Problem 47
Question
Give the slope and \(y\)-intercept of each line whose equation is given. Then graph the linear function. $$g(x)=-\frac{1}{2} x$$
Step-by-Step Solution
Verified Answer
The slope of the line is -\(\frac{1}{2}\) and the \(y\)-intercept is 0. To graph the function, start by marking the \(y\)-intercept at 0 and then use the slope to plot other points and draw a straight line.
1Step 1: Identify the slope
Comparing the equation \(g(x)=-\frac{1}{2} x\) with \(y = mx + c\), it can be seen that \(m = -\frac{1}{2}\). Therefore, the slope of the line is -\frac{1}{2}.
2Step 2: Identify the y-intercept
Again comparing the equation with \(y = mx + c\), There is no constant term which implies \(c = 0\). Therefore, the y-intercept of the line is at 0.
3Step 3: Plot the graph
Start by marking the \(y\)-intercept which is at point (0,0). As the slope is -\frac{1}{2}, start at the \(y\)-intercept and move one unit down and two units to the right, because the slope \( -\frac{1}{2} \) means a rise of -1 and a run of 2. Continue plotting several points in this manner and draw a line through these points.
Key Concepts
Understanding Slope and Y-InterceptPlotting Linear Equations Step by StepBreaking Down the Slope-Intercept Form
Understanding Slope and Y-Intercept
The slope and y-intercept of a line are two critical components that define its position and angle on a coordinate plane. Let's make sense of these terms using a real example from the exercise provided.
The slope of a line measures its steepness and direction. When we look at the equation provided in the exercise, \(g(x)=-\frac{1}{2}x\), we immediately notice that there is a negative sign, which indicates the line slopes downward. The number \(-\frac{1}{2}\) represents the amount of vertical change (rise) for every unit of horizontal change (run) along the line. In simpler terms, for each step to the right (positive direction along the x-axis), the line goes half a step down (negative direction along the y-axis).
The y-intercept, on the other hand, tells us where the line crosses the y-axis. In this exercise, the function is missing a constant term (often represented as \(c\) in \(y = mx + c\)). This absence means that the y-intercept is at \((0,0)\) or the origin. This point is where the plotting of our line will begin. To summarize, understanding how to identify and interpret the slope and y-intercept lays the foundation for accurately graphing linear functions.
The slope of a line measures its steepness and direction. When we look at the equation provided in the exercise, \(g(x)=-\frac{1}{2}x\), we immediately notice that there is a negative sign, which indicates the line slopes downward. The number \(-\frac{1}{2}\) represents the amount of vertical change (rise) for every unit of horizontal change (run) along the line. In simpler terms, for each step to the right (positive direction along the x-axis), the line goes half a step down (negative direction along the y-axis).
The y-intercept, on the other hand, tells us where the line crosses the y-axis. In this exercise, the function is missing a constant term (often represented as \(c\) in \(y = mx + c\)). This absence means that the y-intercept is at \((0,0)\) or the origin. This point is where the plotting of our line will begin. To summarize, understanding how to identify and interpret the slope and y-intercept lays the foundation for accurately graphing linear functions.
Plotting Linear Equations Step by Step
Graphing a linear equation involves creating a visual representation of the equation on a graph. First, we start with the foundation, the y-intercept, which our exercise identifies as \((0,0)\). Since we're beginning at the origin, you'd place a point right where the x and y-axis cross.
Next, you move according to the slope. Our exercise has a slope of \(-\frac{1}{2}\), so from the origin, you would move 1 unit down (because of the negative sign) and 2 units to the right to find a new point. Placing several points using the same method and then drawing a straight line through these points will graph the linear function associated with our equation. Remember, each point reflects a combination of an x and y value that satisfies the equation of the line.
Next, you move according to the slope. Our exercise has a slope of \(-\frac{1}{2}\), so from the origin, you would move 1 unit down (because of the negative sign) and 2 units to the right to find a new point. Placing several points using the same method and then drawing a straight line through these points will graph the linear function associated with our equation. Remember, each point reflects a combination of an x and y value that satisfies the equation of the line.
Plotting Tips:
- Use a ruler to ensure your line is straight and accurate.
- If a fraction is involved in the slope, break down the movement into manageable steps (like the down 1, right 2 motion we described).
- Plot more than two points before drawing your line to be certain of its accuracy.
Breaking Down the Slope-Intercept Form
The slope-intercept form is a fundamental algebraic structure used for writing linear equations. It's given by the formula \(y = mx + c\), where \(m\) signifies the slope and \(c\) denotes the y-intercept of the line. This format is highly effective for graphing because it directly provides you with the two critical pieces needed to plot a line on a graph.
In the context of our exercise, \(g(x)=-\frac{1}{2}x\) lacks a \(c\) value, which typically accompanies \(x\). Thus, it implies a \(c\) of 0. This function is already in slope-intercept form, with \(m = -\frac{1}{2}\) and \(c = 0\). The negative slope indicates a descending line, and because the y-intercept is 0, the line crosses the origin.
In the context of our exercise, \(g(x)=-\frac{1}{2}x\) lacks a \(c\) value, which typically accompanies \(x\). Thus, it implies a \(c\) of 0. This function is already in slope-intercept form, with \(m = -\frac{1}{2}\) and \(c = 0\). The negative slope indicates a descending line, and because the y-intercept is 0, the line crosses the origin.
Importance of Slope-Intercept Form:
- Provides a quick way to identify the slope and y-intercept, which are pivotal for graphing.
- Easy to use for predicting other points on the line.
- Makes comparing and contrasting different lines straightforward due to its standard format.
Other exercises in this chapter
Problem 47
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 47
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 48
Solve: \(\quad \sqrt{x+7}+5=x\).
View solution Problem 48
Write each English sentence as an equation in two variables. Then graph the equation. The \(y\) -value is the difference between four and twice the \(x\) -value
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