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Magazine Chapter 11: Problem 61
In the following exercises, graph by plotting points. $$ y=-\frac{3}{2} x+2 $$
Short Answer
Expert verified
Plot points (0, 2) and (2, -1), then draw a line through them.
Step by step solution
01
Understand the Equation
The given equation is in the slope-intercept form of a line, which is \( y = -\frac{3}{2} x + 2 \). Here, the slope (m) is \(-\frac{3}{2}\) and the y-intercept (b) is 2. We'll use these to find points for the graph.
02
Find the Y-Intercept
The y-intercept is where the line crosses the y-axis, which occurs when \( x = 0 \). By substituting \( x = 0 \) into the equation, we get \( y = 2 \). So, the point (0, 2) is on the graph.
03
Find Another Point
Choose another value for \( x \) to find a second point. For instance, let鈥檚 choose \( x = 2 \). Substituting \( x = 2 \) into the equation, we get \( y = -\frac{3}{2} (2) + 2 = -3 + 2 = -1 \). So, the point (2, -1) is also on the graph.
04
Plot the Points
On a coordinate plane, plot the points (0, 2) and (2, -1).
05
Draw the Line
Using a ruler, draw a straight line through the points (0, 2) and (2, -1). This line represents the equation \( y = -\frac{3}{2} x + 2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope-intercept form
The slope-intercept form of a linear equation is a simple and standard way to represent straight lines. The general format of this form is \( y = mx + b \). Here, 'm' represents the slope of the line, and 'b' is the y-intercept, which is where the line crosses the y-axis.
For example, if we have an equation like \( y = -\frac{3}{2} x + 2 \), we can immediately see that:
For example, if we have an equation like \( y = -\frac{3}{2} x + 2 \), we can immediately see that:
- The slope (m) is \( -\frac{3}{2} \).
- The y-intercept (b) is 2.
plotting points
When graphing linear equations, a crucial step is plotting points. Plotting points involves finding specific values of x and y that satisfy the equation and marking them on the coordinate plane.
To plot points:
To plot points:
- Pick a value for x.
- Substitute this x value into the equation to solve for y.
- Record the resulting coordinates (x, y).
coordinate plane
A coordinate plane is a two-dimensional grid used for plotting points, lines, and curves. It is divided by a horizontal x-axis and a vertical y-axis.
Key features of the coordinate plane:
Key features of the coordinate plane:
- The point where the x-axis and y-axis intersect is called the origin, denoted as (0, 0).
- The plane is divided into four quadrants, each representing a combination of positive and negative values of x and y.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. It is found by setting x to 0 in the equation of the line.
Identifying the y-intercept:
Identifying the y-intercept:
- In the equation \( y = -\frac{3}{2} x + 2 \), set \( x = 0 \).
- Solving this, we get \( y = 2 \).
- The y-intercept is thus (0, 2).
slope
The slope of a line measures its steepness and direction. It is defined as the ratio of the change in y to the change in x between two points on the line.
The slope formula is: \( m = \frac{\Delta y}{\Delta x} \)
In practical terms:
The slope formula is: \( m = \frac{\Delta y}{\Delta x} \)
In practical terms:
- If the slope (m) is positive, the line rises from left to right.
- If the slope is negative, the line falls from left to right.
- If the slope is 0, the line is horizontal.
