Relations and Functions8-1
A function is a special relation in which each number of the domain or input value (each x) is paired with exactly one member of the rangeor output (y value). Functions may be represented using ordered pairs, tables, or graphs.
One way to determine whether a relation that has been graphed is a function is to use the vertical line test. Using an object to represent a vertical line, maybe your pencil, move the object form left to right across the graph. If for each x value in the domain, the object passes through no more than one y value or one point on the graph, then the graph represents a function.
Example 1
Determine whether the relation is a function. Explain your answer.
[(-5,2), (-5,8), (6,-80), (13,-11), (2,-5)]
______, the relation is not a function. Negative five in the domain is paired with 2 and 8 in the range.
Example 2
Represent the relation as a mapping diagram. Then tell whether the relation is a function. Explain your reasoning.
x / y-1 / 0
0 / -1
1 / -1
3 / 2
2 / 3
4 / 11
INPUT OUTPUT
______, the relation is a function. Each value in the domain or input is paired with only one value in the range or output.
8-1 page 2
Example 3
Tell whether the relation represented by the graph is a function.
______, the relation is not a function because the number 1 in the domain is paired with 2 and -6 in the range. It would fail the vertical line test because two points would appear on the vertical line at the same time.
Linear Equations in Two Variables8-2
A function can be represented with an equation. An equation such as is called a linear equation. A linear equation in two variables is an equation in which the variables appear in separate terms and neither variable contains an exponent other than 1.
Linear Equations…
Nonlinear Equations…
In the third example, x has a power of -1.
In the fourth example, x and y are not in separate terms.
Solutions of a linear equation are ordered pairs that make the equation true. One way to find solutions is to make a table.
Example 1
Find solutions for the equation .
x / / y / (x, y)A linear equation can also be represented by a graph. The coordinates of all points that lie on a line are solutions that make the equation true. To graph a linear equation, find ordered pair solutions (as we did in the previous example), plot the corresponding points, and draw a line through them.
8-2 page 2
The graphs of all linear equations are straight lines. Although linear equations are usually also functions, a vertical line is not a function. So, although the equation is linear, it is not a function. An equation that has been solved for y is in function form.You may find it helpful to first write an equation in function form before graphing it.
Example2:
Write the equation in function form. Then graph the equation.
Next, create a function table.
x / / y / (x, y)Then, graph the ordered pairs and connect them by drawing a line through them.
Using Intercepts8-3
The x-intercept is the x-coordinate of a point where a graph crosses the x-axis. The y-coordinate of this point is always 0. To find the x-intercept, substitute y with 0 in the equation and solve for the corresponding x value.
The y-intercept is the y-coordinate of a point where a graph crosses the y-axis. The x-coordinate of this point must be 0. To find the y-intercept, substitute the x with 0 in the equation and solve for the corresponding y value.
Example 1:
So, (6, 0) is the x-intercept and (0, 3) is the y-intercept.
8-3 page 2
Example 2:
The x-intercept is at (2, 0). There is no y-intercept. No matter what value you choose for y, it doesn’t effect x.
**********************************************************
A vertical line has no y-intercept and a horizontal line has no x-intercept. **********************************************************
The Slope of a Line8-4
Slope describes the steepness of a line. It is the ratio of the rise, or vertical change, to the run, or horizontal change of a line:
Slope (m) is the same between any two points on a straight line and can be found by using the coordinates of any two points on that line:(Subtract the change in y on the top. Subtract the change in x on the bottom.)
m= , where .
Slope is the ratio of the difference in y-coordinates to the corresponding difference in x-coordinates.
Example 1:
To find the slope of a graph, count to find the amount of change that has occurred between any two points y coordinates and then, count to find the distance to the right between those same two points.
So, the slope is ______.
Examples 2:
If a line passes through A (2,3) and B (-5,4), find the slope.
m So, the slope = .
______
Example 3:
If a line passes through R (7,-4) and S (7,3), find the slope.
m So, the slope is ______.Therefore, this is a ______line.
______
Example 4:
If a line passes through P (2,5) and Q (6,5) find the slope.
m So, the slope is ____.Therefore, this is a ______line.
Slope-Intercept Form8 – 5
Equations written in the form of , where m is the slope and b is the y-intercept, are linear equations in slope-intercept form.
Example 1:
The graph of y = 5x – 6 is a line that has a slope of 5 and crosses the y-axis at (0,-6) because m is 5 and b is (- 6).
Sometimes you must first rewrite an equation using inverse operations to get it into slope-intercept form before you can find the slope and y-intercept.
Example 2:
The equation 2x + 3y = 15 can be expressed in slope-intercept form by subtracting 2x from each side and then dividing by 3:
which reveals a slope of and a y-intercept of ____.
You can use the slope-intercept form of an equation to graph a line easily. Graph the y-intercept and use the slope to find another point on the line, and then connect the two points with a line.
Example 3: Example 4:
slope =slope =
y-intercept =y-intercept =
8 – 5 page 2
Two lines that are not vertical but are parallel have the same slope.
Two lines that are not vertical but are perpendicular have slopes that are opposite reciprocals. (Hint, a number and its negative reciprocal will have a product of negative one.)
Example 5:
For the line with the equation, find the slope of a parallel line and the slope of a perpendicular line.
This line has a slope of , so a parallel line would have a slope of and a perpendicular line would have a slope of .
Writing Linear Equations8 – 6
If you know the slope and y-intercept, you can write the equation of a line by substituting these values in to y = mx + b.
Example 1: Write the equation of a line that has a slope of 2 and a y-intercept of
(- 3).
You can also use the graph of a line to write an equation.
Locate the y-intercept and choose another point on the line.
Find the slope between the two points.
Then, write the equation for the line.
Example 2:
Looking at the graph, you can identify the y-intercept of 3 and count between two points that the slope is .
So, the equation for this line is y = x + 3.
You can also write an equation for a line if you know the coordinates of two points on the line by using the definition of slope: m = and the slope, y-intercept equation, y = mx + b.
8 – 6 page 2
Example 3:
Find the equation of the line which contains (0, 2) and (-3, 14).
(You will also need to be familiar with the examples from pages 420 and 421 in the textbook. They are for Writing Equations of Parallel or Perpendicular Lines, Writing an Equation from a Table, and Approximating a Best-Fitting Line.)
Function Notation8 – 7
In lesson 8 – 2 you practiced writing equations in “function form”. In this lesson, you will turn such equations into function notation. In example 2 of lesson 8 – 2 we rewrote as . This function could be rewritten in function notation as . The symbol replaces y and is read as “f of x”.
We can find the value of a function for a particular value of x.
We can also find the x for a particular value of a function.
Example 1:
Let. Findwhen x = 6, and find x when = .
In order to graph a function, follow the same procedure you followed to graph equations.
Example 2:
Graph the function .
Use .
The y intercept is . So you would plot (0,).
The slope is 2. So, starting at (0,) go up 2 spaces and right one to graph another point.
Finally, draw a line through the two points you’ve graphed.
8 – 7 page 2
If for a function g, then you know that the graph of the function would have to go through the point .
Example 3:
Write a linear function h given that .
Because we know that the y intercept is at (0, ).
So, in the form of
Systems of Linear Equations8 – 8
Two or more equations together on one coordinate plane are called a system of equations. The solution of a system of equations is the ordered pair that is a solution of both equations. The system of equations may have one solution, no solution, or an infinite number of solutions.
One method for solving a system of equations is to graph the equations on one coordinate plane. The coordinates of the location where the lines cross or intersect is the solution of the system of equations.
Example 1: Solve the system of equations by graphing.
y = -2x + 1
y = x - 2
The point of intersection of the two lines is the solution to the system of equations. So, the solution for this system of equations is ( , ).
*****Parallel lines have ______because they never cross.****
*****Identical lines have an ______number of solutions.*****
A more accurate way to solve a system of equations is by using a method called substitution. There are two steps to this method;
1. Rewrite one or both of the equations in terms of y, or x.
2. Replace a variable in one equation with the stated value of it from the other equation.
8-8 page 2
Example 1:Example 2:Example 3:
Example 4: Example 5:
Graphs of Linear Inequalities8 – 9
A linear inequality is very similar to a linear equation except the equal sign is replaced with an inequality symbol, An ordered pair which creates a true statement when the x and y variables are replaced with the x and y coordinates is a solution for the inequality.
Example 1:
Tell whether the ordered pair (0, 3) is a solution for.
So, since (0, 3) did not make a true statement, it is not a solution for the inequality.
To graph an inequality, first graph the related equation, this is the boundary. Second, shade the region, or half plane, which contains all the points that are solutions of the inequality.
If an inequality contains the symbol or , then use a solid line to indicate that the boundary is included as part of the solution and graph.
If an inequality contains the symbol < or >, then use a dashed line to indicate that the boundary is not included in the solution or graph.
Example 2:
- Using a dotted line, graph the equation.
- Choose a point not on the boundary,for example, (0, 0). Plug it into the inequality to see if it forms a true statement.
This point does not satisfy the inequality. So, shade the opposite side of
the boundary, the side that doesn’t contain the point (0, 0).
8 – 9 page 2
Example 3:
- Using a solid line, graph the equation.
- Choose a point not on the boundary,for example, (1, 0). Plug it into the inequality to see if it forms a true statement.
This point does not satisfy the inequality. So, shade the opposite side of the boundary,the side that doesn’t contain the point (1, 0).
Example 4:
- Using a solid line, graph the equation.
- Choose a point not on the boundary,
for example, (2, 0). Plug it into the inequality to see if it forms a true statement.
This is a true statement. So, shade this side of the boundary,the side that does contain the point (2, 0).