Problem 122 State whether the function is li... [FREE SOLUTION]

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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 122

State whether the function is linear or quadratic. $$f(x)=4-5 x^{2}$$

Short Answer

Expert verified

The function $ f(x) = 4 - 5x^2 $ is quadratic.

Step by step solution

Understand the Function

First, examine the given function. The function given is$ f(x) = 4 - 5x^2 $.

Identify the Highest Power of x

Identify the highest power of the variable x in the function. In this case, the highest power is 2 since we have a term $ -5x^2 $.

Determine the Type of Function

Linear functions have the highest power of x equal to 1, forming equations of the type $ ax + b $. Quadratic functions have the highest power of x equal to 2, forming equations of the type $ ax^2 + bx + c $.

Classify the Function

Since the given function $ f(x) = 4 - 5x^2 $ has the highest power of x equal to 2, it is classified as a quadratic function.

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Key Concepts

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

linear functions

Linear functions are a fundamental concept in algebra. They have the form $ ax + b $, where $a$ and $b$ are constants, and $x$ is the variable. These functions create straight lines when graphed.

Some characteristics of linear functions include:

The highest power of $ x $ is 1.
The graph is a straight line.
The rate of change is constant.

Linear functions are used to model relationships with a constant rate of change, such as speed or cost. For example, $ f(x) = 2x + 3 $ is a linear function. The slope $2 $ shows the change in $ y $ for each unit increase in $ x$, and the $ + 3 $ represents the $ y $-intercept.

power of x

The 'power of x' (also known as the exponent) in a function helps to determine the type of function we are dealing with. The power of $ x $ refers to its highest exponent in the function. The function's behavior changes with different powers:

For $ x^1 $, it is a linear function.
For $ x^2 $, it is a quadratic function.
For $ x^3 $ or higher, it could be cubic, quartic, or higher-degree polynomial functions.

The power of $ x $ affects the shape of the graph and the complexity of the function. In the given function, $ f(x) = 4 - 5x^2 $, the highest power of $ x $ is 2, classifying it as a quadratic function.

function classification

Classifying functions is important in understanding their behavior and solving algebraic problems. There are several types of functions, each with distinct characteristics:

Linear Functions: Highest power of $ x $ is 1, forming a straight line graph.
Quadratic Functions: Highest power of $ x $ is 2, forming a parabolic graph.
Cubic Functions: Highest power of $ x $ is 3, forming an S-shaped graph.
Polynomial Functions: These include higher powers of $ x $ and can have more complex shapes.

By identifying the highest power of $ x $ in a function, we can classify it and predict its general shape on a graph. This helps in solving equations and modeling real-life scenarios. In the example function, $ f(x) = 4 - 5x^2 $, we observe the highest power of $ x $ is 2, leading us to classify it as a quadratic function.

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91影视

State whether the function is linear or quadratic. $$f(x)=4-5 x^{2}$$

Short Answer

Step by step solution

Understand the Function

Identify the Highest Power of x

Determine the Type of Function

Classify the Function

Key Concepts

linear functions

power of x

function classification

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