Problem 3 - Algebra A Combined Function

Question

If \(P(x)=x^{2}+x+1\) and \(Q(x)=5 x^{2}-1,\) find each function value. $$ Q(-10) $$

Step-by-Step Solution

Answer

The value is 499.

Step 1: Learn the Given Expression

We need to find the value of the function \(Q(x)\) at \(x = -10\). The function \(Q(x) = 5x^2 - 1\) is provided, so our task is to substitute \(-10\) into this function.

Step 2: Substitute the Value into the Function

Substitute \(-10\) into the function \(Q(x) = 5x^2 - 1\). This gives: \[Q(-10) = 5(-10)^2 - 1\]

Step 3: Calculate the Square of -10

Calculate the square of \(-10\): \[(-10)^2 = 100\]

Step 4: Multiply by the Coefficient

Multiply the square of \(-10\) by the coefficient 5 in the function: \[5 \times 100 = 500\]

Step 5: Complete the Calculation

Subtract 1 from the result obtained in the previous step to complete the calculation of \(Q(-10)\): \[Q(-10) = 500 - 1 = 499\]

Key Concepts

Polynomial Functions

Polynomial functions are a crucial concept in algebra. They consist of variables raised to different power, usually non-negative integers, and combined with constants. A polynomial is expressed as a sum of these terms. For example, the general form of a polynomial function is:

• \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)

where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants, and \(n\) is a non-negative integer This representation makes polynomial functions quite flexible, capable of modeling a variety of real-world situations.

Polynomials can be categorized by their degree, which is the highest power of the variable in the function. In the exercise example, the function \(Q(x) = 5x^2 - 1\) is a quadratic polynomial since its highest degree is 2. Remember, the degree of a polynomial helps us predict the graph's general shape.

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. This involves replacing every occurrence of the variable with the provided number and then performing the necessary calculations. Function evaluation is essential when determining the specific value of a function at a point. This is exactly what we do when we find \(Q(-10)\) for the polynomial \(Q(x) = 5x^2 - 1\) in the original exercise.

The steps to evaluate a function such as \(Q(x)\) are:

• Identify the variable in the function.
• Substitute the given number for the variable.
• Carry out all mathematical operations involved, respecting order of operations: parentheses, exponents, multiplication, division, and addition/subtraction (PEMDAS).

Accurate function evaluation helps us understand how the values of a polynomial equation change and respond under specific conditions.

Quadratic Equations

Quadratic equations are a specific type of polynomial equation distinguished by the highest exponent of 2. They generally take the form \(ax^2 + bx + c = 0\). This specific structure gives rise to a distinct parabolic graph when plotted on a coordinate plane.

Key features of quadratic equations include:

• They can have two solutions, one solution, or no real solution at all.
• Their graph is a parabola, opening upwards if \(a > 0\) and downwards if \(a < 0\).
• Solutions can be found using different methods such as factorization, completing the square, or the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\).

Understanding quadratic equations is vital because they frequently appear in numerous fields like physics, engineering, economics, and other sciences where relationships are best described by a parabolic path or shape.