Week 4-6
What are the five steps for problem solving?
1. Familiarize yourself with the problem situation 2. Translate the problem into an equation 3. Solve the equation 4. Check the answer in the original problem 5. State the answer to the problem clearly
What is a graph of an equation?
A graph of an equation is a drawing that represents all of is solution pairs. Meaning, all the points that make the equation true are represented by the drawing. This can take the form of many recognizable patterns like lines, parabolas, and other geometric shapes.
What is the difference between solving an equation and solving an inequality?
Equations provide you with the opportunity to solve for missing quantities that will make the equation balanced or equal on both sides. Inequalities allow you to solve for sets (or ranges) of numbers that are larger or smaller than another quantity. Inequalities and equations can be solved similarly, except you have to account for negatives.
Do graphs of linear equations have to have x and y intercepts?
No. Vertical lines (with 1 exception) only have x intercepts. Horizontal lines (with 1 exception) only have y intercepts.
What does a basic graph of an inequality look like? More importantly, what does it mean?
Notice that you mark the starting point with either a filled in circle or an open circle. The filled in circle signifies that the solutions include the starting number, while the open circle means that the starting point is a boundary that you do not include as a solution. The graph gives you a picture of all the numbers that will make the inequality true.
What does a positive slope mean compared to a negative slope?
Positive slope is when the x and y values are either both increasing or both decreasing. A negative slope indicates that the x and y values are changing differently (one increasing while the other is decreasing or vice versa).
Why represent the slope as a fraction instead of a decimal?
Slope, represented as a fraction is really just tradition. But, it does tell you how much the graph changes left to right (x) and how much it changes up an down (y). It is valid to represent it as a decimal, you just cannot as easily get useful information from this representation.
What is point-slope form for linear equations?
The point slope form comes from solving the slope formula for y-y1. It will help you find the equation of a line if you are given a point on the line and the slope of the line.
What is the slope, y-intercept, and x-intercept of a line?
The slope is (by tradition) given in fraction form where the top shows how far up or down you travel from point to point, and the bottom of the fraction represents how far left or right from point to point you travel. The y-intercept is where the graph of the line crosses the y-axis. The x-intercept is where the graph of the line crosses the x-axis.
What is the slope formula? Better question, what does slope mean?
The slope is really just a ratio, or a quickly referenced comparison between the changes in y (up-down) an the changes in x (left-right).
What does the graph of a linear equation look like?
The straight lines represent all of the points that make the specific equation true. For example, on the blue line (0,5) is a point on that line which when plugged in 5 = -0 + 5 or 5=5 is true.
When and why was the cartesian plane (x-y axis) invented? What does it tell you and how does it work?
There is much debate about the actual origin of the cartesian plane, though it is named for René Descartes. Many believe that the current 2-axis (x, y) coordinate system wasn't invented until later as a means of making sense of what René Descartes wrote in his book La Géométrie. The basic idea is to show location by using a single point and then using signed numbers (+ and -) as directions.
What does a circle graph or pie chart look like? What are they best used for?
These are best used for displaying the percentages of smaller categories compared to the whole in a manner that is appealing and enables quick size comparisons and references.
What is the difference between set notation (interval notation) and graphing inequalities?
These are just two different ways of representing the data. One is a visual while the other is written. Both valid for providing solutions for the given inequality.
What is the standard form of a linear equation?
This form is just a way of organizing the various parts of a linear equation. For this form, you have all of the variables on one side equal to the constant of the equation.
What is the slope-intercept form for linear equations?
This is a form that linear two variable equations can be placed into. When set equal to the dependent variable (y) you can easily get valuable information like the slope (m) and the y-intercept (b). This makes for quick and efficient graphing by hand.
Why do we need all these different forms for linear equations?
When you are looking for specific information, or you are given specific information, it is better to use the various forms. You really don't need all of them, but to be able to transfer between the various forms will help you do certain problems more quickly if you are interested in efficiency.
What is a major difference between solving inequalities compared to solving equations?
You can do many if not all of the operations for solving inequalities that you are used to from solving equations. The one catch is that inequalities produce an error when concerning negative numbers. Due to this error, it is up to the operator (you) to manually correct the error each time you either divide or multiply by a negative number.
What does it mean to have a slope that is undefined and a slope that is zero?
Zero slope is a horizontal line. You get a slope of zero when the change in y values is equal to zero, or the top of the slope fraction is zero. Undefined slope is a vertical line. You get a slope of undefined when the change in the x values is equal to zero, or the bottom of a fraction is equal to zero. This is termed undefined, because dividing by zero is not possible in terms of mathematical operations.