Determine The Function Value For F ( X ) = 4 X 2 F(x) = 4x^2 F ( X ) = 4 X 2 When X ≤ 0 X \leq 0 X ≤ 0 .

Introduction

In mathematics, functions are used to describe the relationship between variables. Quadratic functions, in particular, are a type of function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on determining the function value for the quadratic function $f(x) = 4x^2$ when $x \leq 0$.

Understanding Quadratic Functions

Quadratic functions are a fundamental concept in algebra and are used to model a wide range of real-world phenomena. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.

Properties of Quadratic Functions

Quadratic functions have several important properties that are useful to know when working with them. Some of the key properties of quadratic functions include:

• Domain and Range: The domain of a quadratic function is all real numbers, while the range is all real numbers greater than or equal to the minimum value of the function.
• Vertex: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa.
• Axis of Symmetry: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex and divides the graph into two symmetrical parts.

Determining Function Values

To determine the function value for a quadratic function, we need to substitute the given value of $x$ into the function and simplify. In the case of the function $f(x) = 4x^2$, we need to substitute $x \leq 0$ into the function.

Step 1: Substitute $x \leq 0$ into the function

To determine the function value for $f(x) = 4x^2$ when $x \leq 0$, we need to substitute $x \leq 0$ into the function. This gives us:

$f(x) = 4(x \leq 0)^2$

Step 2: Simplify the expression

To simplify the expression, we need to expand the squared term:

$f(x) = 4(x^2)$

Step 3: Evaluate the expression

To evaluate the expression, we need to substitute $x \leq 0$ into the function:

$f(x) = 4(0)$

Conclusion

In conclusion, to determine the function value for $f(x) = 4x^2$ when $x \leq 0$, we need to substitute $x \leq 0$ into the function and simplify. This gives us $f(x) = 4(0)$, which is equal to 0.

Example Problems

Here are some example problems that illustrate how to determine function values for quadratic functions:

Example 1

Determine the function value for $f(x) = 3x^2$ when $x = -2$.

Solution

To determine the function value for $f(x) = 3x^2$ when $x = -2$, we need to substitute $x = -2$ into the function:

$f(x) = 3(-2)^2$

$f(x) = 3(4)$

$f(x) = 12$

Example 2

Determine the function value for $f(x) = 2x^2$ when $x = 1$.

Solution

To determine the function value for $f(x) = 2x^2$ when $x = 1$, we need to substitute $x = 1$ into the function:

$f(x) = 2(1)^2$

$f(x) = 2(1)$

$f(x) = 2$

Practice Problems

Here are some practice problems that you can use to practice determining function values for quadratic functions:

Problem 1

Determine the function value for $f(x) = 5x^2$ when $x = -3$.

Problem 2

Determine the function value for $f(x) = 4x^2$ when $x = 2$.

Problem 3

Determine the function value for $f(x) = 3x^2$ when $x = -1$.

Conclusion

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Introduction

In our previous article, we discussed how to determine the function value for quadratic functions. In this article, we will provide a Q&A section to help you better understand the concept and address any questions you may have.

Q: What is a quadratic function?

A: A quadratic function is a type of function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

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

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

Q: What is the domain and range of a quadratic function?

A: The domain of a quadratic function is all real numbers, while the range is all real numbers greater than or equal to the minimum value of the function.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa.

Q: How do I determine the function value for a quadratic function?

A: To determine the function value for a quadratic function, you need to substitute the given value of $x$ into the function and simplify.

Q: What if the value of $x$ is negative?

A: If the value of $x$ is negative, you need to substitute $x \leq 0$ into the function and simplify.

Q: Can you provide an example of how to determine the function value for a quadratic function?

A: Yes, here is an example:

Determine the function value for $f(x) = 4x^2$ when $x = -2$.

To determine the function value for $f(x) = 4x^2$ when $x = -2$, we need to substitute $x = -2$ into the function:

$f(x) = 4(-2)^2$

$f(x) = 4(4)$

$f(x) = 16$

Q: What if the value of $x$ is a fraction?

A: If the value of $x$ is a fraction, you need to substitute the fraction into the function and simplify.

Q: Can you provide an example of how to determine the function value for a quadratic function with a fraction?

A: Yes, here is an example:

Determine the function value for $f(x) = 3x^2$ when $x = \frac{1}{2}$.

To determine the function value for $f(x) = 3x^2$ when $x = \frac{1}{2}$, we need to substitute $x = \frac{1}{2}$ into the function:

$f(x) = 3(\frac{1}{2})^2$

$f(x) = 3(\frac{1}{4})$

$f(x) = \frac{3}{4}$

Q: What if the value of $x$ is a decimal?

A: If the value of $x$ is a decimal, you need to substitute the decimal into the function and simplify.

Q: Can you provide an example of how to determine the function value for a quadratic function with a decimal?

A: Yes, here is an example:

Determine the function value for $f(x) = 2x^2$ when $x = 0.5$.

To determine the function value for $f(x) = 2x^2$ when $x = 0.5$, we need to substitute $x = 0.5$ into the function:

$f(x) = 2(0.5)^2$

$f(x) = 2(0.25)$

$f(x) = 0.5$

Conclusion

In conclusion, determining function values for quadratic functions is an important concept in algebra that can be used to model a wide range of real-world phenomena. By following the steps outlined in this article, you can determine the function value for any quadratic function. Remember to substitute the given value of $x$ into the function and simplify to get the final answer.

Practice Problems

Here are some practice problems that you can use to practice determining function values for quadratic functions:

Problem 1

Determine the function value for $f(x) = 5x^2$ when $x = -3$.

Problem 2

Determine the function value for $f(x) = 4x^2$ when $x = 2$.

Problem 3

Determine the function value for $f(x) = 3x^2$ when $x = -1$.

Answer Key

Here are the answers to the practice problems:

Problem 1

$f(x) = 5(-3)^2$

$f(x) = 5(9)$

$f(x) = 45$

Problem 2

$f(x) = 4(2)^2$

$f(x) = 4(4)$

$f(x) = 16$

Problem 3

$f(x) = 3(-1)^2$

$f(x) = 3(1)$

$f(x) = 3$