Problem 48
Question
Give the slope and \(y\) -intercept of each line Whose equation is given. Then graph the r function. \(g(x)--\frac{1}{3} x\)
Step-by-Step Solution
Verified Answer
The slope of the function \(g(x) = -\frac{1}{3}x\) is \(-\frac{1}{3}\) and the y-intercept is \(0\). The graph of the function is a straight line starting at the origin and going downwards to the right.
1Step 1: Identify the slope and the y-intercept
The equation is \(g(x) = -\frac{1}{3}x\). This is in the form of \(y = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept. In this case, the slope \(m\) is \(-\frac{1}{3}\) and since there is no constant being added or subtracted, the y-intercept \(c\) is \(0\).
2Step 2: Plot the y-intercept
The y-intercept is the point where the line cuts the y-axis. As we found out, our y-intercept is \(0\). Hence, the line cuts the y-axis at the origin \((0,0)\)
3Step 3: Use the slope to find the next point and plot the line
The slope of the line is \(-\frac{1}{3}\). The slope is the ratio of 'rise over run'. This means we go down 1 unit (as it's negative) and run 3 units to the right. Starting from the y-intercept \((0,0)\) and using the slope \(-\frac{1}{3}\), we end up at the point \((3,-1)\). We continue this pattern to get more points if necessary and then draw the line passing through these points.
Key Concepts
Slope-Intercept FormSlope of a LineY-InterceptPlotting Points on a Coordinate Plane
Slope-Intercept Form
Understanding the slope-intercept form of a line is a fundamental concept in algebra that can greatly simplify the process of graphing linear equations. The general equation for a line in slope-intercept form is given by \( y = mx + b \), where \( m \) represents the slope, and \( b \) denotes the y-intercept. The beauty of this form is that it directly provides you with the two crucial pieces of information needed to sketch the graph: the steepness and direction of the line (slope) and the point at which the line crosses the y-axis (y-intercept).
When you come across an equation like \( g(x) = -\frac{1}{3}x \), you can immediately identify that the slope \( m \) is -1/3, and because there is no added constant, the y-intercept \( b \) is 0. This equation is already in the slope-intercept form, telling us how to start plotting the line on a coordinate grid.
When you come across an equation like \( g(x) = -\frac{1}{3}x \), you can immediately identify that the slope \( m \) is -1/3, and because there is no added constant, the y-intercept \( b \) is 0. This equation is already in the slope-intercept form, telling us how to start plotting the line on a coordinate grid.
Slope of a Line
The slope of a line is a measure of its steepness and direction. It is calculated as the 'rise' over the 'run' between any two points on the line. A positive slope means the line is ascending from left to right, while a negative slope indicates the line is descending.
In the practice problem, the slope is \( -\frac{1}{3} \), which means for every three units the line moves horizontally to the right (the run), it moves down one unit vertically (the rise). The negative sign tells us the line slopes downward. Remembering this, you can find additional points on the line after the y-intercept, which is particularly helpful if the line doesn't cross other integer points on the axes.
In the practice problem, the slope is \( -\frac{1}{3} \), which means for every three units the line moves horizontally to the right (the run), it moves down one unit vertically (the rise). The negative sign tells us the line slopes downward. Remembering this, you can find additional points on the line after the y-intercept, which is particularly helpful if the line doesn't cross other integer points on the axes.
Y-Intercept
The y-intercept is where a line crosses the y-axis. This is the point at which the value of x is zero. So, in the equation \( y = mx + b \), the y-intercept is the value of \( b \). When graphing, you always start by plotting the y-intercept, which serves as a starting point for your line.
In the given example, the y-intercept is identified as \( 0 \), meaning the line crosses the y-axis at the origin point \( (0,0) \). This point is crucial because it helps anchor your line on the graph and gives you a reference for applying the slope to plot subsequent points.
In the given example, the y-intercept is identified as \( 0 \), meaning the line crosses the y-axis at the origin point \( (0,0) \). This point is crucial because it helps anchor your line on the graph and gives you a reference for applying the slope to plot subsequent points.
Plotting Points on a Coordinate Plane
Plotting points correctly on a coordinate plane is a basic skill that is necessary for graphing any function. The coordinate plane consists of two perpendicular number lines that intersect at the origin: the horizontal x-axis and the vertical y-axis. Each point is defined by a pair of numbers called coordinates, which show location on the plane.
For instance, starting from the origin (0,0), move along the axes, either horizontally or vertically, to reach the new point defined by the slope. In our exercise, from the point \( (0,0) \), you would move three units to the right and one unit down to get to the point \( (3,-1) \). By plotting this point and joining it with the y-intercept, you create a visual representation of the line on the graph. Practice plotting different lines using different slopes and y-intercepts to gain confidence in your graphing skills!
For instance, starting from the origin (0,0), move along the axes, either horizontally or vertically, to reach the new point defined by the slope. In our exercise, from the point \( (0,0) \), you would move three units to the right and one unit down to get to the point \( (3,-1) \). By plotting this point and joining it with the y-intercept, you create a visual representation of the line on the graph. Practice plotting different lines using different slopes and y-intercepts to gain confidence in your graphing skills!
Other exercises in this chapter
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