Problem 5 - Algebra A Combined Function

Question

If \(P(x)=x^{2}+x+1\) and \(Q(x)=5 x^{2}-1,\) find each function value. $$ P(0) $$

Step-by-Step Solution

Answer

P(0) = 1

Step 1: Understand the Function P(x)

The function given is \(P(x) = x^2 + x + 1\). This is a quadratic function in terms of \(x\). To find the function value at a specific \(x\), substitute the value of \(x\) into the function.

Step 2: Substitute x with 0

Since we are asked to find \(P(0)\), replace \(x\) in the function \(P(x) = x^2 + x + 1\) with 0. This gives us:\[P(0) = 0^2 + 0 + 1\]

Step 3: Calculate the Expression

Now, calculate the expression we formed in the previous step. It simplifies to \[P(0) = 0 + 0 + 1 = 1\].

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function characterized by the highest exponent of the variable being 2. The general form of a quadratic function is given by \[ P(x) = ax^2 + bx + c \] where \(a, b,\) and \(c\) are constants, with \(a eq 0\). This function creates a parabola when graphed on a coordinate plane.

• Vertex: It's the peak or the lowest point of the parabola, depending on its orientation.
• Axis of Symmetry: This is a vertical line through the vertex, dividing the parabola into two mirror images.
• Roots: Also called the zeros, these are the points where the parabola crosses the x-axis.

In the function \(P(x) = x^2 + x + 1\), the coefficients are \(a = 1\), \(b = 1\), and \(c = 1\). This specific function forms a parabola that opens upward due to the positive value of \(a\). The function’s vertex can be calculated with the vertex formula \(x = -\frac{b}{2a}\). Understanding quadratic functions is crucial because they frequently appear in various real-world scenarios, such as modeling the trajectory of an object or optimizing areas.

Substitution Method

The substitution method is a straightforward technique used to evaluate functions or solve equations. It involves replacing the variable in an expression with a specific value or another expression. This method is particularly handy when dealing with polynomial functions like quadratics.

• Identify: First, recognize the variable that needs replacement, like \(x\) in our example \(P(x) = x^2 + x + 1\).
• Substitute: Replace every occurrence of the variable \(x\) in the expression with a given number, such as zero in the case of finding \(P(0)\).
• Calculate: Solve the resulting numerical expression to find the function value.

Using substitution in our example, we set \(x = 0\) in \(P(x) = x^2 + x + 1\), leading us to evaluate \(P(0) = 0^2 + 0 + 1 = 1\). This quick process allows you to find specific outputs based on particular inputs, making it highly valuable for evaluating expressions.

Evaluating Expressions

Evaluating expressions involves simplifying an algebraic expression to find its numerical value, which is essential when working with functions. This process typically follows the substitution of variables step.

• Substitution: Insert specific numbers in place of variables, as performed in the previous substitution method.
• Order of Operations: Follow the algebraic order of operations - Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) - often abbreviated as PEMDAS.
• Simplify: Combine and simplify the resulting terms to reach the final value.

In the example given, after substituting \(x = 0\) into the expression \(x^2 + x + 1\), we evaluate and simplify the expression to get \(0^2 + 0 + 1 = 1\). Through this systematic approach, evaluating expressions becomes a methodical task, allowing for precise calculation of outputs for given inputs.