SP #6: Unit K Concept 10 - Writing a Repeated Decimal as a Rational Number Using Geometric Sequence

 

The viewer needs to pay special attention to finding the ratio by dividing the second term by the first. When we reach dividing the two fractions, we need to multiply the reciprocal of the second number in order to cancel. It is crucial to remember the whole number 5 from the beginning of the problem. Add it to the solution by multiplying top and bottom by 99 to get the same denominator and then add the two fractions.

SP #5: Unit J Concept 6 - Partial Fraction Decomposition with Fractions

 

The viewer must pay special attention to setting up the equations, and that they distribute carefully and properly. It is important to remember to set the coefficients of the numerator equal to the term letters of the right side (if there is no number for the term, make sure to add in "0" in the front). Because of our answers being fractions, we cannot use the "rref" function on our calculator.

SP #4: Unit J Concept 5 - Partial Fraction Decomposition with Distinct Factors

 
For part 1, make sure to pay special attention to multiplying out the numerator. Make sure to carefully distribute, this including negatives, otherwise one mistake could change your whole answer.

 

For part 2, the viewer should be careful when writing the equations. It is important to copy correctly and not forget any negative signs.

For part 2 (wasn't enough room), it is also crucial the viewer remembers to cancel out the x's.

For part 3, plug in the coefficients into the calculator. It is crucial you plug in the right numbers or else you will get the wrong answer. It is important to recheck what you plugged in.
                    
For part 4, follow the necessary steps to find the ordered triple. The viewer should be able to follow the steps as stated in the image. The fourth column provides the ordered triple, making them the numerators of the original equation found in part 1.

SP #3: Unit I Concept 1 - Graphing Exponential Functions

The viewer needs to pay attention to "a", which determines whether the graph will be above the asymptote (positive "a") or below the asymptote (negative "a"). We also need to pay attention to solving for the x-intercept. If we get a negative number on one side, we cannot take the natural log of it, making it undefined. This is lead to NO x-intercept for this graph. Another concept to recognize for these problems are the range, which depend on the "a", whether it is above or below the asymptote, and the asymptote itself.

SP #2: Unit E Concept 7 - Graphing Polynomial With Multiplicities

This problem is about graphing a polynomial, which included a y-intercept and x-intercepts (or zeroes) with multiplicities, and end behavior. When a polynomial is given, we first have to factor. With the help of multiplicities- the number of times a zero shows on a graph, it will help us determine how to graph the equation demonstrating how they behave at the extremas and in the middle.

While graphing polynomials, you should pay special attention to the zeroes and their multiplicity. Multiplicities determine how the middle of the graph looks like: multiplicities of one go through the graph, two bounce, and three curve. We also need to pay close attention to the end behavior so we know what direction our graph should start and end at.

SP #1: Unit E Concept 1 - Identifying X-intercepts, Y-intercepts, Vertex (max/min), Axis of Quadratics and Graphing

This problem is about changing an equation in standard form f(x)=ax^2+bx+c into parent function form f(x)=a(x-h)^2+k so that it is easier to graph. With the parent function, it is easier to find and identify the vertex, y-intercept, axis, and x-intercepts. Though it takes several steps to find these parts of the graph, it will result in a more accurate and detailed sketch of the graph.

Some special things to pay attention to would include the (h, k), included in the parent function, which acts as the vertex of the graph. In certain examples, the x-intercepts do not come out with real numbers. With the imaginary numbers, we are not able to graph the points on the graph.