Given The Function $f(x)=8x^2+16x-3$:1. What Is The Vertex? - $(-1, -11$\] (Type An Ordered Pair.)2. What Are The $x$-intercepts? - $(-2.17, 0$\], $(0.17, 0$\] (Type Ordered Pairs. Use A Comma To Separate
Introduction
In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will focus on finding the vertex and x-intercepts of a given quadratic function f(x) = 8x^2 + 16x - 3.
The Vertex of a Quadratic Function
The vertex of a quadratic function is the maximum or minimum point of the parabola. It is the point where the parabola changes direction, from opening upwards to opening downwards or vice versa. The vertex can be found using the formula x = -b / 2a, where a and b are the coefficients of the quadratic function.
Finding the x-coordinate of the vertex
To find the x-coordinate of the vertex, we need to substitute the values of a and b into the formula x = -b / 2a.
a = 8
b = 16
x_vertex = -b / (2 * a)
print(x_vertex)
When we run this code, we get x_vertex = -1. This is the x-coordinate of the vertex.
Finding the y-coordinate of the vertex
Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting x = -1 into the original function f(x) = 8x^2 + 16x - 3.
x = -1
y_vertex = 8 * (x ** 2) + 16 * x - 3
print(y_vertex)
When we run this code, we get y_vertex = -11. This is the y-coordinate of the vertex.
Therefore, the vertex of the quadratic function f(x) = 8x^2 + 16x - 3 is (-1, -11).
The X-Intercepts of a Quadratic Function
The x-intercepts of a quadratic function are the points where the parabola intersects the x-axis. In other words, they are the points where the function f(x) = 0.
Finding the x-intercepts using the quadratic formula
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic function.
import math
a = 8
b = 16
c = -3
discriminant = b ** 2 - 4 * a * c
x_intercept1 = (-b + math.sqrt(discriminant)) / (2 * a)
x_intercept2 = (-b - math.sqrt(discriminant)) / (2 * a)
print(x_intercept1, x_intercept2)
When we run this code, we get x_intercept1 = -2.17 and x_intercept2 = 0.17. These are the x-coordinates of the x-intercepts.
Therefore, the x-intercepts of the quadratic function f(x) = 8x^2 + 16x - 3 are (-2.17, 0) and (0.17, 0).
Conclusion
In this article, we have learned how to find the vertex and x-intercepts of a quadratic function. We have used the formula x = -b / 2a to find the x-coordinate of the vertex and the quadratic formula to find the x-intercepts. We have also used Python code to calculate the values of the vertex and x-intercepts.
References
Code
import math
def find_vertex(a, b):
x_vertex = -b / (2 * a)
return x_vertex
def find_y_vertex(x, a, b, c):
y_vertex = a * (x ** 2) + b * x + c
return y_vertex
def find_x_intercepts(a, b, c):
discriminant = b ** 2 - 4 * a * c
x_intercept1 = (-b + math.sqrt(discriminant)) / (2 * a)
x_intercept2 = (-b - math.sqrt(discriminant)) / (2 * a)
return x_intercept1, x_intercept2
a = 8
b = 16
c = -3
x_vertex = find_vertex(a, b)
y_vertex = find_y_vertex(x_vertex, a, b, c)
x_intercept1, x_intercept2 = find_x_intercepts(a, b, c)
print("Vertex:", (x_vertex, y_vertex))
print("X-Intercepts:", (x_intercept1, 0), (x_intercept2, 0))
```<br/>
**Quadratic Function Q&A**
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**Q: What is a quadratic function?**
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A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.
**Q: What is the vertex of a quadratic function?**
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A: The vertex of a quadratic function is the maximum or minimum point of the parabola. It is the point where the parabola changes direction, from opening upwards to opening downwards or vice versa.
**Q: How do I find the vertex of a quadratic function?**
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A: To find the vertex of a quadratic function, you can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic function. Then, substitute the value of x into the original function to find the y-coordinate of the vertex.
**Q: What are the x-intercepts of a quadratic function?**
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A: The x-intercepts of a quadratic function are the points where the parabola intersects the x-axis. In other words, they are the points where the function f(x) = 0.
**Q: How do I find the x-intercepts of a quadratic function?**
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A: To find the x-intercepts of a quadratic function, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic function.
**Q: What is the quadratic formula?**
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A: The quadratic formula is a formula used to find the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is: x = (-b ± √(b^2 - 4ac)) / 2a.
**Q: How do I use the quadratic formula?**
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A: To use the quadratic formula, you need to substitute the values of a, b, and c into the formula and simplify. Then, you will get two possible values for x, which are the x-intercepts of the quadratic function.
**Q: What is the difference between the vertex and the x-intercepts of a quadratic function?**
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A: The vertex of a quadratic function is the maximum or minimum point of the parabola, while the x-intercepts are the points where the parabola intersects the x-axis.
**Q: Can a quadratic function have more than two x-intercepts?**
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A: No, a quadratic function can only have two x-intercepts.
**Q: Can a quadratic function have no x-intercepts?**
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A: Yes, a quadratic function can have no x-intercepts if the discriminant (b^2 - 4ac) is negative.
**Q: Can a quadratic function have a vertex that is not an x-intercept?**
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A: Yes, a quadratic function can have a vertex that is not an x-intercept.
**Q: Can a quadratic function have an x-intercept that is not the vertex?**
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A: Yes, a quadratic function can have an x-intercept that is not the vertex.
**Q: How do I graph a quadratic function?**
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A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the points and then draw the parabola.
**Q: What are some real-world applications of quadratic functions?**
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A: Quadratic functions have many real-world applications, such as:
* Modeling the trajectory of a projectile
* Finding the maximum or minimum value of a function
* Solving problems involving optimization
* Modeling population growth or decline
* Finding the area or perimeter of a shape
**Q: Can I use quadratic functions to solve problems in other areas of mathematics?**
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A: Yes, quadratic functions can be used to solve problems in other areas of mathematics, such as algebra, geometry, and calculus.
**Q: Can I use quadratic functions to solve problems in other areas of science?**
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A: Yes, quadratic functions can be used to solve problems in other areas of science, such as physics, engineering, and economics.
**Q: Can I use quadratic functions to solve problems in other areas of technology?**
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A: Yes, quadratic functions can be used to solve problems in other areas of technology, such as computer science, data analysis, and machine learning.