In Exercises 39–48, give the slope and y-intercept of each line whose equation is given. Then graph the linear function. f(x) = -2x+1

Verified step by step guidance

1. Identify the equation of the line in the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
2. Compare the given function \( f(x) = -2x + 1 \) with the slope-intercept form \( y = mx + b \).
3. Determine the slope \( m \) by identifying the coefficient of \( x \), which is \(-2\).
4. Determine the y-intercept \( b \) by identifying the constant term, which is \(1\).
5. To graph the linear function, start by plotting the y-intercept \((0, 1)\) on the graph, then use the slope \(-2\) to find another point by moving down 2 units and right 1 unit from the y-intercept.

Key Concepts

Slope

The slope of a line measures its steepness and direction, represented as 'm' in the slope-intercept form of a linear equation, y = mx + b. In the equation f(x) = -2x + 1, the slope is -2, indicating that for every unit increase in x, the value of f(x) decreases by 2 units. A negative slope signifies that the line descends from left to right.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis, represented as 'b' in the slope-intercept form. In the equation f(x) = -2x + 1, the y-intercept is 1, meaning the line intersects the y-axis at the point (0, 1). This value is crucial for graphing the line, as it provides a starting point.

Graphing Linear Functions

Graphing a linear function involves plotting points that satisfy the equation and connecting them to form a straight line. For f(x) = -2x + 1, you can start by plotting the y-intercept (0, 1) and using the slope to find another point, such as moving down 2 units and right 1 unit to (1, -1). This process illustrates the relationship between x and f(x) visually.