Word Problems and Table Building Slope as a Rate of Change Slope of a Line Using Slope to Graph a Line Converting to Slope-Intercept Form and Graphing Linear Parent Graph and Transformations Slope of Line is Tripled… Writing Equations of Lines Writing Equations of Lines Using Tables Applications of Direct Variation and Linear Functions Solving Systems by Graphing Solving Systems by Addition Solving Systems by Substitution Number and Value Word Problems Wind and Current Word Problems Digit Word Problems The Product Rule The Power Rule The Quotient Rule Numerical Bases and Exponents of Zero Combining Exponent Rules Monomials, Polynomials, Binomials, Trinomials First Outer Inner Last Multiplying Binomials, Two Variables Greatest Common Factor Factoring out the Greatest Common Factor Factoring Trinomials with Positive Constants Factoring Trinomials with Negative Constants Difference of Two Squares Beginning Polynomial Equations Intermediate Polynomial Equations Simplifying Rational Expressions Multiplying and Dividing Rational Expressions Adding Rational Expressions Subtracting Rational Expressions Binomials and Trinomials in Denominators Binomials and Trinomials in Denominators F.
Cube Root of Negative 8. Linear or Quadratic Functions Graphs of Quadratic Functions and Beginning Transformations Graphs of Quadratic Functions and Advanced Transformations Solving Quadratic Equations by Graphing Taking the Square Root of Both Sides Completing The Square Variables, Expressions, and Equations Translating English to Algebra Using Manipulatives to Model Algebraic Equations Finding the nth Term in a Pattern Finite Differences to Generalize a Rule Dependency and Correlational Relationships Recognizing and Evaluating Functions Functions in Problem Situations Points on the plane are designated by ordered pairs of numbers written in parentheses with a comma between them, such as 5,7.
This is called an ordered pair because the order in which the numbers are written is important. The ordered pair 5,7 is not the same as the ordered pair 7,5. Points are located on the plane in the following manner. First, start at the origin and count left or right the number of spaces designated by the first number of the ordered pair.
Second, from the point on the x-axis given by the first number count up or down the number of spaces designated by the second number of the ordered pair. Ordered pairs are always written with x first and then y, x,y. The numbers represented by x and y are called the coordinates of the point x,y.
The first number of the ordered pair always refers to the horizontal direction and the second number always refers to the vertical direction. Check each one to determine how they are located. What are the coordinates of the origin? Find several ordered pairs that make a given linear equation true. Locate these points on the Cartesian coordinate system. Draw a straight line through those points that represent the graph of this equation. A graph is a pictorial representation of numbered facts.
There are many types of graphs, such as bar graphs, circular graphs, line graphs, and so on. You can usually find examples of these graphs in the financial section of a newspaper. Graphs are used because a picture usually makes the number facts more easily understood.
In this section we will discuss the method of graphing an equation in two variables. In other words, we will sketch a picture of an equation in two variables. All possible answers to this equation, located as points on the plane, will give us the graph or picture of the equation.
A sketch can be described as the "curve of best fit. Remember, there are infinitely many ordered pairs that would satisfy the equation. Solution We wish to find several pairs of numbers that will make this equation true.
We will accomplish this by choosing a number for x and then finding a corresponding value for y. A table of values is used to record the data.
In the top line x we will place numbers that we have chosen for x. Then in the bottom line y we will place the corresponding value of y derived from the equation. Of course, we could also start by choosing values for y and then find the corresponding values for x. These values are arbitrary. We could choose any values at all. Notice that once we have chosen a value for x, the value for y is determined by using the equation.
These values of x give integers for values of y. Thus they are good choices. We now locate the ordered pairs -3,9 , -2,7 , -1,5 , 0,3 , 1,1 , 2,-1 , 3,-3 on the coordinate plane and connect them with a line. The line indicates that all points on the line satisfy the equation, as well as the points from the table.
The arrows indicate the line continues indefinitely. The graphs of all first-degree equations in two variables will be straight lines. This fact will be used here even though it will be much later in mathematics before you can prove this statement.
Such first-degree equations are called linear equations. Equations in two unknowns that are of higher degree give graphs that are curves of different kinds. You will study these in future algebra courses.
Since the graph of a first-degree equation in two variables is a straight line, it is only necessary to have two points. However, your work will be more consistently accurate if you find at least three points. Mistakes can be located and corrected when the points found do not lie on a line. We thus refer to the third point as a "checkpoint. You will be surprised how often you will find an error by locating all three points. Solution First make a table of values and decide on three numbers to substitute for x.
We will try 0, 1,2. Again, you could also have started with arbitrary values of y. The answer is not as easy to locate on the graph as an integer would be. Sometimes it is possible to look ahead and make better choices for x.
We will readjust the table of values and use the points that gave integers. This may not always be feasible, but trying for integral values will give a more accurate sketch. We can do this since the choices for x were arbitrary. How many ordered pairs satisfy this equation? Associate the slope of a line with its steepness. Write the equation of a line in slope-intercept form. Graph a straight line using its slope and y-intercept.
We now wish to discuss an important concept called the slope of a line. Intuitively we can think of slope as the steepness of the line in relationship to the horizontal. Following are graphs of several lines. Study them closely and mentally answer the questions that follow. What seems to be the relationship between the coefficient of x and the steepness Which graph would be steeper: Which graph would be steeper: If m as the value of m increases, the steepness of the line decreases and the line rises to the left and falls to the right.
In other words, in an equation of the form y - mx, m controls the steepness of the line. In mathematics we use the word slope in referring to steepness and form the following definition:. Solution We first make a table showing three sets of ordered pairs that satisfy the equation. Remember, we only need two points to determine the line but we use the third point as a check. Example 2 Sketch the graph and state the slope of.
Why use values that are divisible by 3? Compare the coefficients of x in these two equations. Again, compare the coefficients of x in the two equations. Observe that when two lines have the same slope, they are parallel.
The slope from one point on a line to another is determined by the ratio of the change in y to the change in x. If you want to impress your friends, you can write where the Greek letter delta means "change in. We could also say that the change in x is 4 and the change in y is - 1. This will result in the same line. The change in x is 1 and the change in y is 3. If an equation is in this form, m is the slope of the line and 0,b is the point at which the graph intercepts crosses the y-axis.
The point 0,b is referred to as the y-intercept. If the equation of a straight line is in the slope-intercept form, it is possible to sketch its graph without making a table of values. Use the y-intercept and the slope to draw the graph, as shown in example 8.
First locate the point 0, This is one of the points on the line. The slope indicates that the changes in x is 4, so from the point 0,-2 we move four units in the positive direction parallel to the x-axis.
Since the change in y is 3, we then move three units in the positive direction parallel to the y-axis. The resulting point is also on the line. Since two points determine a straight line, we then draw the graph. Always start from the y-intercept. A common error that many students make is to confuse the y-intercept with the x-intercept the point where the line crosses the x-axis.
To express the slope as a ratio we may write -3 as or. If we write the slope as , then from the point 0,4 we move one unit in the positive direction parallel to the x-axis and then move three units in the negative direction parallel to the y-axis. Then we draw a line through this point and 0,4.
Can we still find the slope and y-intercept? The answer to this question is yes. To do this, however, we must change the form of the given equation by applying the methods used in section Section dealt with solving literal equations. You may want to review that section. Solution First we recognize that the equation is not in the slope-intercept form needed to answer the questions asked.
To obtain this form solve the given equation for y. Sketch the graph of here. Sketch the graph of the line on the grid below. These were inequalities involving only one variable. We found that in all such cases the graph was some portion of the number line. Since an equation in two variables gives a graph on the plane, it seems reasonable to assume that an inequality in two variables would graph as some portion or region of the plane.
This is in fact the case. To summarize, the following ordered pairs give a true statement. The following ordered pairs give a false statement. If one point of a half-plane is in the solution set of a linear inequality, then all points in that half-plane are in the solution set.
This gives us a convenient method for graphing linear inequalities. To graph a linear inequality 1. Replace the inequality symbol with an equal sign and graph the resulting line. Check one point that is obviously in a particular half-plane of that line to see if it is in the solution set of the inequality. If the point chosen is in the solution set, then that entire half-plane is the solution set. If the point chosen is not in the solution set, then the other half-plane is the solution set.
Why do we need to check only one point? The point 0,0 is not in the solution set, therefore the half-plane containing 0,0 is not the solution set. Since the line itself is not a part of the solution, it is shown as a dashed line and the half-plane is shaded to show the solution set. The solution set is the half-plane above and to the right of the line. Since the point 0,0 is not in the solution set, the half-plane containing 0,0 is not in the set. Hence, the solution is the other half-plane.
Therefore, draw a solid line to show that it is part of the graph. The solution set is the line and the half-plane below and to the right of the line.
Next check a point not on the line. Notice that the graph of the line contains the point 0,0 , so we cannot use it as a checkpoint. The point - 2,3 is such a point. When the graph of the line goes through the origin, any other point on the x- or y-axis would also be a good choice.
Sketch the graphs of two linear equations on the same coordinate system. Determine the common solution of the two graphs. Example 1 The pair of equations is called a system of linear equations. We have observed that each of these equations has infinitely many solutions and each will form a straight line when we graph it on the Cartesian coordinate system.
We now wish to find solutions to the system. In other words, we want all points x,y that will be on the graph of both equations. Solution We reason in this manner: In this table we let x take on the values 0, 1, and 2.
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6 Lessons in Chapter 3: Inequalities: Homework Help Chapter Practice Test Solving and graphing absolute value inequalities brings a lot of different skills together in one place. The practice. Homework Help; Specialized Programs. ADD/ADHD Tutoring Programs; Math Review of Inequalities If the variable is isolated on the left side of the inequality symbol, then the graph on the number line can point in the same direction as the inequality symbol. For example, 3 > x can be rewritten as x.
The Working With Inequalities chapter of this High School Precalculus Homework Help course helps students complete their inequalities homework and. Math homework help. Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. Online tutoring available for math help. Graphing Inequalities in One Variable; Graphing Linear Equations; Graphing Linear Inequalities in Two Variables; Graphing Logarithmic Functions;.