Problem 83 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \begin{aligned} &\text { If } f(x)=a x^{2}+b x+c, \text { find } f(0), f(1), f(-1), f(a) \text { and } f(b) . \text { Ans. } c, a+b+c, a-b+c,\\\ &\left.a^{3}+a b+c, a b^{2}+b^{2}+c\right\\} \end{aligned} $$

Step-by-Step Solution

Answer

In summary, the short answer for the required values of the function is: \(f(0) = c\), \(f(1) = a + b + c\), \(f(-1) = a - b + c\), \(f(a) = a^3 + ab + c\), and \(f(b) = ab^2 + b^2 + c\).

Step 1: Write down the function

The quadratic function we are given is: \(f(x) = ax^2 + bx + c\) Now, we plug in the values 0, 1, -1, a, and b.

Step 2: Calculate the value of f(0) using the function

To find f(0), substitute 0 for x in the given function: \(f(0) = a(0)^2 + b(0) + c\) The result is: \(f(0) = c\)

Step 3: Calculate the value of f(1) using the function

To find f(1), substitute 1 for x in the given function: \(f(1) = a(1)^2 + b(1) + c\) The result is: \(f(1) = a + b + c\)

Step 4: Calculate the value of f(-1) using the function

To find f(-1), substitute -1 for x in the given function: \(f(-1) = a(-1)^2 + b(-1) + c\) The result is: \(f(-1) = a - b + c\)

Step 5: Calculate the value of f(a) using the function

To find f(a), substitute a for x in the given function: \(f(a) = a(a)^2 + b(a) + c\) The result is: \(f(a) = a^3 + ab + c\)

Step 6: Calculate the value of f(b) using the function

To find f(b), substitute b for x in the given function: \(f(b) = a(b)^2 + b(b) + c\) The result is: \(f(b) = ab^2 + b^2 + c\)

Step 7: List the answers

Now that we have found the values of f(0), f(1), f(-1), f(a), and f(b), we can list the answers: \(f(0) = c\) \(f(1) = a + b + c\) \(f(-1) = a - b + c\) \(f(a) = a^3 + ab + c\) \(f(b) = ab^2 + b^2 + c\)

Key Concepts

Polynomial Evaluation

When you work with a quadratic function like \(f(x) = ax^2 + bx + c\), evaluating the polynomial involves finding the value of the function for different inputs. This is often referred to as polynomial evaluation.
To evaluate a polynomial, simply substitute a number for \(x\) in the equation and perform the necessary calculations. In the example provided, we evaluated \(f(x)\) at several points including \(f(0), f(1), f(-1), f(a)\), and \(f(b)\).

• At \(f(0)\): Substitute \(x = 0\), so \(f(0) = a(0)^2 + b(0) + c = c\).
• At \(f(1)\): Substitute \(x = 1\), resulting in \(f(1) = a(1)^2 + b(1) + c = a + b + c\).
• At \(f(-1)\): Substitute \(x = -1\), thus \(f(-1) = a(-1)^2 + b(-1) + c = a - b + c\).
• At \(f(a)\): Substitute \(x = a\), giving us \(f(a) = a(a)^2 + b(a) + c = a^3 + ab + c\).
• At \(f(b)\): Substitute \(x = b\), yielding \(f(b) = a(b)^2 + b^2 + c = ab^2 + b^2 + c\).

Polynomial evaluation is a foundational skill in algebra that helps you find specific values of a function, crucial for graphing and analyzing patterns in data.

Substitution Method

The substitution method is a simple yet powerful way to solve expressions and equations by replacing variables with numbers or other expressions. In mathematical contexts, substitution allows you to simplify issues of finding unknown values within functions and equations.
In the context of quadratic functions like \(f(x) = ax^2 + bx + c\), substitution is a straightforward process:

• Identify the variable you wish to evaluate. For example, start with \(f(x)\) and choose a particular value for \(x\).
• Replace the variable with the chosen value in the polynomial, simplifying each term accordingly.
• Perform arithmetic operations including squaring or multiplying, as necessary, to find the polynomial's value.

Using substitution helps in verifying the behavior or properties of a polynomial function at specific points, crucial for understanding and interpreting mathematical models.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Unlike equations, they do not have an equality sign, but expressions can be part of equations when evaluating functions like quadratic functions.
An algebraic expression in the form of \(ax^2 + bx + c\) is a fundamental component of polynomial equations, where:

• \(a\), \(b\), and \(c\) are coefficients - constant values that multiply with the variable terms.
• \(x\) represents a variable, the unknown that can take different values in context.
• \(ax^2 + bx\) represents the degree of the expression, with - in our case - the highest degree being 2, indicating a quadratic function.

Understanding and manipulating algebraic expressions is crucial in algebra. It enables us to simplify, evaluate and solve complex mathematical scenarios, paving the way for more advanced topics like calculus and beyond.