-intercept. This occurs when the quadratic equation has two real solutions.
Q: Can a parabola have no x-intercept?
A: Yes, a parabola can have no x-intercept. This occurs when the quadratic equation has no real solutions.
Q: Can a parabola have only one x-intercept?
A: Yes, a parabola can have only one x-intercept. This occurs when the quadratic equation has one real solution.
Q: How do I determine the direction of the parabola?
A: To determine the direction of the parabola, you need to 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: Can a parabola have a vertex that is not on the x-axis?
A: No, a parabola cannot have a vertex that is not on the x-axis. The vertex of a parabola is always on the x-axis.
Q: Can a parabola have a vertex that is not on the y-axis?
A: Yes, a parabola can have a vertex that is not on the y-axis. This occurs when the parabola is not a function.
Conclusion
In this article, we answered some frequently asked questions about finding the x-intercept(s) and the coordinates of the vertex for a parabola. We hope that this article has been helpful in clarifying any confusion you may have had about these concepts.
Example Problems
Problem 1
Find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 4x - 5.
Solution
To find the x-intercept(s), we need to set y = 0 and solve for x:
0 = x^2 + 4x - 5
Using the quadratic formula, we get:
x = (-(4) ± √((4)^2 - 4(1)(-5))) / 2(1) x = (-4 ± √(16 + 20)) / 2 x = (-4 ± √36) / 2 x = (-4 ± 6) / 2
Simplifying this expression, we get two possible values for x:
x = (-4 + 6) / 2 = 1 x = (-4 - 6) / 2 = -5
Therefore, the x-intercept(s) of the parabola y = x^2 + 4x - 5 are x = 1 and x = -5.
To find the coordinates of the vertex, we need to use the formula:
x = -b / 2a
In this case, a = 1 and b = 4. Plugging these values into the formula, we get:
x = -4 / 2(1) x = -4 / 2 x = -2
To find the y-coordinate of the vertex, we need to plug the value of x into the equation of the parabola:
y = x^2 + 4x - 5 y = (-2)^2 + 4(-2) - 5 y = 4 - 8 - 5 y = -9
Therefore, the coordinates of the vertex of the parabola y = x^2 + 4x - 5 are (-2, -9).
Problem 2
Find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 - 2x - 3.
Solution
To find the x-intercept(s), we need to set y = 0 and solve for x:
0 = x^2 - 2x - 3
Using the quadratic formula, we get:
x = (2 ± √((-2)^2 - 4(1)(-3))) / 2(1) x = (2 ± √(4 + 12)) / 2 x = (2 ± √16) / 2 x = (2 ± 4) / 2
Simplifying this expression, we get two possible values for x:
x = (2 + 4) / 2 = 3 x = (2 - 4) / 2 = -1
Therefore, the x-intercept(s) of the parabola y = x^2 - 2x - 3 are x = 3 and x = -1.
To find the coordinates of the vertex, we need to use the formula:
x = -b / 2a
In this case, a = 1 and b = -2. Plugging these values into the formula, we get:
x = -(-2) / 2(1) x = 2 / 2 x = 1
To find the y-coordinate of the vertex, we need to plug the value of x into the equation of the parabola:
y = x^2 - 2x - 3 y = (1)^2 - 2(1) - 3 y = 1 - 2 - 3 y = -4
Therefore, the coordinates of the vertex of the parabola y = x^2 - 2x - 3 are (1, -4).
Final Thoughts
In this article, we answered some frequently asked questions about finding the x-intercept(s) and the coordinates of the vertex for a parabola. We hope that this article has been helpful in clarifying any confusion you may have had about these concepts.