Write down the equation of the quadratic function

Oct 13, 2025 by miodeya.com 24 views

Introduction to Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upward or downward. The $y$ -intercept of a quadratic function is the point at which the graph intersects the $y$ -axis, which is the value of $f(x)$ when $x = 0$ .

Finding the $$y$$-intercept of a Quadratic Function

To find the $y$ -intercept of a quadratic function, we need to substitute $x = 0$ into the equation of the function. This will give us the value of $f(0)$ , which is the $y$ -intercept.

Example: Finding the $$y$$-intercept of $f(x) = x^2 - 8x + 12$

To find the $y$ -intercept of the quadratic function $f(x) = x^2 - 8x + 12$ , we need to substitute $x = 0$ into the equation.

$f(0) = (0)^2 - 8(0) + 12$

$f(0) = 0 - 0 + 12$

$f(0) = 12$

Conclusion

The $y$ -intercept of the quadratic function $f(x) = x^2 - 8x + 12$ is 12. This means that the graph of the function intersects the $y$ -axis at the point (0, 12).

Step-by-Step Solution

Write down the equation of the quadratic function: $f(x) = x^2 - 8x + 12$
Substitute $x = 0$ into the equation: $f(0) = (0)^2 - 8(0) + 12$
Simplify the equation: $f(0) = 0 - 0 + 12$
Evaluate the expression: $f(0) = 12$

Common Mistakes to Avoid

Not substituting $x = 0$ into the equation: This is the most common mistake when finding the $y$ -intercept of a quadratic function.
Not simplifying the equation: Make sure to simplify the equation after substituting $x = 0$ .
Not evaluating the expression: Make sure to evaluate the expression after simplifying the equation.

Real-World Applications

Quadratic functions have many real-world applications, including:

Physics: Quadratic functions are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic functions are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic functions are used to model the behavior of economic systems, such as supply and demand curves.

Final Answer

The final answer is $\boxed{12}$ .

Introduction

In the previous article, we discussed how to find the $y$ -intercept of a quadratic function. In this article, we will answer some common questions related to quadratic functions and their $y$ -intercepts.

Q: What is the $y$ -intercept of the quadratic function $f(x) = x^2 + 5x + 6$ ?

A: To find the $y$ -intercept of the quadratic function $f(x) = x^2 + 5x + 6$ , we need to substitute $x = 0$ into the equation.

$f(0) = (0)^2 + 5(0) + 6$

$f(0) = 0 + 0 + 6$

$f(0) = 6$

The $y$ -intercept of the quadratic function $f(x) = x^2 + 5x + 6$ is 6.

Q: How do I find the $y$ -intercept of a quadratic function with a negative leading coefficient?

A: To find the $y$ -intercept of a quadratic function with a negative leading coefficient, we need to substitute $x = 0$ into the equation and simplify the expression.

For example, let's find the $y$ -intercept of the quadratic function $f(x) = -x^2 + 3x - 2$ .

$f(0) = -(0)^2 + 3(0) - 2$

$f(0) = 0 + 0 - 2$

$f(0) = -2$

The $y$ -intercept of the quadratic function $f(x) = -x^2 + 3x - 2$ is -2.

Q: Can the $y$ -intercept of a quadratic function be a fraction?

A: Yes, the $y$ -intercept of a quadratic function can be a fraction. For example, let's find the $y$ -intercept of the quadratic function $f(x) = x^2 - 2x + 1/2$ .

$f(0) = (0)^2 - 2(0) + 1/2$

$f(0) = 0 - 0 + 1/2$

$f(0) = 1/2$

The $y$ -intercept of the quadratic function $f(x) = x^2 - 2x + 1/2$ is 1/2.

Q: How do I find the $y$ -intercept of a quadratic function with a zero coefficient?

A: To find the $y$ -intercept of a quadratic function with a zero coefficient, we need to substitute $x = 0$ into the equation and simplify the expression.

For example, let's find the $y$ -intercept of the quadratic function $f(x) = 2x^2 + 0x + 3$ .

$f(0) = 2(0)^2 + 0(0) + 3$

$f(0) = 0 + 0 + 3$

$f(0) = 3$

The $y$ -intercept of the quadratic function $f(x) = 2x^2 + 0x + 3$ is 3.

Q: Can the $y$ -intercept of a quadratic function be a negative number?

A: Yes, the $y$ -intercept of a quadratic function can be a negative number. For example, let's find the $y$ -intercept of the quadratic function $f(x) = x^2 - 4x - 3$ .

$f(0) = (0)^2 - 4(0) - 3$

$f(0) = 0 - 0 - 3$

$f(0) = -3$

The $y$ -intercept of the quadratic function $f(x) = x^2 - 4x - 3$ is -3.

Q: How do I find the $y$ -intercept of a quadratic function with a complex coefficient?

A: To find the $y$ -intercept of a quadratic function with a complex coefficient, we need to substitute $x = 0$ into the equation and simplify the expression.

For example, let's find the $y$ -intercept of the quadratic function $f(x) = x^2 + 2ix - 3$ .

$f(0) = (0)^2 + 2i(0) - 3$

$f(0) = 0 + 0i - 3$

$f(0) = -3$

The $y$ -intercept of the quadratic function $f(x) = x^2 + 2ix - 3$ is -3.

Conclusion

In this article, we answered some common questions related to quadratic functions and their $y$ -intercepts. We hope that this article has been helpful in understanding how to find the $y$ -intercept of a quadratic function.