Expanding the walls of our classroom. This is an interactive learning ecology for students and parents in our Algebra 2 class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.
Linear Equations in Two Variables A linear equation in two variables can be written in the form Ax+By=C, where A,B, and C are real numbers (A and B both not 0). This form is called Standard Form.
Slopes of Perpendicular lines If neither is vertical, perpendicular lines have slopes that are negative reciprocals; that is, their product is -1. Also, lines with slopes that are negative reciprocals are perpendicular.
Equations of Horizontal and Vertical Lines: The horizontal line through the point (a,b) has equation y=b. The vertical line through the point (a,b) has equation x=a.
Linear Inequality in Two Variables: An inequality that can be written as Ax+bYC, where A,B, and C are real numbers and A and B are not both 0, is a linear inequality in two variables.
Graphing a Linear Inequality: Step 1. Draw a graph of the straight line that is the boundary. Make a line solid if the inequality involves ≥ or≤; make the line dashed if the inequality involves> or<. Step 2. Choose a test point. Choose any point not on the line, and substitute the coordinates of this point in the inequality. Step 3. Shade the appropriate region. Shade the region that includes the test point if it satisfies the original inequality; otherwise, shade the region on the other side of the boundary line.
Function: a function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.
Domain and Range: In a relation, the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable (y) is the range.
Agreement on Domain: The domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.
Variations of the Definition Function: 1. A function is a relation in which , for each value of the first component of the ordered pairs, there is exactly one value of the second component. 2. A function is a set of ordered pairs in which no first component is repeated. 3. A function is a rule or correspondence that assigns exactly one range value to each domain value,.
Solving a Variation Problem: Step 1. Write the variation equation Step 2. Substitute the initial values and solve for k. Step 3. Rewrite the variation equation with the value of k from Step 2. Step 4. Substitute the remaining values, solve for the unknown, and find the required answer.
Inverse Variation: y varies inversely as x if there exists a real number k such that y=k/x Also, y varies inversely as the nth power of x if there exists a real number k such that y=k/x^n
KEY TERMS FOR CHAPTER 4! ordered pair origin x-axis y-axis rectangular coordinate system plot components coordinate quadrant graph of an equation first-degree equation linear equation in two variables x-intercept y-intercept rise run slope linear inequality in two variables boundary line dependent variable independent variable relation function domain range function notation linear function constant function
4.1: The Rectangular Coordinate System
ReplyDeleteLinear Equations in Two Variables
A linear equation in two variables can be written in the form
Ax+By=C,
where A,B, and C are real numbers (A and B both not 0). This form is called Standard Form.
Slopes of Horizontal and Vertical Lines
ReplyDeleteThe slope of a horizontal line is 0.
The slope of a vertical line is undefined.
Slopes of Perpendicular lines
ReplyDeleteIf neither is vertical, perpendicular lines have slopes that are negative reciprocals; that is, their product is -1. Also, lines with slopes that are negative reciprocals are perpendicular.
4.3: Linear Equations in tWO vARIABLES
ReplyDeleteSlope-Intercept Form:
The slope intercept form of the equation of a line with slope m and y-intercept (o,b) is
y=mx+b
Point Slope Form
ReplyDeleteThe point slope form of the equation of a line with slope m passing through the point (x1,y1) is
y-y1=,(x-x1)
Equations of Horizontal and Vertical Lines:
ReplyDeleteThe horizontal line through the point (a,b) has equation y=b.
The vertical line through the point (a,b) has equation x=a.
4.4: Linear Inequalities in Two Variables
ReplyDeleteLinear Inequality in Two Variables:
An inequality that can be written as
Ax+bYC,
where A,B, and C are real numbers and A and B are not both 0, is a linear inequality in two variables.
Graphing a Linear Inequality:
ReplyDeleteStep 1. Draw a graph of the straight line that is the boundary. Make a line solid if the inequality involves ≥ or≤; make the line dashed if the inequality involves> or<.
Step 2. Choose a test point. Choose any point not on the line, and substitute the coordinates of this point in the inequality.
Step 3. Shade the appropriate region. Shade the region that includes the test point if it satisfies the original inequality; otherwise, shade the region on the other side of the boundary line.
4.5: Introduction to Functions
ReplyDeleteRelation: A relation is any set of ordered pairs.
Function: a function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.
ReplyDeleteDomain and Range: In a relation, the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable (y) is the range.
ReplyDeleteAgreement on Domain:
ReplyDeleteThe domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.
Vertical Line Test:
ReplyDeleteIf every vertical line intersects the graph of a relation in no more than one point, then the relation represents a function.
Variations of the Definition Function:
ReplyDelete1. A function is a relation in which , for each value of the first component of the ordered pairs, there is exactly one value of the second component.
2. A function is a set of ordered pairs in which no first component is repeated.
3. A function is a rule or correspondence that assigns exactly one range value to each domain value,.
Finding an Expression for f(x):
ReplyDeleteStep 1. Solve the equation for y.
Step 2. Replace y with f(x).
Linear Function:
ReplyDeleteA function that can be defined by
f(x)=mx+b
for real numbers m and b is a linear function.
4.6: Variation
ReplyDeleteDirect Variation:
y varies directly as x if there exists some constant k such that
y=kx
Solving a Variation Problem:
ReplyDeleteStep 1. Write the variation equation
Step 2. Substitute the initial values and solve for k.
Step 3. Rewrite the variation equation with the value of k from Step 2.
Step 4. Substitute the remaining values, solve for the unknown, and find the required answer.
Direct Variation as a Power:
ReplyDeletey varies directly as the nth power of x if there exists a real number k such that
y=kx^n
Inverse Variation:
ReplyDeletey varies inversely as x if there exists a real number k such that
y=k/x
Also, y varies inversely as the nth power of x if there exists a real number k such that
y=k/x^n
Joint Variation:
ReplyDeletey varies jointly as x and z if there exists a real number k such that
y=kxz.
KEY TERMS FOR CHAPTER 4!
ReplyDeleteordered pair
origin
x-axis
y-axis
rectangular coordinate system
plot
components
coordinate
quadrant
graph of an equation
first-degree equation
linear equation in two variables
x-intercept
y-intercept
rise
run
slope
linear inequality in two variables
boundary line
dependent variable
independent variable
relation
function
domain
range
function notation
linear function
constant function