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Magazine Chapter 4: Problem 51
Graph each polynomial function. Give the domain and range. \(f(x)=-2 x+1\)
Short Answer
Expert verified
Domain: \([-\infty, \infty]\); Range: \([-\infty, \infty]\)
Step by step solution
01
Identify the type of function
The given function is a linear polynomial function of the form \(f(x) = ax + b\). In this case, the function is \(f(x) = -2x + 1\).
02
Find the Slope and Y-Intercept
The slope (\(a\)) is -2, and the y-intercept (\(b\)) is 1.This means the function crosses the y-axis at the point (0, 1).
03
Plot the Y-Intercept
On a graph, mark the point (0, 1), as this is where the line intersects the y-axis.
04
Use the Slope to Find Another Point
The slope -2 means that for every unit increase in \(x\), \(f(x)\) decreases by 2 units. Starting from (0, 1), move one unit right to \(x = 1\), and move down 2 units to \(y = -1\). This gives another point on the line: (1, -1).
05
Plot and Draw the Line
Plot the second point (1, -1) on the graph. Then, draw a straight line through the points (0, 1) and (1, -1), extending it in both directions.
06
Determine the Domain and Range
For a linear function like \(f(x) = -2x + 1\), the domain is all real numbers (\(-\infty, \infty\)). Since the function can output any real number, the range is also all real numbers (\(-\infty, \infty\)).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. A simple example of a polynomial function is a linear function like the one in our exercise, which has the form f(x) = ax + b. Here, 'a' and 'b' are constants; 'a' is the slope of the line, and 'b' is the y-intercept.
Linear functions are the simplest type of polynomial functions. They create straight lines when graphed. Other types of polynomial functions like quadratic (f(x) = ax^2 + bx + c) and cubic (f(x) = ax^3 + bx^2 + cx + d) create curves. As the degree of the polynomial increases, the shape of the graph becomes more complex.
Understanding higher-degree polynomial functions is crucial for tackling more advanced math topics. But mastering linear polynomial functions lays down a solid foundation for these future concepts.
Linear functions are the simplest type of polynomial functions. They create straight lines when graphed. Other types of polynomial functions like quadratic (f(x) = ax^2 + bx + c) and cubic (f(x) = ax^3 + bx^2 + cx + d) create curves. As the degree of the polynomial increases, the shape of the graph becomes more complex.
Understanding higher-degree polynomial functions is crucial for tackling more advanced math topics. But mastering linear polynomial functions lays down a solid foundation for these future concepts.
Slope-Intercept Form
The slope-intercept form of a linear equation is y = mx + c, where 'm' is the slope, and 'c' is the y-intercept.
The slope tells you how steep the line is. A positive slope means the line goes up as you move right, while a negative slope means it goes down. In the given exercise, the slope is -2. This means the line falls two units for every one unit it moves to the right.
The y-intercept is the point where the line crosses the y-axis. It's the value of 'y' when 'x' is 0. For the function f(x) = -2x + 1, the y-intercept is 1. We plot this point first when graphing, then use the slope to find another point.
This form allows us to quickly graph a linear equation and identify its key characteristics. Knowing the slope and y-intercept helps you understand how the line behaves and where it is positioned.
The slope tells you how steep the line is. A positive slope means the line goes up as you move right, while a negative slope means it goes down. In the given exercise, the slope is -2. This means the line falls two units for every one unit it moves to the right.
The y-intercept is the point where the line crosses the y-axis. It's the value of 'y' when 'x' is 0. For the function f(x) = -2x + 1, the y-intercept is 1. We plot this point first when graphing, then use the slope to find another point.
This form allows us to quickly graph a linear equation and identify its key characteristics. Knowing the slope and y-intercept helps you understand how the line behaves and where it is positioned.
Domain and Range
The domain of a function is the set of all possible input values (x-values). For linear functions like f(x) = -2x + 1, the domain is all real numbers, denoted as \(\text{-鈭�, 鈭瀩\). This is because you can input any real number for 'x', and the function will spit out a corresponding 'y' value.
The range is the set of all possible output values (y-values). Similar to the domain, for linear functions, the range is also all real numbers \(\text{-鈭�, 鈭瀩\). No matter what x-value you choose, the equation will give you a real number for 'y'.
To summarize:
The range is the set of all possible output values (y-values). Similar to the domain, for linear functions, the range is also all real numbers \(\text{-鈭�, 鈭瀩\). No matter what x-value you choose, the equation will give you a real number for 'y'.
To summarize:
- Domain: \(\text{-鈭�, 鈭瀩\)
- Range: \(\text{-鈭�, 鈭瀩\)
