Problem 44
Question
Give the slope and \(y\) -intercept of each line Whose equation is given. Then graph the r function. \(f(x)-\frac{3}{4} x-3\)
Step-by-Step Solution
Verified Answer
The slope of the function \(f(x) = -\frac{3}{4}x - 3\) is \(-\frac{3}{4}\) and the y-intercept is \(-3\).
1Step 1: Identify the slope
The slope of the line is the coefficient of \(x\) in the linear equation. Here, the slope \(m\) is \(-\frac{3}{4}\).
2Step 2: Identify the y-intercept
The y-intercept is the constant term in the linear equation. In this case, the y-intercept \(c\) is \(-3\). It is the point on the y-axis where the line intersects it.
3Step 3: Graph the function
Begin by marking the y-intercept (\(0, -3\)) on the graph. From there, use the slope to determine the rise and run. Since the slope is -3/4, for every 4 units move to the right along the x-axis, move 3 units down along the y-axis. Draw the line that passes through these points to complete the graph.
Key Concepts
Slope of a LineY-InterceptLinear Function
Slope of a Line
Understanding the slope of a line is foundational in graphing linear equations. The slope is a measure of how steep a line is and is often represented by the letter 'm'. It tells us how much the 'y' value of a point on the line changes for a one unit increase in the 'x' value. In simpler terms, it’s the 'rise over run'.
For example, consider the linear equation from the exercise, which is written as \(f(x) = -\frac{3}{4}x - 3\). The number in front of 'x', which is \( -\frac{3}{4} \), represents the slope of the line. This means for every four units you move horizontally to the right on the graph (the run), the line will move three units downwards (the rise).
If you are plotting this on a graph, start at any point on the line and measure. A negative slope indicates that the line is falling from left to right, which can also serves as a reminder that negative emotions might drop or 'fall' down, making the concept easier to remember.
For example, consider the linear equation from the exercise, which is written as \(f(x) = -\frac{3}{4}x - 3\). The number in front of 'x', which is \( -\frac{3}{4} \), represents the slope of the line. This means for every four units you move horizontally to the right on the graph (the run), the line will move three units downwards (the rise).
If you are plotting this on a graph, start at any point on the line and measure. A negative slope indicates that the line is falling from left to right, which can also serves as a reminder that negative emotions might drop or 'fall' down, making the concept easier to remember.
Y-Intercept
The y-intercept is a specific point on a graph where the line crosses the y-axis. It's significant because it is the value of 'y' when 'x' equals zero. This point is often expressed as \( (0, b) \), with 'b' being the y-intercept.
In our given exercise, the equation of the line tells us that the y-intercept is \( -3 \), which means that the line crosses the y-axis at the point \( (0, -3) \). When drawing the graph, you start plotting the line at this point. It's like meeting a friend at a known location before starting a walk together—this 'location' (the y-intercept) is where the function begins its journey on the graph.
In our given exercise, the equation of the line tells us that the y-intercept is \( -3 \), which means that the line crosses the y-axis at the point \( (0, -3) \). When drawing the graph, you start plotting the line at this point. It's like meeting a friend at a known location before starting a walk together—this 'location' (the y-intercept) is where the function begins its journey on the graph.
Linear Function
A linear function is a type of function that creates a straight line when graphed. These functions are generally written in the form \( y = mx + b \), where 'm' is the slope, and 'b' is the y-intercept. The 'x' and 'y' correspond to the coordinates of any point on the line.
The equation from the exercise, \( f(x) = -\frac{3}{4}x - 3 \), is an example of a linear function. It is a rule that for every 'x' value input, the output 'y' will be calculated based on the slope and y-intercept. Graphing it involves plotting the y-intercept and then using the slope to find other points on the line to form that distinctive straight path across the graph.
To remember this, think of a linear function as a path on a treasure map: starting from the ‘treasure spot’ which is the y-intercept, you follow a straight direction (determined by the slope) to continue the journey across the graph.
The equation from the exercise, \( f(x) = -\frac{3}{4}x - 3 \), is an example of a linear function. It is a rule that for every 'x' value input, the output 'y' will be calculated based on the slope and y-intercept. Graphing it involves plotting the y-intercept and then using the slope to find other points on the line to form that distinctive straight path across the graph.
To remember this, think of a linear function as a path on a treasure map: starting from the ‘treasure spot’ which is the y-intercept, you follow a straight direction (determined by the slope) to continue the journey across the graph.
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