# A is for Algebra

### How to complete the square and graph rationals

## Completing the Square

## About:

Completing the square is a process used to solve for x in a quadratic equation (ax^2 + bx + c = 0). The process is in many cases, quicker than using the quadratic formula. It is especially useful when the cross and box method just doesn't work out (for example, no two integers can both multiply into the top number of the cross, and add into the bottom number).

## Steps:

- Move the constant to the right.
- Divide by the coefficient if it isn't 1.
- Half the B term, then square it ; add this to both sides of the equation.
- Write the equation as a perfect square on the left.
- Solve.

## Step 1 The constant is moved to the right (10 is subtracted on both sides). | ## Step 2 The coefficient was not 1, so we divided the entire equation by it (everything was divided by 5). | ## Step 3 The B term (4) was divided by 2. From this, it was squared, resulting in 4. This was added to both sides of the equation. |

## Step 2

The coefficient was not 1, so we divided the entire equation by it (everything was divided by 5).

## Step 4 The equation was then written as a perfect square (on the left). (x+2)^2 is still x^2 + 4x +4. | ## Step 5 This was then solved by square rooting each side to get x+2 alone. From there, 2 was subtracted from each side to get x alone. | ## Answer The answer is x = -2 + root3 and x = -2 - root3 |

## Graphing Rationals

## Background information:

The parent

*rational function*is**f(x) = (1/x)**. This is known as a reciprocal function. A*transformation function*is**f(x) = (a / (x - h)) + k**.## Example #1:

## Given We are given a transformation function to graph. | ## Step 1: Their is only one part of this equation that needed to be factored, the denominator. To do this, we found the greatest common factor, which was 5. 5(x+3) = 5x+15. | ## Step 2: Here, the denominator was solved, giving us our vertical asymptote at x=-3. From this, we know that our graph will never touch this line. As a visual, a dotted line was drawn in to show the vertical asymptote. |

## Step 1:

Their is only one part of this equation that needed to be factored, the denominator. To do this, we found the greatest common factor, which was 5.

**5(x+3) = 5x+15**. ## Step 3: In this equation, because we are given the K value of -2, we know that the horizontal asymptote is y=-2. Once again, the graph will not touch this line, so as a visual, a dotted line was drawn in at y=-2 to show the asymptote. | ## Step 4: Knowing that from the parent function (shown to right), the graph passes through the point 1 left and 1 down, as well as the point 1 right and 1 up from the center of the graph, these points were drawn in. From there, the lines of the graph were drawn in, never touching the horizontal or vertical asymptotes, but passing through the points made. | ## Parent function From this function, y = 1/x, we can see that when the asymptotes are y=0 and x=0, the graph passes through points (1,1), and (-1,-1). |

## Step 3:

In this equation, because we are given the K value of -2, we know that the horizontal asymptote is y=-2. Once again, the graph will not touch this line, so as a visual, a dotted line was drawn in at y=-2 to show the asymptote.

## Step 4:

Knowing that from the parent function (shown to right), the graph passes through the point 1 left and 1 down, as well as the point 1 right and 1 up from the center of the graph, these points were drawn in. From there, the lines of the graph were drawn in, never touching the horizontal or vertical asymptotes, but passing through the points made.

## Example #2:

## Given We are given the function to graph. | ## Step 1: The only thing factorable is the numerator. Using the cross and box method, we find that (x^2 - 8x +15) = (x=5) (x=3). | ## Step 2: Here we solve for the vertical asymptote, and find that it will be x=-1. This is found by solving for the denominator. |

## Step 3: In looking through the three solutions (as we are not given a K value), we see that there's no horizontal asymptote, as P is greater than Q. | ## Step 4: With no horizontal asymptote, we know that there is a slant. From this, we can conclude that solving the equation for a slant is necessary. To do this, we divide the numerator by the denominator using long devision. If their is a remainder, it is invalid. As shown, the slant equation was solved to be y=x-9. | ## Step 5: Here, the zeros are solved, and the equation is graphed. Upon graphing, dashed lines were drawn to show both the vertical asymptote and the slant. We know that our graph will never touch these lines. Next, the zeros were solved using the numerator, which from step 2 we know was (x-5)(x-3). It was found that the zeros are (5,0) and (3,0), meaning that the graph passes through these points. |

## Step 3:

In looking through the three solutions (as we are not given a K value), we see that there's no horizontal asymptote, as P is greater than Q.

## Step 4:

With no horizontal asymptote, we know that there is a slant. From this, we can conclude that solving the equation for a slant is necessary. To do this, we divide the numerator by the denominator using long devision. If their is a remainder, it is invalid. As shown, the slant equation was solved to be y=x-9.

## Step 5:

Here, the zeros are solved, and the equation is graphed. Upon graphing, dashed lines were drawn to show both the vertical asymptote and the slant. We know that our graph will never touch these lines. Next, the zeros were solved using the numerator, which from step 2 we know was (x-5)(x-3). It was found that the zeros are (5,0) and (3,0), meaning that the graph passes through these points.

## Overview:

- Completing the square is a concept used to factor (and solve for x).
- Completing the square is good to use when the cross and box method doesn't work out.
- A rational function is a reciprocal function.
- The x intercepts in a rational function can be found by solving for X in the numerator.

- If P>Q than there is no x intercept (slant)
- If P<Q than y=0
- If P=Q than y=(3/2)

## Upcoming Quizes and Tests:

Tuesday December 8th: Quiz 4

Thursday December 10th: Test 2

## Citations:

McGraw Hill Education textbook (connectedED)

desmos.com