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Graphing Linear Equations: Slope & yintercept (y= mx + b)
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 We'll learn how to graph an equation in the form y = mx + b.
 We will also learn how to find the equation of a line on a coordinate plane.
 And we'll see how this knowledge can help us ride GoKarts, visit a theme park and go skiing!

Discussion Questions

Before VideoHow do you graph a point on a coordinate plane?ANSWER

Points are given using ordered pairs (x, y) so the first number is the placement on the xaxis and the second is the placement on the yaxis. The point is where the x and ycoordinates meet.

A relation is a rule that relates two variables. We often represent relations with graphs, tables of values, and equations.

A rate is a ratio between two measurements with different units, like km/h or dollars/lb.

A constant is a value that is fixed and doesn’t change, like the height of an adult, the cost to book a taxi, or the base fee to hold a special event in a venue.

The line would be steep, because that is how you show something moving a large distance in a short time.


After VideoWhat is slope?ANSWER

The slope is the [ggfrac]rise/run[/ggfrac]. It is a rate that relates the two variables on a coordinate plane.

A linear relation with a positive slope shows that as one variable increases the other also increases. This is the opposite of a negative slope, whereas one variable increases, the other decreases.

The slope is a rate, so is multiplied by the xvariable. Since 3 is multiplied by x, 3 is the slope. The yintercept is a constant that is added in the equation. Since –10 is added, the yintercept is −10. Note that y=3x10 is the same as y=3x+(10).

The rate is the amount she sells each glass for, written as $2/glass. The constant is the cost of making the lemonade. Since the cost is something Molly needs to spend, it is negative. So, the constant is –$10.

The slope is a rate, so the slope is $2/glass. The yintercept is a constant, so the yintercept is –10. This makes the equation y=2x10, where x represents the number of glasses sold and y represents the amount of profit. I would label the horizontal axis “Number of glasses sold” and the vertical axis “Profit.” This is because Molly uses the number of glasses sold to calculate her profit. The number of glasses sold is the independent variable, and the profit is the dependant variable.



Vocabulary

Coordinate plane
DEFINE
Two number lines that intersect at 0 and are perpendicular to each other, where the horizontal axis is the xaxis, and the vertical axis is the yaxis

Origin
DEFINE
The point where the two number lines cross on a coordinate plane: (0, 0).

Ordered pair
DEFINE
Describes the position of a point on a coordinate plane. An ordered pair is written (x, y) where x is the distance from the origin on the xaxis, and y is the distance from the origin on the yaxis.

Slope
DEFINE
A rate that describes how y changes in relation to x.
slope = [ggfrac]rise/run[/ggfrac] = [ggfrac]change in y/change in x[/ggfrac] = [ggfrac]( y₂  y₁ )/( x₂  x₁ )[/ggfrac]. 
yintercept
DEFINE
The point where a line crosses the yaxis is called the yintercept. The yintercept is often represented as “b” in the slopeintercept form of a line.

Slopeintercept form of a line
DEFINE
The equation of a linear relation written in the form y=mx+b.

Table of values
DEFINE
A table with two columns, with the left column labeled “x” and the right column labeled “y.” Each row in a table of values represents an ordered pair.

Coordinate plane
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT IS A LINEAR EQUATION?A linear equation is an equation that makes a straight line when it is graphed. Linear equations can have positive or negative slope and do not have to cross through the origin.
To better understand graphing linear equations and slope…
WHAT IS A LINEAR EQUATION?. A linear equation is an equation that makes a straight line when it is graphed. Linear equations can have positive or negative slope and do not have to cross through the origin. To better understand graphing linear equations and slope…LET’S BREAK IT DOWN!
Graph a linear equation using a table of values.
A table of values is a chart that relates the x and y coordinates for a set of points in a relation. Each row in the table represents an ordered pair you can use to draw a point on the coordinate plane. When you graph a linear relationship from an equation, choose a set of xvalues and calculate their corresponding yvalues to make the basis for the line. Let’s take the equation y=2x+4 for example. If you want to graph the point with an xcoordinate of 1, substitute 1 for x and then simplify the equation to find y. So, y=2x+4 becomes y=2(1)+4, which becomes y=6. Record the value 6 in the y column in the row where x = 1. Graph the point (1, 6) on the coordinate plane. Once you are done graphing all the points, connect them all with a straight line. This is the graph of the linear relation. Try this yourself: Write a table of values for y=3x+5 with x values {−2, −1, 0, 1, 2}, and then graph the linear relation from the table.
Graph a linear equation using a table of values. A table of values is a chart that relates the x and y coordinates for a set of points in a relation. Each row in the table represents an ordered pair you can use to draw a point on the coordinate plane. When you graph a linear relationship from an equation, choose a set of xvalues and calculate their corresponding yvalues to make the basis for the line. Let’s take the equation y=2x+4 for example. If you want to graph the point with an xcoordinate of 1, substitute 1 for x and then simplify the equation to find y. So, y=2x+4 becomes y=2(1)+4, which becomes y=6. Record the value 6 in the y column in the row where x = 1. Graph the point (1, 6) on the coordinate plane. Once you are done graphing all the points, connect them all with a straight line. This is the graph of the linear relation. Try this yourself: Write a table of values for y=3x+5 with x values {−2, −1, 0, 1, 2}, and then graph the linear relation from the table.Graph a linear equation using the slope and yintercept.
An easier way to graph a linear relation is by using the slope and yintercept. The slopeintercept form of a line is y=mx+b, where m is the slope and b is the yintercept. The yintercept is a constant and is the point where the line crosses the yaxis. Graph this point first. The slope = [ggfrac]changeiny/changeinx[/ggfrac] . The numerator is how many units up you move, and the denominator is how many units right you move. Remember that any number is a fraction over 1. In our example y=2x+4, the slope is 2 and the yintercept is 4. To graph this, first look at the yaxis and mark a point at the yintercept, (0, 4). Next, move from that point 1 unit right and 2 units up to mark the next point. Repeat this as many times as you want before connecting all the points with a ruler to finish the graph. Try this yourself: Graph y=3x+5 using the slope and yintercept.
Graph a linear equation using the slope and yintercept. An easier way to graph a linear relation is by using the slope and yintercept. The slopeintercept form of a line is y=mx+b, where m is the slope and b is the yintercept. The yintercept is a constant and is the point where the line crosses the yaxis. Graph this point first. The slope = [ggfrac]changeiny/changeinx[/ggfrac] . The numerator is how many units up you move, and the denominator is how many units right you move. Remember that any number is a fraction over 1. In our example y=2x+4, the slope is 2 and the yintercept is 4. To graph this, first look at the yaxis and mark a point at the yintercept, (0, 4). Next, move from that point 1 unit right and 2 units up to mark the next point. Repeat this as many times as you want before connecting all the points with a ruler to finish the graph. Try this yourself: Graph y=3x+5 using the slope and yintercept.Understand slope in a linear equation.
Slope can be positive or negative. This doesn’t mean good or bad, it just tells us whether the line slants up or down as it moves from left to right along the xaxis. Lines with positive slope slant upwards left to right, whereas lines with negative slope slant downwards. You can graph any linear relationship if you know 2 points on the graph and can connect them with your ruler. Similarly, you can calculate the slope of a graph using the coordinates of any two points on the line. slope = [ggfrac]rise/run[/ggfrac] = [ggfrac]changeiny/changeinx[/ggfrac] = [ggfrac](y₂y₁)/(x₂x₁)[/ggfrac] = m, where (x₁ =y₁) and ( x₂ = y₂) are points on the line. Try this yourself: Find the slope of the line that goes through (–2, 2) and (4, 7).
Understand slope in a linear equation. Slope can be positive or negative. This doesn’t mean good or bad, it just tells us whether the line slants up or down as it moves from left to right along the xaxis. Lines with positive slope slant upwards left to right, whereas lines with negative slope slant downwards. You can graph any linear relationship if you know 2 points on the graph and can connect them with your ruler. Similarly, you can calculate the slope of a graph using the coordinates of any two points on the line. slope = [ggfrac]rise/run[/ggfrac] = [ggfrac]changeiny/changeinx[/ggfrac] = [ggfrac](y₂y₁)/(x₂x₁)[/ggfrac] = m, where (x₁ =y₁) and ( x₂ = y₂) are points on the line. Try this yourself: Find the slope of the line that goes through (–2, 2) and (4, 7).Determine an equation from a graph.
Just as you can graph a linear relation from the equation, you can find the equation from the graph. First, check for the yintercept on the graph. Look for where the line crosses the yaxis and record that as “b.” Next, find any other point on the graph and determine the rise and run between the yintercept and that point. Points that land exactly on the increments on the coordinate plane (lattice points) are best because you can see for sure what the coordinates are, and you don’t need to estimate any values. Use the rise and run to calculate the slope, “m.” Now that you have the slope and yintercept, you can substitute them into the slopeintercept form equation y=mx+b. Try this yourself: Draw a linear relation on graphing paper, and then exchange your graph with a partner. Write the equation for the graph your partner drew and then check your work with your partner.
Determine an equation from a graph. Just as you can graph a linear relation from the equation, you can find the equation from the graph. First, check for the yintercept on the graph. Look for where the line crosses the yaxis and record that as “b.” Next, find any other point on the graph and determine the rise and run between the yintercept and that point. Points that land exactly on the increments on the coordinate plane (lattice points) are best because you can see for sure what the coordinates are, and you don’t need to estimate any values. Use the rise and run to calculate the slope, “m.” Now that you have the slope and yintercept, you can substitute them into the slopeintercept form equation y=mx+b. Try this yourself: Draw a linear relation on graphing paper, and then exchange your graph with a partner. Write the equation for the graph your partner drew and then check your work with your partner. 
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Science & Math$_{/yr}

Science Only$_{/yr}
Identify the slope and yintercept in the equation y=3x4.
Graph the linear equation y=2x+5.
[Show students the graph of y = 3x – 3.] Identify the slope and yintercept of the line. Then write the equation that represents the line.
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