Problem 41
Question
Give the slope and \(y\)-intercept of each line whose equation is given. Then graph the linear function. $$f(x)=-2 x+1$$
Step-by-Step Solution
Verified Answer
The slope of the line is -2 and the y-intercept is 1.
1Step 1: Identify the Slope
The slope \( m \) of the line is the coefficient of \( x \) in the equation of the line. Looking at the equation \( f(x) = -2x + 1 \), the coefficient of \( x \) is -2. Hence the slope of the line is -2.
2Step 2: Identify the y-Intercept
The y-intercept \( c \) is the constant term in the equation of the line. From the equation \( f(x) = -2x + 1 \), the constant term is 1. Hence the y-intercept of the line is 1.
3Step 3: Plot the Line
Start by plotting the y-intercept, a point at \( y = 1 \) on the y-axis. Then, from this point, move down 2 units (negative slope means move down) and to the right 1 unit (since the slope is essentially 'rise over run' which equals -2/1) and plot another point. Connect those two points with a straight line to make the graph of the linear function.
Key Concepts
Slope of a LineY-interceptGraphing Linear Functions
Slope of a Line
The slope of a line is a measure of its steepness. It tells us how much the line goes up or down for a given horizontal movement. For the linear equation of the form \( y = mx + c \), the slope \( m \) is the number that appears before the \( x \) term. It indicates how the vertical distance ("rise") changes with the horizontal distance ("run"). In simpler terms, it tells us the change in \( y \) for each change in \( x \).
In the equation \( f(x) = -2x + 1 \), the slope is \(-2\). This means that for every unit increase in \( x \), \( y \) decreases by two units. The negative sign indicates that the line is downward sloping.
- Positive slope: The line rises from left to right.
- Negative slope: The line falls from left to right.
- Zero slope: The line is horizontal.
- Undefined slope: The line is vertical.
In the equation \( f(x) = -2x + 1 \), the slope is \(-2\). This means that for every unit increase in \( x \), \( y \) decreases by two units. The negative sign indicates that the line is downward sloping.
Y-intercept
The y-intercept is where the line crosses the y-axis. The value at this point provides a starting point for graphing a line on the coordinate plane. It is represented by the constant \( c \) in the equation \( y = mx + c \). To find the y-intercept, set \( x = 0 \) in the equation, and the value of the function is the y-intercept.
For the equation \( f(x) = -2x + 1 \), the y-intercept is \( 1 \). This implies that when \( x = 0 \), \( y = 1 \), meaning the line will pass through the point \( (0,1) \) on the y-axis. This is the first point that is typically plotted when graphing a line because it gives an easy visual starting point.
For the equation \( f(x) = -2x + 1 \), the y-intercept is \( 1 \). This implies that when \( x = 0 \), \( y = 1 \), meaning the line will pass through the point \( (0,1) \) on the y-axis. This is the first point that is typically plotted when graphing a line because it gives an easy visual starting point.
Graphing Linear Functions
Graphing linear functions involves displaying the equation \( y = mx + c \) on a coordinate plane. It's a straightforward process that only requires a few points to define the whole line. Start by plotting the y-intercept and then use the slope to find another point to draw the line.
Graphing provides a visual representation of the linear function, making it easier to understand how changes in \( x \) affect \( y \).
Steps for Graphing
- Begin with plotting the y-intercept on the y-axis. For \( f(x) = -2x + 1 \), it is the point \( (0, 1) \).
- Use the slope to determine the direction and steepness of the line. From \( (0, 1) \), move down 2 units and to the right 1 unit (since the slope is \(-2/1\), which is rise over run).
- Plot this second point.
- Draw a straight line through these points, extending in both directions. This line represents the graph of the function \( f(x) = -2x + 1 \).
Graphing provides a visual representation of the linear function, making it easier to understand how changes in \( x \) affect \( y \).
Other exercises in this chapter
Problem 41
Determine whether each function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the \(y\) -axis, the origin,
View solution Problem 41
Graph the given functions, \(f\) and \(g,\) in the same rectangular coordinate system. Select integers for \(x,\) starting with -2 and ending with \(2 .\) Once
View solution Problem 42
Find \(f+g, f-g,\) fg, and \(\frac{f}{x}\). Determine the domain for each function. $$f(x)=6-\frac{1}{x}, g(x)=\frac{1}{x}$$
View solution Problem 42
a. Find an equation for \(f^{-1}(x)\) b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and t
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