Problem 2 - Algebra A Combined Function

Question

If \(P(x)=x^{2}+x+1\) and \(Q(x)=5 x^{2}-1,\) find each function value. $$ Q(4) $$

Step-by-Step Solution

Answer

Q(4) = 79.

Step 1: Understand the Problem

We need to find the value of the function \(Q(x)\) when \(x = 4\) using the given function \(Q(x) = 5x^2 - 1\).

Step 2: Substitute the Value of x

Replace \(x\) with 4 in the function \(Q(x) = 5x^2 - 1\). This gives: \(Q(4) = 5(4)^2 - 1\).

Step 3: Calculate the Square

Compute \(4^2\) to get 16. So the expression becomes: \(Q(4) = 5(16) - 1\).

Step 4: Compute the Product

Multiply 5 by 16 to get 80. Now the expression is: \(Q(4) = 80 - 1\).

Step 5: Final Calculation

Subtract 1 from 80 to find the value of \(Q(4)\), which is 79.

Key Concepts

Understanding Polynomials

Polynomials are a fundamental concept in mathematics. They are expressions made up of variables and coefficients, linked together by addition, subtraction, and multiplication operations. Each term in a polynomial has a coefficient and a variable raised to a non-negative integer power. Here's what makes up a polynomial:

• Terms: Building blocks, like \(5x^2\)
• Coefficients: Numbers multiplying the variables, such as 5 in \(5x^2\)
• Degree: The highest power of the variable, which determines the polynomial's order. For example, \(x^2 + x + 1\) has a degree of 2.

By understanding polynomials, students can easily manipulate equations and solve problems using arithmetic operations. In our exercise, we had two polynomial functions, \(P(x) = x^2 + x + 1\) and \(Q(x) = 5x^2 - 1\). We used the second function to calculate its value at a specific point by substituting a number for the variable.

Exploring the Substitution Method

The substitution method is an elementary algebraic technique utilized to evaluate functions. It involves replacing the variable in a function with a given number, thereby finding the specific output at that point. Here's a simple breakdown of how it works:

• Identify the function: Recognize the function you need to evaluate. In our exercise, it was \(Q(x) = 5x^2 - 1\).
• Substitute: Replace every occurrence of the variable with the given number. For instance, we replaced \(x\) with 4, thus changing the function to \(Q(4) = 5(4)^2 - 1\).
• Simplify: Carry out arithmetic operations to simplify the expression to a single numerical value. For instance, by calculating \(4^2 = 16\), we simplify to \(Q(4) = 5(16) - 1\).

Using this substitution process helps in solving specific values of functions, especially when dealing with polynomials. It also helps in understanding how changes in input affect outputs.

Comprehending Quadratic Functions

Quadratic functions are a particular type of polynomial function with a degree of 2. They often appear in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). Key features of quadratic functions include:

• Parabolic Shape: Quadratic functions graph as a parabola — a U-shaped curve that may open upwards or downwards based on the sign of \(a\).
• Vertex: The highest or lowest point of the parabola. It represents the maximum or minimum value of the function.
• Roots: Points where the graph intersects the x-axis, also known as solutions or zeros of the function.

In our given exercise, both functions provided are quadratic. The function \(Q(x) = 5x^2 - 1\) is quadratically expanded, and evaluating it at \(x = 4\) demonstrated how quadratic functions can be solved to find specific outputs using simple polynomial arithmetic.