Write A Quadratic Function In Standard Form Whose Graph Satisfies The Given Condition:The Zeros Of The Function Are -1 And 3, And The Graph Opens Downward.

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Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we will focus on writing a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are -1 and 3, and the graph opens downward.

Understanding Quadratic Functions

A quadratic function can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, which is a U-shaped curve. The zeros of a quadratic function are the values of x that make the function equal to zero. In other words, they are the points where the graph intersects the x-axis.

Graphing Quadratic Functions

To graph a quadratic function, we need to find the zeros of the function and the vertex of the parabola. The vertex of a parabola is the point where the graph changes direction. If the graph opens upward, the vertex is the minimum point, and if the graph opens downward, the vertex is the maximum point.

Writing a Quadratic Function in Standard Form

To write a quadratic function in standard form whose graph satisfies the given condition, we need to find the values of a, b, and c. Since the zeros of the function are -1 and 3, we know that the factored form of the function is f(x) = a(x + 1)(x - 3). To find the value of a, we need to determine the direction of the graph.

Determining the Direction of the Graph

Since the graph opens downward, we know that the coefficient of the x^2 term, which is a, must be negative. Therefore, we can write the function as f(x) = -a(x + 1)(x - 3).

Finding the Value of a

To find the value of a, we can use the fact that the graph passes through the point (0, 6). This means that when x = 0, f(x) = 6. Substituting x = 0 into the function, we get:

f(0) = -a(0 + 1)(0 - 3) f(0) = -a(1)(-3) f(0) = 3a

Since f(0) = 6, we can set up the equation:

3a = 6

Dividing both sides by 3, we get:

a = 2

Writing the Quadratic Function in Standard Form

Now that we have found the value of a, we can write the quadratic function in standard form:

f(x) = -2(x + 1)(x - 3)

To expand the function, we can use the distributive property:

f(x) = -2(x^2 - 3x + x - 3) f(x) = -2(x^2 - 2x - 3) f(x) = -2x^2 + 4x + 6

Conclusion

In this article, we have written a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are -1 and 3, and the graph opens downward. We have found the values of a, b, and c, and we have expanded the function to write it in standard form. This example illustrates the importance of understanding quadratic functions and how to write them in standard form.

Examples and Applications

Quadratic functions have numerous applications in various fields, including physics, engineering, and economics. Here are a few examples:

  • Projectile Motion: The trajectory of a projectile under the influence of gravity is a quadratic function.
  • Optimization: Quadratic functions are used to optimize functions, such as finding the maximum or minimum value of a function.
  • Signal Processing: Quadratic functions are used in signal processing to filter signals and remove noise.

Tips and Tricks

Here are a few tips and tricks to help you write quadratic functions in standard form:

  • Use the factored form: When writing a quadratic function in standard form, it is often helpful to start with the factored form of the function.
  • Determine the direction of the graph: The direction of the graph is determined by the coefficient of the x^2 term.
  • Use the distributive property: To expand the function, use the distributive property to multiply the terms.

Practice Problems

Here are a few practice problems to help you practice writing quadratic functions in standard form:

  • Problem 1: Write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are 2 and 4, and the graph opens upward.
  • Problem 2: Write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are -2 and 1, and the graph opens downward.
  • Problem 3: Write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are 3 and -1, and the graph opens upward.

Conclusion

In conclusion, writing a quadratic function in standard form is an important skill in mathematics. By understanding the properties of quadratic functions and using the factored form, we can write quadratic functions in standard form. This article has provided a step-by-step guide on how to write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are -1 and 3, and the graph opens downward.

Introduction

In our previous article, we discussed how to write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are -1 and 3, and the graph opens downward. In this article, we will answer some frequently asked questions about quadratic functions.

Q&A

Q1: What is the difference between a quadratic function and a linear function?

A1: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. A linear function, on the other hand, is a polynomial function of degree one, which means the highest power of the variable is one.

Q2: How do I determine the direction of the graph of a quadratic function?

A2: To determine the direction of the graph of a quadratic function, you need to look at the coefficient of the x^2 term. If the coefficient is positive, the graph opens upward. If the coefficient is negative, the graph opens downward.

Q3: What is the vertex of a quadratic function?

A3: The vertex of a quadratic function is the point where the graph changes direction. If the graph opens upward, the vertex is the minimum point, and if the graph opens downward, the vertex is the maximum point.

Q4: How do I find the zeros of a quadratic function?

A4: To find the zeros of a quadratic function, you need to set the function equal to zero and solve for x. This can be done using factoring, the quadratic formula, or other methods.

Q5: What is the difference between a quadratic function and a polynomial function?

A5: A quadratic function is a specific type of polynomial function, where the highest power of the variable is two. A polynomial function, on the other hand, is a function that can have any degree, including zero.

Q6: How do I write a quadratic function in standard form?

A6: To write a quadratic function in standard form, you need to expand the factored form of the function using the distributive property. This will give you the coefficients of the x^2, x, and constant terms.

Q7: What is the significance of the coefficient of the x^2 term in a quadratic function?

A7: The coefficient of the x^2 term in a quadratic function determines the direction of the graph. If the coefficient is positive, the graph opens upward. If the coefficient is negative, the graph opens downward.

Q8: How do I use the quadratic formula to find the zeros of a quadratic function?

A8: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic function. To use the quadratic formula, you need to plug in the values of a, b, and c and solve for x.

Q9: What is the difference between a quadratic function and a rational function?

A9: A quadratic function is a polynomial function of degree two, while a rational function is a function that can be written as the ratio of two polynomials.

Q10: How do I graph a quadratic function?

A10: To graph a quadratic function, you need to find the zeros of the function and the vertex of the parabola. You can then use this information to plot the graph of the function.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We hope that this information has been helpful in understanding quadratic functions and how to work with them.

Practice Problems

Here are a few practice problems to help you practice working with quadratic functions:

  • Problem 1: Write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are 2 and 4, and the graph opens upward.
  • Problem 2: Write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are -2 and 1, and the graph opens downward.
  • Problem 3: Write a quadratic function in standard form whose graph satisfies the given condition: the zeros of the function are 3 and -1, and the graph opens upward.

Tips and Tricks

Here are a few tips and tricks to help you work with quadratic functions:

  • Use the factored form: When working with quadratic functions, it is often helpful to start with the factored form of the function.
  • Determine the direction of the graph: The direction of the graph is determined by the coefficient of the x^2 term.
  • Use the distributive property: To expand the function, use the distributive property to multiply the terms.
  • Use the quadratic formula: The quadratic formula is a useful tool for finding the zeros of a quadratic function.

Conclusion

In conclusion, quadratic functions are an important topic in mathematics, and understanding how to work with them is essential for success in mathematics and other fields. We hope that this article has been helpful in understanding quadratic functions and how to work with them.