Select The Quadratic Function With A Graph That Has The Following Features:- X-intercept At { (8, 0)$}$- Y-intercept At { (0, -32)$}$- Maximum Value At { (6, 4)$}$- Axis Of Symmetry At { X = 6$} A . \[ A. \[ A . \[ F(x) =
===========================================================
Introduction
Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra, geometry, and calculus. In this article, we will focus on selecting the quadratic function that has a graph with specific features, including an x-intercept, y-intercept, maximum value, and axis of symmetry.
Understanding Quadratic Functions
A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic function is:
f(x) = ax^2 + bx + c
where a, b, and c are constants, and a cannot be zero.
Graph Features of Quadratic Functions
The graph of a quadratic function is a parabola, which is a U-shaped curve. The key features of a quadratic function's graph include:
- X-intercept: The point where the graph intersects the x-axis, i.e., where y = 0.
- Y-intercept: The point where the graph intersects the y-axis, i.e., where x = 0.
- Maximum/Minimum Value: The highest or lowest point on the graph, which occurs at the vertex of the parabola.
- Axis of Symmetry: The vertical line that passes through the vertex of the parabola and divides the graph into two symmetrical parts.
Selecting the Quadratic Function
We are given the following features of the graph:
- X-intercept at (8, 0)
- Y-intercept at (0, -32)
- Maximum value at (6, 4)
- Axis of symmetry at x = 6
To select the quadratic function, we need to use the given features to determine the values of a, b, and c in the general form of the quadratic function.
Using the X-Intercept
The x-intercept is the point where the graph intersects the x-axis, i.e., where y = 0. Since the x-intercept is at (8, 0), we can substitute x = 8 and y = 0 into the general form of the quadratic function to get:
f(8) = a(8)^2 + b(8) + c f(8) = 64a + 8b + c
Since the y-coordinate is zero, we can set the equation equal to zero:
64a + 8b + c = 0
Using the Y-Intercept
The y-intercept is the point where the graph intersects the y-axis, i.e., where x = 0. Since the y-intercept is at (0, -32), we can substitute x = 0 and y = -32 into the general form of the quadratic function to get:
f(0) = a(0)^2 + b(0) + c f(0) = c
Since the y-coordinate is -32, we can set the equation equal to -32:
c = -32
Using the Maximum Value
The maximum value is the highest point on the graph, which occurs at the vertex of the parabola. Since the maximum value is at (6, 4), we can substitute x = 6 and y = 4 into the general form of the quadratic function to get:
f(6) = a(6)^2 + b(6) + c f(6) = 36a + 6b + c
Since the y-coordinate is 4, we can set the equation equal to 4:
36a + 6b + c = 4
Using the Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex of the parabola. Since the axis of symmetry is at x = 6, we can substitute x = 6 into the general form of the quadratic function to get:
f(6) = a(6)^2 + b(6) + c f(6) = 36a + 6b + c
Since the x-coordinate is 6, we can set the equation equal to the value of the function at x = 6:
36a + 6b + c = f(6)
Solving the System of Equations
We now have a system of four equations with four unknowns (a, b, c, and f(6)). We can solve this system of equations to find the values of a, b, and c.
From the equation c = -32, we can substitute c into the other equations:
64a + 8b - 32 = 0 36a + 6b - 32 = 4
Simplifying the equations, we get:
64a + 8b = 32 36a + 6b = 36
Dividing the first equation by 8, we get:
8a + b = 4
Dividing the second equation by 6, we get:
6a + b = 6
Subtracting the second equation from the first equation, we get:
2a = -2
Dividing both sides by 2, we get:
a = -1
Substituting a = -1 into the equation 8a + b = 4, we get:
-8 + b = 4
Adding 8 to both sides, we get:
b = 12
Substituting a = -1 and b = 12 into the equation c = -32, we get:
c = -32
The Quadratic Function
We have now found the values of a, b, and c. The quadratic function is:
f(x) = -x^2 + 12x - 32
This is the quadratic function that has a graph with the given features.
Conclusion
In this article, we have selected the quadratic function that has a graph with specific features, including an x-intercept, y-intercept, maximum value, and axis of symmetry. We have used the given features to determine the values of a, b, and c in the general form of the quadratic function. The quadratic function is f(x) = -x^2 + 12x - 32.
==========================
Introduction
In our previous article, we discussed how to select the quadratic function that has a graph with specific features, including an x-intercept, y-intercept, maximum value, and axis of symmetry. In this article, we will answer some frequently asked questions about quadratic functions and provide additional information to help you better understand these functions.
Q: What is a quadratic function?
A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic function is:
f(x) = ax^2 + bx + c
where a, b, and c are constants, and a cannot be zero.
Q: What are the key features of a quadratic function's graph?
The key features of a quadratic function's graph include:
- X-intercept: The point where the graph intersects the x-axis, i.e., where y = 0.
- Y-intercept: The point where the graph intersects the y-axis, i.e., where x = 0.
- Maximum/Minimum Value: The highest or lowest point on the graph, which occurs at the vertex of the parabola.
- Axis of Symmetry: The vertical line that passes through the vertex of the parabola and divides the graph into two symmetrical parts.
Q: How do I find the x-intercept of a quadratic function?
To find the x-intercept of a quadratic function, you can set the function equal to zero and solve for x. This will give you the x-coordinate of the x-intercept.
Q: How do I find the y-intercept of a quadratic function?
To find the y-intercept of a quadratic function, you can substitute x = 0 into the function and solve for y. This will give you the y-coordinate of the y-intercept.
Q: How do I find the maximum or minimum value of a quadratic function?
To find the maximum or minimum value of a quadratic function, you can find the vertex of the parabola. The vertex is the point where the parabola changes direction, and it is the maximum or minimum value of the function.
Q: How do I find the axis of symmetry of a quadratic function?
To find the axis of symmetry of a quadratic function, you can find the x-coordinate of the vertex. The axis of symmetry is the vertical line that passes through the vertex and divides the graph into two symmetrical parts.
Q: Can a quadratic function have more than one x-intercept?
No, a quadratic function can only have one x-intercept. However, it can have more than one y-intercept.
Q: Can a quadratic function have more than one maximum or minimum value?
No, a quadratic function can only have one maximum or minimum value. However, it can have more than one axis of symmetry.
Q: How do I graph a quadratic function?
To graph a quadratic function, you can use the following steps:
- Find the x-intercept(s) of the function.
- Find the y-intercept(s) of the function.
- Find the vertex of the parabola.
- Draw the parabola using the x-intercept(s), y-intercept(s), and vertex.
Q: What are some common applications of quadratic functions?
Quadratic functions have many applications in real-world problems, including:
- Projectile motion: Quadratic functions can be used to model the trajectory of a projectile under the influence of gravity.
- Optimization: Quadratic functions can be used to find the maximum or minimum value of a function.
- Physics: Quadratic functions can be used to model the motion of an object under the influence of a force.
- Engineering: Quadratic functions can be used to design and optimize systems.
Conclusion
In this article, we have answered some frequently asked questions about quadratic functions and provided additional information to help you better understand these functions. We hope this article has been helpful in your studies and applications of quadratic functions.