Understanding Linear Functions and Graphing

In this video, I'm going to go over some basic questions relating to linear functions. So let's start with this problem. What is the slope of the line passing through the points 2, negative 3 and 4, 5? So to calculate the slope between two points, you need to use this formula. It's y2 minus y1 divided by x2 minus x1. The first number is x, the second one is y. So let's call 2, negative 3, x1, y1. and 4,5x2y2. So y2 in this example is 5, and y1 is negative 3. x2 is 4, x1 is 2. Now 5 minus negative 3 is the same as 5 plus 3, and 5 plus 3 is 8. 4 minus 2 is 2, and 8 divided by 2 is 4. So that's the slope that passes through these two points. Number two, what is the slope and y-intercept of the line y equals 2x minus 3? Now, you need to know that this equation is in slope-intercept form, which is y equals mx plus b. So, M represents the slope, and B is the y-intercept. So, we can see that the slope is equal to 2, and the y-intercept is negative 3. And so that's it for this question. Now the y-intercept, you can write it as an ordered pair if you want. You could say it's 0, negative 3, or simply the y-intercept is equal to negative 3. You can write it both ways. But that's it for this problem. Number three, graph the equations x equals 2 and y equals 3. So let's start with x equals 2. How can we graph this particular equation? Now, whenever x is equal to a number, the type of graph you're going to get, it's going to be a line, but specifically a vertical line at x equals 2. Now for the other one, y equals 3, the graph is simply a horizontal line where y is 3. And that's it. That's all you need to do in order to graph these two equations. Number four, graph the equation using the slope-intercept method. So we have the equation y is equal to 3x minus 2. So we can see that the slope is equal to 3, and the y-intercept is negative 2. I'm just going to write the order up here. So we have the point 0, negative 2. So here's the first point at 0, negative 2. Now the slope is 3, which represents the rise over the run. So we need to rise 3 units and go over 1 unit to the right to get the next point. So the next point is going to be at 1, 1. and then we could do it again. Rise 3 over 1. So the next point is going to be 2, 4. And then we can just connect these points with a straight line. And so that's the rough estimate of... the graph y equals 3x minus 2. Number 5. Graph the equation 2x minus 3y is equal to 6 using the x and y intercepts. Now, to find the x-intercept, replace y with 0. Negative 3 times 0 is simply 0. And so 0 is nothing, so this just disappears. So 2x is equal to 6. We divide both sides by 2, we can see that x is equal to 3. So that's the x-intercept, which means we have the point. Now to calculate the y-intercept, we need to replace x with 0 and solve for the x-intercept. for y. So 2 times 0 is nothing, and so we have negative 3y is equal to 6, and now let's divide both sides by negative 3. So then positive 6, excuse me, divided by negative 3, that's negative 2, and so that's the y-intercept. So we have the point 0, negative 2. At this point, we can make the graph. So the first point is 3, 0. That's the x-intercept. The next one is 0, negative 2. That's the y-intercept. And then just connect these two points with a straight line. And it appears I missed it. So there it is. So that's how you can graph a linear equation in standard form. There's three forms you need to be familiar with. This is known as slope-intercept form. This is called standard form. And then this equation is the point-slope form. Number six, write the equation of the line passing through the point 2, 5 with a slope of 3. So whenever you're given the point and the slope, it's best to use the point-slope formula to write the linear equation. So x1 is 2, y1 is 5, and m is 3. So it's going to be y minus 5 is equal to 3 times x minus 2. So that's the linear equation in point-slope form. Now let's convert it to slope-intercept form. So I'm going to distribute the 3. 3 times x, that's 3x. And then it's 3 times negative 2, which is negative 6. So now let's add 5 to both sides. Negative 6 plus 5 is negative 1. So this is the linear equation in slope intercept form. And that's the answer. Number 7. Write the equation of the line passing through the points negative 3 comma 1 and 2 comma negative 4. Now we can't use the point slope formula yet because I don't have the slope. So we need to calculate the slope first. So it's y2 minus y1 divided by x2 minus x1. So this is going to be x1 and that's going to be y1. and this is X 2 and Y 2 so Y 2 is negative 4 Y 1 is positive 1 X 2 is 2 X 1 is negative 3 so negative 4 minus 1 is negative 5 minus negative 3 is the same as 2 plus 3, that's 5. Negative 5 divided by 5 is negative 1. So that's the slope. It's equal to negative 1. Now we can use the point-slope formula. So I'm going to use the first point, negative 3 comma 1. So y minus y1 is equal to m times x minus x1. So y1 is 1, m is negative 1, and x1 is negative 3. So this is y minus 1 equals to negative 1 times x plus 3. These two negative signs will become positive. Now let's distribute the negative 1. So it's going to be negative x minus 3. And then let's add 1 to both sides. So y is equal to negative x, and negative 3 plus 1 is negative 3. negative 2. So this is the final answer. It's negative x minus 2. Number 8. Write the equation of the line passing through the point 3, negative 2 and parallel to the line 2x plus 5y minus 3. So what you need to know is that parallel lines have the same slope. So if the slope here is 2, then the slope of the other line will also be 2. So we already have the point passing through, or that's part of that line. We need to find the slope of this line, which will be the slope of the line of the equation that we're looking to find. So let's turn this equation and change it into its slope-intercept form. So first, I'm going to move the 3 to this side. So it's going to be positive 3 on the right side. And then I'm going to take the 2x and move it to that side. where it's going to be negative 2x. So I have 5y is equal to negative 2x plus 3. And then divide every term by 5. So y is equal to negative 2 over 5 times x plus 3 over 5. So the slope of this line is the number in front of x when it's in slope intercept form. So the slope is negative 2 over 5. So now I can use the point-slope formula to write the equation of the line. So this is going to be x1. and y1. So y1, that's negative 2. M is negative 2 over 5. X1 is 3. So this is going to be Y plus 2. And then let's distribute the negative 2 over 5 to X minus 3. Now, negative 2 over 5 times negative 3. Negative 2 times negative 3 is 6. So this is going to be 6 over 5. And now let's subtract both sides by 2. now 2 over 1 i'm going to multiply this by 5 over 5 to get common denominators so negative 2 is the same as negative 10 over 5 now 6 minus 10 is 4 this is going to be negative 4 over 5 And this is the answer. So that's the equation of the line in slope intercept form. Number 9. Write the equation of the line passing through the point negative 4, negative 3, and perpendicular to the line 3x minus 4y plus 5. So first, we need to find the slope of this equation. So let's get y by itself. I'm going to move the 3x and the 5 to the other side. So it's going to be negative 3x and negative 5 on the right side. Now, I need to divide by the square root of y. each term by negative 4 so Y is equal to 3 over 4 times X plus 5 over 4 whenever you divide two negative numbers you're going to get a positive positive result. So the slope of this line is 3 over 4. Now let's say if we have two lines, line K and line L. And let's say that these two lines are perpendicular, which means that they meet at right angles or at a 90 degree angle. Let's say that the slope of line K is 2 over 5. What do you think is the slope of line L? Perpendicular lines have a slope that is the negative reciprocal of each other. So this slope is positive, the other one will be negative. And then you need to flip the fraction. So the reciprocal of 2 over 5 is 5 over 2. And so that tells us that these two lines are perpendicular. So, the slope for this equation is 3 over 4. So, therefore, the slope of the perpendicular line that we want to find is going to be the negative reciprocal of that fraction, which is negative 4 divided by 3. Now, let's finish this problem. So, let's use the point-slope formula. So x1 is going to be negative 4, y1 is negative 3. So it's going to be y minus negative 3, and the slope is negative 4 over 3, and then x1 is negative 4. So this becomes y plus 3, and that's equal to negative 4 divided by 3 times x plus 4. So that's the answer in point-slope form. Now let's distribute the negative 4 over 3. Let's convert it to slope-intercept form. So this is going to be negative 4 over 3 times x, and then negative 4 times 4 is negative 16. Now let's subtract both sides by 3. Now we need to get common denominators, so I'm going to multiply this 3 by 3 over 3. So it becomes negative 9 over 3. Negative 16 minus 9 is negative 25. So this is the final answer. It's negative 4 over 3x minus 25 over 3.

So to calculate the slope between two points, you need to use this formula. It's y2 minus y1 divided by x2 minus x1. The first number is x, the second one is y. So let's call 2, negative 3, x1, y1.

and 4,5x2y2. So y2 in this example is 5, and y1 is negative 3. x2 is 4, x1 is 2. Now 5 minus negative 3 is the same as 5 plus 3, and 5 plus 3 is 8. 4 minus 2 is 2, and 8 divided by 2 is 4. So that's the slope that passes through these two points. Number two, what is the slope and y-intercept of the line y equals 2x minus 3? Now, you need to know that this equation is in slope-intercept form, which is y equals mx plus b.

So, M represents the slope, and B is the y-intercept. So, we can see that the slope is equal to 2, and the y-intercept is negative 3. And so that's it for this question. Now the y-intercept, you can write it as an ordered pair if you want.

You could say it's 0, negative 3, or simply the y-intercept is equal to negative 3. You can write it both ways. But that's it for this problem. Number three, graph the equations x equals 2 and y equals 3. So let's start with x equals 2. How can we graph this particular equation?

Now, whenever x is equal to a number, the type of graph you're going to get, it's going to be a line, but specifically a vertical line at x equals 2. Now for the other one, y equals 3, the graph is simply a horizontal line where y is 3. And that's it. That's all you need to do in order to graph these two equations. Number four, graph the equation using the slope-intercept method. So we have the equation y is equal to 3x minus 2. So we can see that the slope is equal to 3, and the y-intercept is negative 2. I'm just going to write the order up here.

So we have the point 0, negative 2. So here's the first point at 0, negative 2. Now the slope is 3, which represents the rise over the run. So we need to rise 3 units and go over 1 unit to the right to get the next point. So the next point is going to be at 1, 1. and then we could do it again.

Rise 3 over 1. So the next point is going to be 2, 4. And then we can just connect these points with a straight line. And so that's the rough estimate of... the graph y equals 3x minus 2. Number 5. Graph the equation 2x minus 3y is equal to 6 using the x and y intercepts.

Now, to find the x-intercept, replace y with 0. Negative 3 times 0 is simply 0. And so 0 is nothing, so this just disappears. So 2x is equal to 6. We divide both sides by 2, we can see that x is equal to 3. So that's the x-intercept, which means we have the point. Now to calculate the y-intercept, we need to replace x with 0 and solve for the x-intercept.

for y. So 2 times 0 is nothing, and so we have negative 3y is equal to 6, and now let's divide both sides by negative 3. So then positive 6, excuse me, divided by negative 3, that's negative 2, and so that's the y-intercept. So we have the point 0, negative 2. At this point, we can make the graph. So the first point is 3, 0. That's the x-intercept. The next one is 0, negative 2. That's the y-intercept.

And then just connect these two points with a straight line. And it appears I missed it. So there it is.

So that's how you can graph a linear equation in standard form. There's three forms you need to be familiar with. This is known as slope-intercept form. This is called standard form. And then this equation is the point-slope form.

Number six, write the equation of the line passing through the point 2, 5 with a slope of 3. So whenever you're given the point and the slope, it's best to use the point-slope formula to write the linear equation. So x1 is 2, y1 is 5, and m is 3. So it's going to be y minus 5 is equal to 3 times x minus 2. So that's the linear equation in point-slope form. Now let's convert it to slope-intercept form. So I'm going to distribute the 3. 3 times x, that's 3x.

And then it's 3 times negative 2, which is negative 6. So now let's add 5 to both sides. Negative 6 plus 5 is negative 1. So this is the linear equation in slope intercept form. And that's the answer. Number 7. Write the equation of the line passing through the points negative 3 comma 1 and 2 comma negative 4. Now we can't use the point slope formula yet because I don't have the slope.

So we need to calculate the slope first. So it's y2 minus y1 divided by x2 minus x1. So this is going to be x1 and that's going to be y1. and this is X 2 and Y 2 so Y 2 is negative 4 Y 1 is positive 1 X 2 is 2 X 1 is negative 3 so negative 4 minus 1 is negative 5 minus negative 3 is the same as 2 plus 3, that's 5. Negative 5 divided by 5 is negative 1. So that's the slope. It's equal to negative 1. Now we can use the point-slope formula.

So I'm going to use the first point, negative 3 comma 1. So y minus y1 is equal to m times x minus x1. So y1 is 1, m is negative 1, and x1 is negative 3. So this is y minus 1 equals to negative 1 times x plus 3. These two negative signs will become positive. Now let's distribute the negative 1. So it's going to be negative x minus 3. And then let's add 1 to both sides.

So y is equal to negative x, and negative 3 plus 1 is negative 3. negative 2. So this is the final answer. It's negative x minus 2. Number 8. Write the equation of the line passing through the point 3, negative 2 and parallel to the line 2x plus 5y minus 3. So what you need to know is that parallel lines have the same slope. So if the slope here is 2, then the slope of the other line will also be 2. So we already have the point passing through, or that's part of that line. We need to find the slope of this line, which will be the slope of the line of the equation that we're looking to find.

So let's turn this equation and change it into its slope-intercept form. So first, I'm going to move the 3 to this side. So it's going to be positive 3 on the right side. And then I'm going to take the 2x and move it to that side. where it's going to be negative 2x.

So I have 5y is equal to negative 2x plus 3. And then divide every term by 5. So y is equal to negative 2 over 5 times x plus 3 over 5. So the slope of this line is the number in front of x when it's in slope intercept form. So the slope is negative 2 over 5. So now I can use the point-slope formula to write the equation of the line. So this is going to be x1.

and y1. So y1, that's negative 2. M is negative 2 over 5. X1 is 3. So this is going to be Y plus 2. And then let's distribute the negative 2 over 5 to X minus 3. Now, negative 2 over 5 times negative 3. Negative 2 times negative 3 is 6. So this is going to be 6 over 5. And now let's subtract both sides by 2. now 2 over 1 i'm going to multiply this by 5 over 5 to get common denominators so negative 2 is the same as negative 10 over 5 now 6 minus 10 is 4 this is going to be negative 4 over 5 And this is the answer. So that's the equation of the line in slope intercept form.

Number 9. Write the equation of the line passing through the point negative 4, negative 3, and perpendicular to the line 3x minus 4y plus 5. So first, we need to find the slope of this equation. So let's get y by itself. I'm going to move the 3x and the 5 to the other side.

So it's going to be negative 3x and negative 5 on the right side. Now, I need to divide by the square root of y. each term by negative 4 so Y is equal to 3 over 4 times X plus 5 over 4 whenever you divide two negative numbers you're going to get a positive positive result.

So the slope of this line is 3 over 4. Now let's say if we have two lines, line K and line L. And let's say that these two lines are perpendicular, which means that they meet at right angles or at a 90 degree angle. Let's say that the slope of line K is 2 over 5. What do you think is the slope of line L?

Perpendicular lines have a slope that is the negative reciprocal of each other. So this slope is positive, the other one will be negative. And then you need to flip the fraction. So the reciprocal of 2 over 5 is 5 over 2. And so that tells us that these two lines are perpendicular. So, the slope for this equation is 3 over 4. So, therefore, the slope of the perpendicular line that we want to find is going to be the negative reciprocal of that fraction, which is negative 4 divided by 3. Now, let's finish this problem.

So, let's use the point-slope formula. So x1 is going to be negative 4, y1 is negative 3. So it's going to be y minus negative 3, and the slope is negative 4 over 3, and then x1 is negative 4. So this becomes y plus 3, and that's equal to negative 4 divided by 3 times x plus 4. So that's the answer in point-slope form. Now let's distribute the negative 4 over 3. Let's convert it to slope-intercept form. So this is going to be negative 4 over 3 times x, and then negative 4 times 4 is negative 16. Now let's subtract both sides by 3. Now we need to get common denominators, so I'm going to multiply this 3 by 3 over 3. So it becomes negative 9 over 3. Negative 16 minus 9 is negative 25. So this is the final answer. It's negative 4 over 3x minus 25 over 3.

Transcript for:Understanding Linear Functions and Graphing

Transcript for:
Understanding Linear Functions and Graphing