Writing the Quadratic Function

Oct 13, 2025 by thesweetdesigns.com 124 views

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It can be written in the form of ${f(x) = ax^2 + bx + c}$ , where ${a}$ , ${b}$ , and ${c}$ are constants, and ${a}$ is the leading coefficient. The zeros of a quadratic function are the values of ${x}$ for which ${f(x) = 0}$ .

Given Zeros: 0 and -5

We are given that the zeros of the quadratic function are 0 and -5. This means that when we substitute ${x = 0}$ and ${x = -5}$ into the function, we should get ${f(x) = 0}$ .

Writing the Quadratic Function

To write the quadratic function, we can use the fact that if ${x = r}$ is a zero of the function, then ${(x - r)}$ is a factor of the function. Since we have two zeros, 0 and -5, we can write the function as:

${f(x) = a(x - 0)(x + 5)}$

Simplifying the Function

We can simplify the function by multiplying the factors:

${f(x) = a(x)(x + 5)}$ ${f(x) = ax(x + 5)}$ ${f(x) = ax^2 + 5ax}$

Standard Form with Leading Coefficient

We are asked to write the function in standard form with a leading coefficient. Since the leading coefficient is ${a}$ , we can rewrite the function as:

${f(x) = ax^2 + 5ax}$

Finding the Value of a

To find the value of ${a}$ , we can use the fact that the function has a zero at ${x = 0}$ . Substituting ${x = 0}$ into the function, we get:

${f(0) = a(0)^2 + 5a(0)}$ ${f(0) = 0}$

This means that the value of ${a}$ is not important, as long as the function has a zero at ${x = 0}$ . However, if we want to find a specific value for ${a}$ , we can choose any value that makes the function easy to work with. For example, if we choose ${a = 1}$ , the function becomes:

${f(x) = x^2 + 5x}$

Conclusion

In conclusion, we have written a quadratic function with zeros 0 and -5. The function is in standard form with a leading coefficient, and we have found a specific value for the leading coefficient ${a}$ . The function is:

${f(x) = x^2 + 5x}$

Example Use Case

The quadratic function can be used to model a variety of real-world situations, such as the height of a projectile or the cost of producing a certain quantity of goods. For example, if we want to model the height of a projectile that is launched from the ground and lands at a distance of 5 units, we can use the function:

${h(x) = x^2 + 5x}$

where ${h(x)}$ is the height of the projectile at a distance of ${x}$ units from the launch point.

Tips and Variations

To find the zeros of a quadratic function, we can set the function equal to zero and solve for ${x}$ .
To find the maximum or minimum value of a quadratic function, we can use the fact that the function has a vertex at the point where the derivative of the function is zero.
To find the equation of a quadratic function that passes through a given point, we can use the fact that the function has a zero at that point.

Common Mistakes

Not using the correct form of the quadratic function.
Not finding the correct value of the leading coefficient.
Not using the correct method to find the zeros of the function.

Conclusion

${f(x) = x^2 + 5x}$

This function can be used to model a variety of real-world situations, and it can be used to find the zeros of the function, the maximum or minimum value of the function, and the equation of the function that passes through a given point.

Understanding Quadratic Functions

Frequently Asked Questions

Q: What are the zeros of a quadratic function?

A: The zeros of a quadratic function are the values of ${x}$ for which ${f(x) = 0}$ . In other words, they are the points where the graph of the function intersects the x-axis.

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

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

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is ${f(x) = ax^2 + bx + c}$ , where ${a}$ , ${b}$ , and ${c}$ are constants, and ${a}$ is the leading coefficient.

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

A: To write a quadratic function in standard form, you can start with the factored form of the function and multiply out the factors. For example, if the factored form is ${f(x) = a(x - r)(x - s)}$ , you can multiply out the factors to get ${f(x) = a(x^2 - rx - sx + rs)}$ .

Q: What is the leading coefficient of a quadratic function?

A: The leading coefficient of a quadratic function is the coefficient of the ${x^2}$ term, which is denoted by ${a}$ .

Q: How do I find the value of the leading coefficient?

A: To find the value of the leading coefficient, you can use the fact that the function has a zero at a certain point. For example, if the function has a zero at ${x = 0}$ , you can substitute ${x = 0}$ into the function and solve for ${a}$ .

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the graph of the function changes direction. It is also the point where the function has a maximum or minimum value.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the fact that the vertex is located at the point where the derivative of the function is zero. You can find the derivative of the function using the power rule, and then set it equal to zero and solve for ${x}$ .

Q: What is the equation of a quadratic function that passes through a given point?

A: To find the equation of a quadratic function that passes through a given point, you can use the fact that the function has a zero at that point. You can substitute the coordinates of the point into the function and solve for ${a}$ , ${b}$ , and ${c}$ .

Common Mistakes

Not using the correct form of the quadratic function.
Not finding the correct value of the leading coefficient.
Not using the correct method to find the zeros of the function.

Conclusion

In conclusion, we have answered some frequently asked questions about quadratic functions. We have discussed the zeros of a quadratic function, how to find the zeros, the standard form of a quadratic function, and how to write a quadratic function in standard form. We have also discussed the leading coefficient of a quadratic function, how to find the value of the leading coefficient, the vertex of a quadratic function, and how to find the vertex. Finally, we have discussed the equation of a quadratic function that passes through a given point.