Question

3 sketch the graph of the line: $y = -\frac{5}{2}x + 3$

Explanation:

Step1: Identify the slope and y-intercept

The equation of the line is in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the line \(y=-\frac{5}{2}x + 3\), the slope \(m =-\frac{5}{2}\) and the y - intercept \(b = 3\).

Step2: Plot the y - intercept

The y - intercept is the point where the line crosses the y - axis. When \(x = 0\), \(y=3\). So we plot the point \((0,3)\) on the coordinate plane.

Step3: Use the slope to find another point

The slope \(m=-\frac{5}{2}\) can be thought of as \(\frac{\text{rise}}{\text{run}}\). A slope of \(-\frac{5}{2}\) means that for every 2 units we move to the right (run = 2) along the x - axis, we move down 5 units (rise=- 5) along the y - axis. Starting from the point \((0,3)\), if we move 2 units to the right (to \(x = 0+2=2\)) and 5 units down (to \(y=3 - 5=-2\)), we get the point \((2,-2)\). We can also move in the opposite direction: for every 2 units we move to the left (run=-2), we move up 5 units (rise = 5). Starting from \((0,3)\), moving 2 units to the left (to \(x=0 - 2=-2\)) and 5 units up (to \(y = 3+5 = 8\)) gives the point \((-2,8)\).

Step4: Draw the line

Using a straightedge, draw a line that passes through the two (or more) points we have plotted (e.g., \((0,3)\) and \((2,-2)\) or \((0,3)\) and \((-2,8)\)).

(Note: Since this is a sketching problem, the key is to identify the y - intercept and use the slope to find additional points to draw the line. The actual graph will be a straight line with a negative slope, crossing the y - axis at (0,3) and passing through other points determined by the slope.)

Answer:

To sketch the graph of \(y =-\frac{5}{2}x+3\):

1. Plot the y - intercept \((0,3)\).
2. Use the slope \(-\frac{5}{2}\) to find another point (e.g., from \((0,3)\), move 2 units right and 5 units down to get \((2, - 2)\) or 2 units left and 5 units up to get \((-2,8)\)).
3. Draw a straight line through the plotted points.