Problem 126 State whether the function is li... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 126

State whether the function is linear or quadratic. $$f(x)=2-x-x^{2}$$

Short Answer

Expert verified

The function is quadratic.

Step by step solution

Identify the highest power of x

Examine the given function: \[ f(x) = 2 - x - x^2 \]Identify the term with the highest exponent. Here, the highest power of $ x $ is 2, which corresponds to $ -x^2 $.

Determine the type of function

A linear function has the highest power of $ x $ as 1, and it follows the form $ f(x) = ax + b $. A quadratic function has the highest power of $ x $ as 2, and it follows the form $ f(x) = ax^2 + bx + c $. Since the highest power in $ f(x) = 2 - x - x^2 $ is 2, it conforms to a quadratic function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Identification

To determine the type of a function, you first need to understand how different functions are categorized. Functions like the one given in your exercise, \[f(x) = 2 - x - x^2\], can generally be categorized into linear functions or quadratic functions.

**Linear functions**: These functions are in the form \[ f(x) = ax + b\], where the highest power of the variable $ x $ is 1. They produce straight-line graphs.

**Quadratic functions**: These functions are in the form \[ f(x) = ax^2 + bx + c\], and the highest power of the variable $ x $ is 2. They produce parabolic graphs.

By identifying the function type, you can predict the shape of its graph and understand its general behavior better.
Looking at $ f(x) = 2 - x - x^2 $, you can determine that it's a quadratic function because the highest power of $ x $ is 2.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

**General form**: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \], where $ a_n, a_{n-1}, ..., a_1, a_0 $ are constants.
**Degrees**: The 'degree' of the polynomial is the highest power of the variable in the polynomial. For example, in $ 2 - x - x^2 $, the degree is 2.

When a polynomial function has a degree of 2 (i.e., the highest power of $ x $ is 2), it is specifically called a quadratic polynomial. These polynomials often have the general form: \[ f(x) = ax^2 + bx + c \].
For the given function $ 2 - x - x^2 $, you can see that it matches this form and can be identified as a quadratic polynomial.

Highest Power of x

Understanding how to identify the highest power of $ x $ in a function is crucial for classifying the function type. Follow these steps to determine it:

**Step 1**: Look at all the terms in the function.
**Step 2**: Identify the term with the largest exponent on $ x $

In the function provided, $ f(x) = 2 - x - x^2 $, you have three terms: $ 2, -x, $ and $ -x^2 $.
The highest exponent is 2 (from the $ x^2 $ term).
Identifying the highest power of $ x $ helps classify the function because:

If the highest power is 1, the function is linear.
If the highest power is 2, the function is quadratic.

In your function, since the highest power is 2, it's a quadratic function.

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91影视

State whether the function is linear or quadratic. $$f(x)=2-x-x^{2}$$

Short Answer

Step by step solution

Identify the highest power of x

Determine the type of function

Key Concepts

Function Identification

Polynomial Functions

Highest Power of x

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