Question

2 - 32. when yoshi graphed the lines y = 2x + 3 and y = 2x - 2, she got the graph shown below.
a. one of the lines above matches the equation y = 2x+3, and the other matches y = 2x - 2. which line matches which equation?
b. yoshi wants to add the line y = 2x + 11 to her graph. predict where it would lie and sketch a graph to show its position. justify your prediction.
c. where would the line y=-2x + 11 lie? again, justify your prediction and add the graph of this line to your graph from part (b).

Explanation:

Step1: Analyze the slope - intercept form

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. For $y = 2x+3$, slope $m = 2$ and y - intercept $b = 3$. For $y = 2x - 2$, slope $m = 2$ and y - intercept $b=-2$. For $y = 2x + 11$, slope $m = 2$ and y - intercept $b = 11$. Lines with the same slope are parallel.

Step2: Determine the position of lines

Since all lines $y = 2x+3$, $y = 2x - 2$ and $y = 2x + 11$ have a slope of 2, they are parallel to each other. The line $y = 2x+3$ has a y - intercept of 3, $y = 2x - 2$ has a y - intercept of - 2 and $y = 2x + 11$ has a y - intercept of 11. A larger positive y - intercept means the line is higher on the y - axis.

Step3: Answer part a

The line $y = 2x+3$ will be above the line $y = 2x - 2$ because its y - intercept (3) is greater than the y - intercept of $y = 2x - 2$ (-2). The line $y = 2x+11$ will be above both $y = 2x+3$ and $y = 2x - 2$ since its y - intercept (11) is the largest among them.

Step4: Answer part b

The line $y = 2x+11$ has a y - intercept of 11. It will be above the line $y = 2x+3$ (y - intercept 3) and $y = 2x - 2$ (y - intercept - 2). To graph $y = 2x+11$, start at the point (0,11) on the y - axis (the y - intercept). Then, since the slope $m = 2=\frac{2}{1}$, from the point (0,11), move 1 unit to the right and 2 units up to find another point on the line, and draw the line through these points.

Step5: Answer part c

The line $y=-2x + 11$ has a slope of - 2. The lines $y = 2x+3$, $y = 2x - 2$ and $y = 2x+11$ have a slope of 2. A line with a negative slope will have a different direction than lines with a positive slope. The line $y=-2x + 11$ will intersect the lines $y = 2x+3$, $y = 2x - 2$ and $y = 2x+11$ because it has a different slope. To graph $y=-2x + 11$, start at the y - intercept (0,11). Since the slope $m=-2=\frac{-2}{1}$, from the point (0,11), move 1 unit to the right and 2 units down to find another point on the line, and draw the line through these points.

Answer:

a. The line $y = 2x+3$ is above $y = 2x - 2$ and $y = 2x+11$ is above both $y = 2x+3$ and $y = 2x - 2$ because of the y - intercept values.
b. The line $y = 2x+11$ will be above $y = 2x+3$ and $y = 2x - 2$. Start at (0,11) (y - intercept) and use slope 2 (move 1 right, 2 up) to graph it.
c. The line $y=-2x + 11$ will intersect the other lines as it has a negative slope (-2) while the others have positive slope (2). Start at (0,11) (y - intercept) and use slope - 2 (move 1 right, 2 down) to graph it.