1. A review of graphing lines, finding slope, and finding equations of lines from 2 points and a perpendicular line.
2. Review how to multiply and Factor all types of Polynomials. I cover Common Factoring, Factor by grouping, Trinomial factoring, and difference and sum of cubes!
3. I show you how to solve all types of linear equations from basic linear equations to more complicated ones with fractions and variables on both sides. There are also the ones that end up with no solutions and all reals as a solution.
4. The first step into learning how to solve a quadratic is by factoring. Here I show you how and explain why.
5. In addition we will look at the domains and restrictions of Rational Expressions.
6. How to Add and Subtract Rational Expressions. I show you how to find common denominators so you can simplify.
7. How to simplify Complex Fractions. I show you two different methods.
The systems in this section will consist of two linear equations and two unknowns Given linear equations, we are asked to find out if they have simultaneous solutions. In other words, where do the two lines intersect? This question brings up three cases:
Most of the time the linear system will have a common point, (x, y). The point where they cross is the solution to the system.
However, not all linear systems have on ordered pair solution; some have no common points and others have infinitely many. Imagine that you were asked to solve the system consisting of two parallel lines, where do they cross? In this case, there is no simultaneous solution and the system of two parallel lines is inconsistent. In the case where the system consists of two lines that happen to be the same line, there are infinitely many common points. This system is dependent and solutions can be presented in the form (x, y) where x can be any real number and y = mx + b.
Solve the systems given the graph:
Is the ordered pair a solution to the system?
The next example shows how to find the common point, the point where the two lines intersect, if we are given a linear system in standard form.
Solve the system:
Step 1: Place the linear equations in slope-intercept form.
Step 2: Graph the lines and use the graph to find the common point.
Step 3: Check your answer and present it as an ordered pair.
Accuracy here is important, use graph paper and a straight edge when using the graphing method to solve linear systems.
Solve the system using the graphing method:
Graph the lines using slope-intercept form.
Solve the system using the graphing method:
Graph the lines using slope-intercept form.
Take extra care with the scale when graphing to find the intersection. Make sure all your tick marks are equal in size. This will make your graphs more accurate and easier to read. Check to see that your answer works for BOTH equations.
Solve the system using the graphing method:
Place lines in slope-intercept form.
And then graph them.
Solve the system using the graphing method:
Place lines in slope-intercept form and then graph them.
Dependent systems seem to give beginning algebra students the most trouble. Remember that we are looking for points where the two lines intersect. If the lines are the same, well then they will cross at infinitely many points. Because of this we have to use special notation to indicate an infinite set. Notice that we have already put the line in y-intercept form, y = mx + b, so it is not a big leap to write the set final ordered pair solutions in the form (x, mx + b).
You might see different notation in other texts such as,
One way to graph logarithmic functions is to first graph its inverse exponential. Then use your knowledge about the symmetry of inverses to graph the logarithm.
Recall that inverses are symmetric about the line y = x. For example,
Step 1: Find some points on the exponential f(x). The more points we plot the better the graph will look.
Step 2: Switch the x and y values to obtain points on the inverse.
Step 3: Determine the asymptote.
In practice, we use a combination of techniques to graph logarithms. We can use our knowledge of transformations, techniques for finding intercepts, and symmetry to find as many points as we can to make these graphs. General guidelines follow:
1. Graph the vertical asymptote. All logarithmic functions of the form
have a vertical asymptote at x = h.
2. Find the x- and y-intercepts if they exist. To find x-intercepts set y = f(x) to zero and to find y-intercepts set x = 0.
3. Plot a few more points and graph it.
Graph the following logarithmic functions. State the domain and range.
In the previous solved problem, make a note of the rigid transformations. If we start with the basic graph y = log(x) then the first translation is a shift to the left 3 units y = log(x+3). Next we see a vertical shift up 2 units y = log(x+3)+2 .
In the above problem there was a reflection about the x-axis as well as a shift to the left 3 units.
