Linear Functions
The parent function of a linear equation is y=mx+b. This is the simplest form of a linear equation. It makes a straight line that continues in both directions infinitely.
A long as there are no restrictions on the graph, all real numbers are possible solutions for x and y.
The domain is all the solutions for x. (Input) The range is all the solutions for y. (Output)
This equation can take several forms, while remaining the same graph.
The first form it can take is its parent function, called slope-intercept form.
A long as there are no restrictions on the graph, all real numbers are possible solutions for x and y.
The domain is all the solutions for x. (Input) The range is all the solutions for y. (Output)
This equation can take several forms, while remaining the same graph.
The first form it can take is its parent function, called slope-intercept form.
Slope-Intercept Form
y=mx+b
The m stands for the slope of the line. You can find this by choosing two points on the graph, and counting the number of spaces upward and sideways that you have to move to get from one point to the next. You can put these numbers in a fraction:
Rise
Run
This goes in place of m in the equation.
The b stands for the y-intercept. This is where the line hits the y-axis. You can find this by making x zero, and finishing the math from there. The number you get is the y-intercept.
Slope-intercept form is very useful for graphing, since the y-intercept and slope are easily found. This quality also can be helpful with word problems.
The m stands for the slope of the line. You can find this by choosing two points on the graph, and counting the number of spaces upward and sideways that you have to move to get from one point to the next. You can put these numbers in a fraction:
Rise
Run
This goes in place of m in the equation.
The b stands for the y-intercept. This is where the line hits the y-axis. You can find this by making x zero, and finishing the math from there. The number you get is the y-intercept.
Slope-intercept form is very useful for graphing, since the y-intercept and slope are easily found. This quality also can be helpful with word problems.
Standard Form
Standard form is Ax+By=C. A, B, and C are integers that go with x and y. A can't be a negative number, and if it is, you need to multiply all the parts of the equation by negative one to change it.
Standard form is useful because it allows us to write equations for vertical lines, which we'll talk more about later. We can't use slope-intercept form for these lines because their slopes are undefined.
Standard form is useful because it allows us to write equations for vertical lines, which we'll talk more about later. We can't use slope-intercept form for these lines because their slopes are undefined.
Point-Slope Form
Point slope form is written:
y-y1=m(x-x1)
In this equation, you use a single point, (x1, y1) and you plug it into the equation at the corresponding places. You can use both points in the slope formula shown below to find the slope, which is m.
Point slope form is very useful when you are given two points on a line. You can plug in the points in your equation to find the y-intercept. You can find the slope of a line when only given two coordinates by using this formula:
m=x-x1
y-y1
When given a graph, you can see and count the rise over run points to find the slope. You can see where the y-intercept is to find b. You can easily plug these numbers into the point slope formula.
y-y1=m(x-x1)
In this equation, you use a single point, (x1, y1) and you plug it into the equation at the corresponding places. You can use both points in the slope formula shown below to find the slope, which is m.
Point slope form is very useful when you are given two points on a line. You can plug in the points in your equation to find the y-intercept. You can find the slope of a line when only given two coordinates by using this formula:
m=x-x1
y-y1
When given a graph, you can see and count the rise over run points to find the slope. You can see where the y-intercept is to find b. You can easily plug these numbers into the point slope formula.
Special Lines
There are some lines that have special qualities. These qualities can help us understand slopes better.
Horizontal Lines: y=b
A horizontal line can be written, for example, y=-2. The y will always be the same number, no matter what x is. This line is parallel to the x-axis. The line has a slope of zero, meaning that is doesn't rise or fall at all; it's completely flat.
Horizontal Lines: y=b
A horizontal line can be written, for example, y=-2. The y will always be the same number, no matter what x is. This line is parallel to the x-axis. The line has a slope of zero, meaning that is doesn't rise or fall at all; it's completely flat.
Vertical Lines: x=a
A vertical line is one parallel to the y-axis. It's slope is undefined, because it "rises" but it doesn't "run" at all.
Ex.
2
0
You cannot divide anything by zero, so the slope has to be undefined.
Parallel and Perpendicular Lines: Two lines that are parallel to each other have the exact same slope. These lines will never touch. Perpendicular lines touch each other one time, forming 90 degree angles. Their slopes are also related to each other. One of the lines' slopes is the opposite reciprocal of the other. For example. if one line's slope is 3, then the other's is -1/3.
A vertical line is one parallel to the y-axis. It's slope is undefined, because it "rises" but it doesn't "run" at all.
Ex.
2
0
You cannot divide anything by zero, so the slope has to be undefined.
Parallel and Perpendicular Lines: Two lines that are parallel to each other have the exact same slope. These lines will never touch. Perpendicular lines touch each other one time, forming 90 degree angles. Their slopes are also related to each other. One of the lines' slopes is the opposite reciprocal of the other. For example. if one line's slope is 3, then the other's is -1/3.
Extras:
Midpoint: The midpoint on a line is the exact middle point's coordinates between two other points.
The formula is
x1+x2 y1+y2
2 , 2
You just find the average distance between the x coordinates and the y coordinates of each endpoint.
Distance: To find the distance between two points you have to subtract the x values and the y values. When you get those numbers, you put them into the Pythagorean Theorem,a^2+b^2=c^2, as a and b. You can then find c, which is your answer. The formula looks like this:
Midpoint: The midpoint on a line is the exact middle point's coordinates between two other points.
The formula is
x1+x2 y1+y2
2 , 2
You just find the average distance between the x coordinates and the y coordinates of each endpoint.
Distance: To find the distance between two points you have to subtract the x values and the y values. When you get those numbers, you put them into the Pythagorean Theorem,a^2+b^2=c^2, as a and b. You can then find c, which is your answer. The formula looks like this:
Inequalities:
A linear inequality looks the same as a linear function, except it has an inequality symbol instead of an equal sign.
Here are some inequality symbols:
A linear inequality looks the same as a linear function, except it has an inequality symbol instead of an equal sign.
Here are some inequality symbols:
When these signs are there, the graph of a line changes. You make the line in the same place, but if the symbol is < or >, the line is dashed. If the sign is < or >, the line is solid. A "less than" or "than or equal to" means to shade the graph below the line. A "greater than" or "greater than or equal to" sign means to shade above the line. Here is a video to help!