Sign In
| 17 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here is a recommended reading before getting started with this lesson.
In a nonproportional relationship, the line passes through the point (0,b), or the y-intercept. Use the Slope Formula to derive the equation for a nonproportional relationship. m=y_2-y_1/x_2-x_1
Assume that the other point the line passes through is (x,y).The intercepts of a line provide information about the position of the line in the coordinate plane. They can be used to identify the equation of a line from known points or to graph a line from its equation.
The applet shows the graphs of different lines. Identify the intercepts of these lines.
A linear equation is an equation with at least one linear term and any number of constants. No other types of terms may be included. Linear equations in one variable have the following form, where a and b are real numbers and a≠ 0.
ax+b=0 or ax = b
Linear equations in two variables have the form below, where a, b, and c are real numbers and a≠ 0 and b≠ 0.
ax+by+c=0 or ax+by = c
A linear equation can be written in the following form called the slope-intercept form.
y=mx+b
In this form, m is the slope and b is the y-intercept. These are the general characteristics or parameters of the line. They determine the steepness and the position of the line on the coordinate plane. Consider the following graph.
This line has a slope of 2 and a y-intercept of 1. The equation of the line can be written in slope-intercept form using these values.
y= mx+ b ⇓ y= 2x+ 1The following applet generates linear equations. Those equations represent the relation between two variables x and y. Determine whether the generated equation is in slope-intercept form.
The y-intercept b can be used to find the the first point the line passes through. y=2x-3 ⇔ y= 2x+( - 3) The y-intercept b is - 3. Plot the point (0, - 3) on a coordinate plane.
There should be at least two points to draw a line. The second point can be plotted on the coordinate plane by using the slope m. Based on the equation, the slope is 2. y= 2x+( - 3) This means that the rise is 2 and the run is 1. m=rise/run ⇔ 2=2/1 From the first point (0,- 3), move 1 unit right and 2 units up to plot the second point.
Jordan is investigating the fish population in Grassy Lake. The number of fish in the lake was counted as 60 in 2016.
She finds out that the population increases by 120 after each year. The following equation represents the fish population in the lake. y=120x+60 In this equation, y is the number of fish in the lake after x years.
y-intercept: Number of fish in the year 2016
y= mx+ b ⇓ y= 120x+ 60 The slope is 120 and the y-intercept is 60. This means that the line crosses the y-axis at (0, 60).
Next, move 1 unit to the right and 120 units up from the y-intercept to plot another point.
Finally, the graph of the equation can be competed by drawing a line passing through these two points.
Note that the number of years passed or the number of fish cannot be negative. Therefore, only the first quadrant of the coordinate plane is shown.
Slope &= 120 y-intercept &= 60 In this context, the slope 120 represents the increase in the fish population per year. The y-intercept 60 represents the number of fish in 2016.
The y-intercept b and the slope m of a line must be found to write the equation of the graph of the line in slope-intercept form . y=mx+b Consider the line shown as an example.
The y-intercept is the y-coordinate of the point where the line crosses the y-axis. This line intercepts the y-axis at (0,- 4), which means that the y-intercept is - 4.
The y-intercept can be substituted into the slope-intercept form equation for b. y=mx+ b ⇓ y=mx - 4
Next, the slope of the line must be determined. The slope of a line is the ratio of the rise and run of the line. m=rise/run The rise is the vertical distance between two points and the run is the horizontal distance. Any two points on the line can be used to find the slope.
For this line, the rise is 6 and the run is 2. Substitute these values into the formula to calculate the slope of the line. m=6/2 ⇒ m=3
Finally, substitute m= 3 in to the equation from Step 2 to complete the equation. y= mx-4 ⇓ y= 3x-4 The equation of the line in slope-intercept form is now complete.
Jordan turns her attention to the cheetah population over years.
She finds a graph on the Internet that shows the number of cheetahs over years since 1900.
Jordan wants to represent the graph algebraically to interpret it further.
y-intercept: Number of cheetahs in the year 1900
y= mx+ b In this form, m is the slope and b is the y-intercept. The y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis.
In the given graph, the line intercepts the y-axis at (0,100 000). This means that the y-intercept of the line is 100 000. Substitute 100 000 for b into the slope-intercept form of an equation. y= mx+ b ⇓ y= mx+ 100 000 The slope of a line is the ratio of the rise and run of the line. m=rise/run Any two points on the line can be chosen to determine the rise and run. One point can be the one where the line intercepts the y-axis. The other point can be (50,60 000) for simplicity.
rise= -40 000, run= 50
Put minus sign in front of fraction
Calculate quotient
y= -800x+ 100 000 The slope of the line is negative, which suggests that the number of cheetahs decreases by 800 per year. The y-intercept represents that there were 100 000 cheetahs in 1900.
y= 0
LHS+800x=RHS+800x
.LHS /800.=.RHS /800.
Calculate quotient
The slope m and the y-intercept b of a line must be known to write a linear equation in slope-intercept form. y=mx+b When only two points on the line are known, the following four-step method can be used. For example, the equation of the line that passes through the points (- 4,1) and (8,4) will be written.
Substitute ( - 4, 1) & ( 8,4)
a-(- b)=a+b
Add and subtract terms
Calculate quotient
Now that the value of the slope is known, it can be substituted for m in the slope-intercept form of an equation. y= mx+b ⇓ y= 0.25x+b
x= 8, y= 4
Multiply
LHS-2=RHS-2
Rearrange equation
Lastly, the complete equation in slope-intercept form can be written by substituting the y-intercept into the equation from Step 2. y=0.25x+ b ⇓ y=0.25x+ 2 The equation of the line in slope-intercept form is now complete.
Jordan is interested in endangered animals. She reads a magazine to learn what can be done to save these animals.
Suppose Jordan reads the same number of pages each day. The following table shows the number of pages y left in the magazine after x days.
Number of Days x | 5 | 9 | 15 |
---|---|---|---|
Number of Pages y | 147 | 95 | 17 |
Write an equation for the given situation in slope-intercept form using the Slope Formula.
The given equations are in slope-intercept form. y= mx+ b In this form, m is the slope and b is the y-intercept. These two characteristics need to be determined to write an equation in slope-intercept form. Begin by representing given information in terms of points (x,y).
Number of Days x | 5 | 9 | 15 |
---|---|---|---|
Number of Pages y | 147 | 95 | 17 |
Point (x,y) | ( 5, 147) | ( 9, 95) | ( 15, 17) |
Substitute ( 5, 147) & ( 9, 95)
Subtract terms
Put minus sign in front of fraction
Calculate quotient
x= 15, y= 17
(- a)b = - ab
LHS+195=RHS+195
Rearrange equation
Write an equation in slope-intercept form using the given line, two points, or equation.
Determine the slope or y-intercept the given line based on its graph, points, or its equation. Be careful — the equation may or may not be written in slope-intercept form!
Substitute ( 0, b) & ( x, y)
Subtract term
LHS * x=RHS* x
a/x* x = a
LHS+b=RHS+b
Rearrange equation
Consider a line that has a y-intercept but no x-intercept.
We want to find the slope of a line that has a y-intercept but no x-intercept. Let's first remember what the y-intercept and the x-intercept of a line are.
x-intercept | x-coordinate of the point where the line crosses the x-axis |
---|---|
y-intercept | y-coordinate of the point where the line crosses the y-axis |
If a line has a y-intercept, it means that the line crosses the y-axis. If it does not have an x-intercept, it means that it does not cross the x-axis and the value of y is never equal to 0. This is only the case when our line is parallel to the x-axis — it is a horizontal line. Notice that the line cannot lie on the x-axis.
Let's use the Slope Formula to find the slope of a horizontal line. m=y_2-y_1/x_2-x_1 Every point on a horizontal line has the same y-coordinate. This means that the numerator of the Slope Formula will always be 0, regardless of the x-coordinates of the points. m=y_2-y_1/x_2-x_1 ⇒ m=0/x_2-x_1 Therefore, the slope of a line is always 0 if the line has a y-intercept but no x-intercept.
Tadeo has to graph a line with the equation y= 12x-3. He wants to know what the line will look like before he graphs the line.
Is the line increasing, decreasing, horizontal, or vertical? | Does the line cross the y-axis below or above the origin? | Which quadrants does the line pass through? | |
---|---|---|---|
A | Increasing line | Above the origin | Quadrants I, II, and III |
B | Decreasing line | Below the origin | Quadrants II, III, and IV |
C | Horizontal line | Above the origin | Quadrants I and II |
D | Vertical line | Above the origin | Quadrants III and IV |
E | Increasing line | Below the origin | Quadrants I, III, and IV |
F | Decreasing line | Below the origin | Quadrants I, II, and IV |
We know that Tadeo wants to graph a line with the equation y= 12x - 3. We want to describe how the line will look like, including the quadrants that it will pass through. The equation is given in slope-intercept form, so let's start by focusing on its slope. y= 1/2x - 3 The slope of the line is 12. Remember that when a slope is positive, the relationship between x and y is increasing. When a slope is negative, the relationship is decreasing. The slope is positive in Tadeo's case, so the relationship in increasing. Now let's focus on the y-intercept of the equation. y= 1/2x - 3 ⇔ y= 1/2x + ( - 3) The y-intercept of the line is - 3. The y-intercept is the y-coordinate of the point on a graph where the line crosses the y-axis. Therefore, when the y-intercept is negative, the line must cross the y-axis at a point with a negative y-coordinate. In other words, the line crosses the y-axis below the origin.
The line that Tadeo will graph is increasing and crosses the y-axis below the origin. Let's sketch how these types of graphs generally look.
All lines that are increasing and cross the y-axis below the origin pass through Quadrants I, III, and IV, which means that Tadeo's graph also will pass through Quadrants I, III, and IV. The option above that best describes the graph is E.