The Function Below Represents The Daily Cost Of Running A Factory That Produces Boxing Gloves, Where $x$ Is The Number Of Pairs Of Boxing Gloves Produced Each Day: F ( X ) = 0.08 X 2 − 12 X + 650 F(x) = 0.08x^2 - 12x + 650 F ( X ) = 0.08 X 2 − 12 X + 650 Determine The Symmetry Of The Function:A.

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Introduction

The function f(x)=0.08x2−12x+650f(x) = 0.08x^2 - 12x + 650 represents the daily cost of running a factory that produces boxing gloves, where xx is the number of pairs of boxing gloves produced each day. In this article, we will determine the symmetry of the function.

Understanding the Function

The given function is a quadratic function, which is a polynomial of degree two. The general form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. In this case, a=0.08a = 0.08, b=−12b = -12, and c=650c = 650.

Symmetry of a Quadratic Function

A quadratic function can have three types of symmetry: even symmetry, odd symmetry, and no symmetry. Even symmetry occurs when the function is symmetric about the y-axis, odd symmetry occurs when the function is symmetric about the origin, and no symmetry occurs when the function does not have any symmetry.

Determining the Symmetry of the Function

To determine the symmetry of the function, we need to examine the coefficients of the quadratic function. If the coefficient of the x2x^2 term is positive, the function has even symmetry. If the coefficient of the x2x^2 term is negative, the function has no symmetry. If the coefficient of the x2x^2 term is zero, the function has odd symmetry.

In this case, the coefficient of the x2x^2 term is 0.080.08, which is positive. Therefore, the function has even symmetry.

Even Symmetry

Even symmetry occurs when the function is symmetric about the y-axis. This means that if we reflect the function about the y-axis, the resulting function will be the same as the original function.

To visualize the even symmetry of the function, we can graph the function. The graph of the function will be a parabola that opens upwards, with the vertex at the point (x,y)=(−b2a,f(−b2a))(x, y) = (-\frac{b}{2a}, f(-\frac{b}{2a})). In this case, the vertex is at the point (x,y)=(−−122(0.08),f(−−122(0.08)))=(75,f(75))(x, y) = (-\frac{-12}{2(0.08)}, f(-\frac{-12}{2(0.08)})) = (75, f(75)).

Graphing the Function

To graph the function, we can use a graphing calculator or a computer algebra system. The graph of the function will be a parabola that opens upwards, with the vertex at the point (75,f(75))(75, f(75)).

Conclusion

In conclusion, the function f(x)=0.08x2−12x+650f(x) = 0.08x^2 - 12x + 650 has even symmetry. This means that the function is symmetric about the y-axis, and the graph of the function will be a parabola that opens upwards, with the vertex at the point (75,f(75))(75, f(75)).

References

Further Reading

Q&A: Understanding the Function and Its Symmetry

Introduction

In our previous article, we discussed the function f(x)=0.08x2−12x+650f(x) = 0.08x^2 - 12x + 650 and determined that it has even symmetry. In this article, we will answer some frequently asked questions about the function and its symmetry.

Q: What is the significance of even symmetry in the function?

A: Even symmetry in the function means that the function is symmetric about the y-axis. This means that if we reflect the function about the y-axis, the resulting function will be the same as the original function. This symmetry is important because it helps us understand the behavior of the function and how it changes as the input variable xx changes.

Q: How can we determine the symmetry of a quadratic function?

A: To determine the symmetry of a quadratic function, we need to examine the coefficients of the quadratic function. If the coefficient of the x2x^2 term is positive, the function has even symmetry. If the coefficient of the x2x^2 term is negative, the function has no symmetry. If the coefficient of the x2x^2 term is zero, the function has odd symmetry.

Q: What is the vertex of the function?

A: The vertex of the function is the point at which the function changes from decreasing to increasing or vice versa. In the case of the function f(x)=0.08x2−12x+650f(x) = 0.08x^2 - 12x + 650, the vertex is at the point (x,y)=(−b2a,f(−b2a))=(75,f(75))(x, y) = (-\frac{b}{2a}, f(-\frac{b}{2a})) = (75, f(75)).

Q: How can we graph the function?

A: We can graph the function using a graphing calculator or a computer algebra system. The graph of the function will be a parabola that opens upwards, with the vertex at the point (75,f(75))(75, f(75)).

Q: What is the significance of the vertex in the function?

A: The vertex of the function is significant because it represents the minimum or maximum value of the function. In the case of the function f(x)=0.08x2−12x+650f(x) = 0.08x^2 - 12x + 650, the vertex represents the minimum value of the function.

Q: How can we use the function to model real-world problems?

A: We can use the function to model real-world problems by substituting the input values into the function and solving for the output values. For example, if we want to find the daily cost of running a factory that produces boxing gloves, we can substitute the number of pairs of boxing gloves produced each day into the function and solve for the daily cost.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have many common applications in real-world problems, such as:

  • Modeling the trajectory of a projectile
  • Finding the maximum or minimum value of a function
  • Solving optimization problems
  • Modeling the growth or decay of a population

Conclusion

In conclusion, the function f(x)=0.08x2−12x+650f(x) = 0.08x^2 - 12x + 650 has even symmetry and can be used to model real-world problems. We can determine the symmetry of a quadratic function by examining the coefficients of the quadratic function, and we can graph the function using a graphing calculator or a computer algebra system.

References

Further Reading