Question

Find \(f(3)\) if \(f(x)=-4 x^{2}+5 x\)

Short answer

f(3) = -21

Step-by-step solution

- Understand the Function

The function given is: \[f(x) = -4x^{2} + 5x\] This function defines the relationship between x and f(x).

- Substitute the Given Value

We need to find f(3), so substitute 3 in place of x in the function: \[f(3) = -4(3)^{2} + 5(3)\]

- Calculate the Square of 3

Find the square of 3: \[3^{2} = 9\]

- Multiply by Coefficients

Multiply the squared value by -4: \[-4 \times 9 = -36\] and multiply 3 by 5: \[5 \times 3 = 15\]

- Add the Results

Finally, add the results from step 4: \[-36 + 15 = -21\] Thus, \[f(3) = -21\]

Key concepts

Quadratic Functions

Quadratic functions are a type of polynomial function of degree 2. They typically appear in the form y = ax^{2} + bx + c where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The highest exponent of x is 2 in these functions, which means the graph will always form a parabola. The parabola can either open upwards or downwards depending on the sign of the coefficient 'a'.

In our original exercise, the function given is: f(x) = -4x^{2} + 5xThis is a quadratic function because the highest power of x is 2. Here, 'a' is -4, 'b' is 5, and 'c' is 0 (since 'c' is absent, it's equivalent to 0).

This specific quadratic function will produce a parabola opening downwards because 'a' is negative (-4).

Substitution Method

The substitution method is commonly used to find specific values for functions or in solving systems of equations. It involves replacing the variable with a given number. This makes calculations easier by reducing the number of variables.

In the exercise, we need to find the value of the function f(x) when x is 3. This means substituting 3 for x in the given function:f(x) = -4x^{2} + 5x
f(3) = -4(3)^{2} + 5(3)Here, we replaced 'x' with 3 in the original quadratic expression. This enables us to perform further calculations straightforwardly using simple arithmetic operations.

Algebraic Operations

Algebraic operations are mathematical steps involving addition, subtraction, multiplication, and division to manipulate expressions and solve equations. These operations are crucial for simplifying and solving problems involving functions.

In our solution, algebraic operations involve:

• Squaring: Calculate 3^{2}, which gives 9.
• Multiplication: Multiply this result by -4, yielding -36.
• Another Multiplication: Multiply 3 by 5, resulting in 15.
• Addition: Finally, sum the products from the previous steps, i.e., -36 + 15 = -21.

These simple yet fundamental operations help us to arrive at the correct value of the function for a specific x.

Polynomials

Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. They are categorized based on their degree, which is determined by the highest power of the variable in the expression.

A quadratic function like f(x) = -4x^{2} + 5x is a polynomial of degree 2 since the highest exponent is 2. Polynomials are foundational in algebra and are used extensively across different areas of mathematics and applied sciences.

Understanding polynomials is crucial because:

• They help in modeling various real-world situations where relationships are quadratic.
• Their graphs provide visual insights into behaviors like maxima, minima, and intercepts.
• They form the basis for further studies in calculus, optimization, and numerical methods.

Thus, grasping the basics of polynomial functions enables a deeper understanding of more advanced mathematical concepts.