Linear Graph Equation And Translation Explained

Hey guys! Let's dive into a cool problem involving linear graphs, equations, and translations. We've got a graph of a line, which we'll call 'g', and we need to figure out a couple of things. First, we need to find the equation of the line 'g'. Then, we're going to slide this line around a bit using a translation vector and find the equation of the new line. Sounds like fun, right? Let's break it down step by step.

A. Determining the Equation of Line g

Okay, so the first thing we need to do is determine the equation of the line g. To do this, we'll need to use some fundamental concepts of linear equations. Remember, a linear equation is generally represented in the form y = mx + c, where m is the slope (or gradient) of the line and c is the y-intercept (the point where the line crosses the y-axis). Let's see how we can find these values from the given graph.

Identifying Key Points

To find the equation, we need at least two points on the line. From the graph provided, we can clearly identify a couple of points where the line intersects with the grid. These points will give us the x and y coordinates we need to calculate the slope. Let's say we've identified two points: Point A (x1, y1) and Point B (x2, y2). It's super important to accurately read these coordinates from the graph to avoid any calculation errors later on.

For example, let’s assume Point A is (1, 2) and Point B is (3, 4). These are just example coordinates, guys; you'll need to read the actual coordinates from your graph. Once you've got these points, we can move on to calculating the slope.

Calculating the Slope (m)

The slope, often denoted by m, tells us how steep the line is. It's calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Using our example points A(1, 2) and B(3, 4), we can plug in the values:

m = (4 - 2) / (3 - 1) = 2 / 2 = 1

So, in this example, the slope m is 1. This means that for every one unit we move to the right along the x-axis, the line goes up by one unit along the y-axis. Cool, right? But remember, use the actual coordinates from your graph to get the correct slope for line g.

Finding the Y-Intercept (c)

Now that we've got the slope, we need to find the y-intercept, which is the value of y where the line crosses the y-axis. This is the c in our equation y = mx + c. There are a couple of ways to find this. One way is to look at the graph and see where the line intersects the y-axis directly. If it's a clear intersection point, you can just read off the y-value.

If it's not super clear from the graph, or if you want to be more precise, we can use the slope we just calculated and one of the points on the line. We plug the slope (m), the x-coordinate, and the y-coordinate of one of our points into the equation y = mx + c and solve for c. Let’s use Point A(1, 2) and our slope m = 1 from the previous example:

2 = (1)(1) + c 2 = 1 + c c = 2 - 1 = 1

So, in this example, the y-intercept c is 1. This means the line crosses the y-axis at the point (0, 1). Again, make sure you use your actual slope and points from the graph!

Writing the Equation

We've got everything we need now! We've calculated the slope (m) and the y-intercept (c). We just plug these values into the slope-intercept form of the equation, y = mx + c. Using our example values, where m = 1 and c = 1, the equation of the line would be:

y = 1x + 1 or simply y = x + 1

Remember, this is just an example using hypothetical points. You need to do the calculations with the actual points from the graph provided in your problem. Find the coordinates, calculate the slope, find the y-intercept, and then write the equation in the form y = mx + c. Once you've done that, you've nailed the first part of the problem!

B. Determining the Equation of the Translated Line

Alright, awesome job on finding the equation of the original line! Now, let's move on to the second part: determining the equation of the translated line. Translation, in this context, means moving the line without changing its orientation. We're essentially sliding the line along the coordinate plane. In this case, we're translating line g by the vector (1, 0).

Understanding Translation Vectors

So, what does the translation vector (1, 0) mean? Translation vectors tell us how far to move a point (or in this case, a line) horizontally and vertically. The first number in the vector represents the horizontal shift, and the second number represents the vertical shift. A positive number means moving to the right (horizontal) or upwards (vertical), while a negative number means moving to the left (horizontal) or downwards (vertical).

In our case, the vector (1, 0) means we're moving the line 1 unit to the right and 0 units vertically. So, we're just sliding the line horizontally along the x-axis.

How Translation Affects the Equation

Here's the cool part: translating a line horizontally only affects the y-intercept (c) in the equation y = mx + c. The slope (m) remains the same because the line's steepness isn't changing, just its position. This makes our job a whole lot easier!

To find the new equation, we need to figure out how the y-intercept changes when we shift the line. Think about it this way: if we move the entire line 1 unit to the right, the point where the line crosses the y-axis will also shift. Let’s look at how to calculate this shift.

Finding the New Y-Intercept

Let's say the original equation of line g is y = mx + c. We know the slope m stays the same after the translation. We just need to find the new y-intercept, which we'll call c'. To do this, we can consider what happens to a point on the line when it's translated. Remember, every point on the line moves 1 unit to the right.

Let's take a general point (x, y) on the original line. After the translation, this point will move to (x + 1, y). This new point must satisfy the equation of the translated line, which will have the form y = mx + c'. So, we can substitute x + 1 for x in the original equation and set it equal to the y-value:

y = m(x + 1) + c'

We also know that the original point (x, y) satisfies the original equation y = mx + c. So, we can substitute mx + c for y in the above equation:

mx + c = m(x + 1) + c'

Now, let's expand and simplify:

mx + c = mx + m + c'

Notice that the mx terms cancel out:

c = m + c'

Finally, we can solve for c':

c' = c - m

This is the key formula! The new y-intercept c' is equal to the original y-intercept c minus the slope m. This makes sense because if the line is shifted to the right, the y-intercept will effectively shift downwards (depending on the slope).

Writing the Equation of the Translated Line

Now we have everything we need. We know the original slope m, the original y-intercept c, and we've calculated the new y-intercept c'. The equation of the translated line is simply:

y = mx + c'

Let's go back to our example where the original equation was y = x + 1 (so m = 1 and c = 1). We can calculate the new y-intercept c':

c' = c - m = 1 - 1 = 0

So, the equation of the translated line in our example would be:

y = 1x + 0 or simply y = x

Remember, this is based on our example. Use the actual slope and y-intercept you calculated for the original line g and plug them into the formula c' = c - m to find the correct new y-intercept. Then, write the equation of the translated line using the original slope and the new y-intercept.

Wrapping Up

And there you have it! We've successfully determined the equation of line g and the equation of the translated line. We used key concepts like slope, y-intercept, and translation vectors to solve the problem. Remember, the key is to break down the problem into smaller steps, identify the relevant information, and apply the correct formulas. You guys got this!