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Plot each of the following lines on the same set of axes. (a) $y=2 x$(b) $y-4=2 x(\text { c) } y+4=2 x$ (d) How are these lines related?

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

Missouri State University

McMaster University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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for this problem. We have three equations that we want to graph. And just to make this easier to track, I have written the three equations in three different colors. When I go to graft, um, I will graph them in the same color that I've written equations in. So that first equation that's in red I'm gonna graph his red green to green, blue to blue. Okay, so let's let's start by graphing these and then we're going to see how the lines are related to each other. So the first thing we're going to do for each of these equations were going to put it into slope intercept form. Now they're almost there, but we're going to tweak it just a little bit. Remember, Slope intercept form is a great form from which to graph a line because it gives us both slope that coefficient of the X term plus the y intercept, which is that constant term. Okay, so let's start with our red equation. This one is already in slope intercept form. It has no constant term. There is no plus B, which means that that zero So we know that that red line is gonna go right through the origin as our Y intercept. And it has a slope of to. So that means I'm going up to over one. A positive slope means my line will be going upward. So as I connect these dots, there is my line. Okay? Onto our green line. Well, we're almost in slope intercept form. I just need to actually solve it for why? So I'm gonna add for to both sides. That's two X plus four. Well, that plus four is my y intercept. +1234 And from there, I have the same slope to Yeah, it's a positive slope. So I'm gonna be going upward. And I could just connect these points. Hey, last Let's look at the blue one again. I'm almost in slope intercept form. I'm just going to subtract four from both sides. Like it y equals two X minus four. That gives me a 123 Why? Intercept of negative four, But again, same slope. So again, I'm gonna be rising to for every unit that I go across so to over one positive slope. Okay, so we've graft. Um, now, how are they related to each other? Well, it's easy to see. And if you're using a ruler and graph paper being very careful, the more careful you are, the easier it is to see these lines are indeed parallel. They all have the same slope. I have a two X in each case, so I have a slope of to. But more than that, I can look at the equations and see how these lines are related and what I'm gonna look at here, Um, the two x doesn't change. But what I wanna look at is how the why changes. So we take our red line is kind of like our baseline. Nothing's changing. I just have a plain old why my green equation has a Y minus four. And if you look at the green Line, I have gone up four units that that why intercept has gone up for units. So why minus four is I've gone up. It's a trend. Is a vertical change up for units. The blue line. I have a Y plus four hoops. Let's do blue for that one. Why plus four. Well, if you examine that and see how that relates to our red baseline, I have gone down four units in general. What I'm doing to that? Why variable? I'm subtracting my change. So why minus four? I have a change of four. Why? Plus four? That's like why? Minus negative force. I've got a change of negative for So a subtraction means I'm moving up. The vertical change upward. Why? Plus a constant means. I have a vertical change downward.

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