Graphing Linear Functions: Find Intersections & Relationships

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Graphing Linear Functions: Find Intersections & Relationships

Hey guys! Today, we're diving into the exciting world of linear functions and their graphs. We're going to plot these functions and figure out how they relate to each other on the coordinate plane. So, grab your graph paper (or your favorite graphing tool) and let's get started!

1) Analyzing y = 1.4x + 2 and y = x + 2

Let's begin by understanding linear functions. In the equation y = 1.4x + 2, we have a slope of 1.4 and a y-intercept of 2. This means the line rises 1.4 units for every 1 unit increase in x, and it crosses the y-axis at the point (0, 2). Similarly, for y = x + 2, the slope is 1 (meaning a 1-unit rise for every 1-unit run), and the y-intercept is also 2.

Plotting the Lines

To plot these lines, we need at least two points for each. For y = 1.4x + 2:

  • When x = 0, y = 1.4(0) + 2 = 2. So, we have the point (0, 2).
  • When x = 1, y = 1.4(1) + 2 = 3.4. So, we have the point (1, 3.4).

For y = x + 2:

  • When x = 0, y = 0 + 2 = 2. So, we have the point (0, 2).
  • When x = 1, y = 1 + 2 = 3. So, we have the point (1, 3).

Plot these points and draw the lines. You'll notice that both lines intersect at the point (0, 2), which is their common y-intercept. The line y = 1.4x + 2 is steeper than y = x + 2 because it has a larger slope.

Relative Position

The lines intersect at (0, 2). Since their slopes are different (1.4 and 1), the lines are neither parallel nor the same line. They are intersecting lines.

2) Analyzing y = -x + 1.5 and y = 2x - 3

Now, let's look at these two linear equations. In the equation y = -x + 1.5, the slope is -1 and the y-intercept is 1.5. This tells us the line goes downwards as x increases, crossing the y-axis at (0, 1.5). For y = 2x - 3, the slope is 2 and the y-intercept is -3, indicating an upward sloping line that crosses the y-axis at (0, -3).

Plotting the Lines

Let's find two points for each line:

For y = -x + 1.5:

  • When x = 0, y = -0 + 1.5 = 1.5. So, we have the point (0, 1.5).
  • When x = 1, y = -1 + 1.5 = 0.5. So, we have the point (1, 0.5).

For y = 2x - 3:

  • When x = 0, y = 2(0) - 3 = -3. So, we have the point (0, -3).
  • When x = 1, y = 2(1) - 3 = -1. So, we have the point (1, -1).

Plot these points to draw the lines. These lines clearly intersect. To find the intersection point, we set the two equations equal to each other:

-x + 1.5 = 2x - 3

Solving for x:

3x = 4.5 x = 1.5

Solving for y:

y = -1.5 + 1.5 = 0

So, the intersection point is (1.5, 0).

Relative Position

The lines intersect at the point (1.5, 0). Because the slopes (-1 and 2) are different, the lines are neither parallel nor the same line. They are intersecting lines.

3) Analyzing y = 7 + 9x and y = -9x - 0.9

In this set, we have the linear functions y = 7 + 9x and y = -9x - 0.9. The first equation has a slope of 9 and a y-intercept of 7. The second equation has a slope of -9 and a y-intercept of -0.9. This suggests one line increases rapidly, while the other decreases rapidly.

Plotting the Lines

Let's find our points:

For y = 7 + 9x:

  • When x = 0, y = 7 + 9(0) = 7. So, we have the point (0, 7).
  • When x = -1, y = 7 + 9(-1) = -2. So, we have the point (-1, -2).

For y = -9x - 0.9:

  • When x = 0, y = -9(0) - 0.9 = -0.9. So, we have the point (0, -0.9).
  • When x = 1, y = -9(1) - 0.9 = -9.9. So, we have the point (1, -9.9).

Plot these lines. To find their intersection point, set the equations equal:

7 + 9x = -9x - 0.9

Solving for x:

18x = -7.9 x = -7.9 / 18 ≈ -0.439

Solving for y:

y = 7 + 9(-0.439) ≈ 3.049

So, the intersection point is approximately (-0.439, 3.049).

Relative Position

These lines intersect at approximately (-0.439, 3.049). Their slopes (9 and -9) are different, indicating that the lines are neither parallel nor the same line; they intersect.

4) Analyzing y = x + 2 and y = x - 14

Lastly, let's consider the functions y = x + 2 and y = x - 14. Both equations have a slope of 1. The first line has a y-intercept of 2, while the second has a y-intercept of -14. Because the slopes are the same but the y-intercepts are different, these lines are parallel.

Plotting the Lines

Let’s find some points:

For y = x + 2:

  • When x = 0, y = 0 + 2 = 2. So, we have the point (0, 2).
  • When x = 1, y = 1 + 2 = 3. So, we have the point (1, 3).

For y = x - 14:

  • When x = 0, y = 0 - 14 = -14. So, we have the point (0, -14).
  • When x = 1, y = 1 - 14 = -13. So, we have the point (1, -13).

Relative Position

Plotting these points shows that the lines are parallel. Parallel lines, by definition, never intersect. They have the same slope but different y-intercepts. Therefore, the lines y = x + 2 and y = x - 14 are parallel.

Summary

To recap, when you're graphing linear functions:

  • Identify the slope and y-intercept. The slope tells you how steep the line is and in what direction it goes, while the y-intercept tells you where the line crosses the y-axis.
  • Plot at least two points for each line. This allows you to accurately draw the line.
  • Analyze the slopes and y-intercepts to determine the relative position of the lines.
    • If the slopes are different, the lines intersect.
    • If the slopes are the same but the y-intercepts are different, the lines are parallel.
    • If both the slopes and y-intercepts are the same, the lines are identical.

Understanding linear functions and their graphs is a fundamental concept in algebra. With practice, you'll become more comfortable identifying the relationships between different lines and visualizing them on the coordinate plane. Keep exploring and happy graphing, guys!