Week 18- Top 5 Things I’ve Learned in PreCalculus 11

Graphing:

This year in precalculus, we have mainly been looking at quadratic function graphs and how the different forms give different information to graph the most accurate parabola.

General form:

  • y-intercept
  • slope

Factored form: a (x-x1) (x-x2)

  • x-intercepts
  • axis of symmetry

Completing the square (standard): {y=a(x-p)^2}+q

  • vertex

Absolute value function: 

Before I had an idea of what an absolute value function was, but learning and seeing it on a graph gave me a deeper understanding of absolute values.

I learned that a linear absolute value looks like a V and a quadratic function looks like a W. I also learned that the points of inflection are the parts of the graph where the directions change dramatically, and there are two points of infliction in absolute value functions. Lastly, I learned how to write piecewise notation.

Image result for linear absolute value function

Reciprocal function: 

This unit was one of the most interesting units because I knew that one pair of numbers whose product is 1 when multiplied together is reciprocal, but seeing the visual was something completely new, and I had a lot of fun drawing the graphs for these reciprocal functions, which I had no idea how to do at first.

I learned that invariant points are both 1 and -1 reciprocating into the same number. I also learned that an asymptote is a horizontal, vertical, or slanted line that the lines(hyperbola) approach but never touches, that goes through the x-intercepts.

Image result for reciprocal function graph

Rational Expressions: 

Building off of the knowledge from previous years of fractions, we learned about rational expressions.

The main thing that I’ve learned is that it is not possible to simplify the equation by removing a variable by itself and that rational expressions need a Lowest common denominator when adding or subtracting variables. I also learned about non-permissible values, which is the value that makes the denominator of a fraction become a 0, which is undefined.

Trigonometry: 

This year we learned how to find the lengths of sides and the angles of a non right triangle( a triangle that does not consist a 90 degree angle), through the sine, and cosine law.

Sine law can determine an angle or side length with the information of 1 angle and side opposite from each other and another side length or angle. a/sinA = b/sinB = c/sinC

Cosine law can be used when you have 2 sides and an angle or 3 sides. a2 = b2 + c2 – 2bc cos A

Week 17- Trigonometry

This week in pre calculus 11, we studied our last unit on trigonomitry, which was an add on to what we’ve learned and covered in math 10, but unlike math 10, we learned how to find the angles and sides of a triangle without a 90° angle (right triangle), through sine law and cosine law.

Review:

Soh- Sine= opposite/hypotenuse

Cah- Cosine= adjacent/hypotenuse

Toa- Tangent= opposite/ adjacent

Sides of a right triangle: a² + b² = c²

New:

CAST law explains which functions are positive in the 4 quadrants

The special triangles with a 90° angle will give us the ratios

Sine law:

Finding the Sides: a/sinA = b/sinB = c/sinC 

Finding the Angles: sinA/a = sinB/b = sinC/c

Take the given triangle and plug the numbers into the sine law to find the two fractions that give you 2 angles and 1 side or vice versa, and isolate the missing variable.

-Uppercase letters stand for the angles of the triangle

-Lowercase letters stand for the sides of the triangle

-All three angles add up to make 180°

example:

steps)

  • find the missing angle(we can easily find it because we are given 2 angles)
  • take two of the fractions that make up the sine law(one always has to have both the angle and side length)
  • isolate side b

Cosine Law: can be used when you have 2 sides and an angle or 3 sides

a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos B

c2 = a2 + b2 – 2ab cos C

or  

Isolate the missing variable.

example:

Steps)

  • plug in the numbers into the equation
  • simplify the equation
  • isolate cos C
  • use the inverse cosine to fine the angle

Week 16- Solving Rational Equations

This week in pre-calculus, we learned how to solve rational equations. We can cross multiply when there are two fractions on both sides of the equal sign, but when there are more than two fractions, you would have to find the common denominator and multiply each side to get rid of the denominator to solve the equation. In both situations, remember to double check to verify the answer.

Cross Multiplication

steps

  1. cross multiply (top left with the bottom right and vice versa)
  2. isolate the x value
  3. plug it back into the original equation to confirm

More than 2 Fraction

steps

  1. find the common denominator
  2. cancel out the denominators
  3. isolate the x value
  4. verify the solution by plugging it back into the original equation

Week 15- Adding and Subtracting Rational Expressions

This week, we continued with our unit on rational expressions and learned how to add and subtract rational expressions.

Review:

  • Finding the LCD
  • What you do to the bottom of a fraction, you have to do to the top as well
  • Only adding and subtracting the numerators

Steps:

  1. Factor the denominators if possible
  2. Determine the lowest common denominator
  3. Rewrite each fraction as an equivalent fraction with the lowest common denominator
  4. Combine numerators by adding and subtracting
  5. Reduce if possible

  • Remember to write out the non-permissive values
  • Remember that there is a negative coefficient infront of the second fraction, so it would get distributed

Week 14- Multiplying and Dividing Rational Expressions

This week we learned about rational expressions and learned how to multiply and divide them. 

Rational number: a quotient of two integers

Rational expression: quotient of two polynomials

  • does not have an equal sign
  • simplifying to it’s the simplest form

Rational Equation: an equation that contains at least one fraction whose numerator and denominator are polynomials

  • has an equal sign
  • solving for the x value

Non- permissible value

  • values that cause the fraction to have a denominator with a value of (we cannot divide by zero, because it is undefined)

Solving the Rational expression:

  1. factor if possible
  2. look for anything common

  • when simplifying, you cannot remove a variable by it’s self
  • when dividing, reciprocate the number to multiply

Week 13- Reciprocal Functions

This week we learned how to solve absolute value equations, and then we learned about linear and quadratic reciprocal functions. You can find that there are some patterns that exist when trying to graph reciprocal functions.

reciprocal: one pair of numbers whose product is 1 when multiplied together

  • a positive(+) number reciprocates into a positive number
  • a negative(-) number reciprocates into a negative number
  • both 1 and -1 reciprocates into the same number(doesn’t change), which are called the invariant points
  • when 0 is reciprocated it becomes undefined
  • big number reciprocate into smaller numbers
  • smaller numbers reciprocate into bigger numbers

asymptote: horizontal, vertical, or slanted line that the lines(hyperbola) approach but never touches

hyperbola: two curve lines that are similar to each other, but have opposite reciprocal function points

1) Graph the first linear line, which in is case would be y=2x + 5

  • You know the y intercept is 5 and the rise/run is 2/1

2) Notice the where the line intercepts with the x axis, which is where the vertical asymptote of the reciprocal is located

3) Mark the invariant points which are the 1, -1 values on the y axis

4) Graph the Hyperbolas onto the graph based on the recipricals

Week 12- Absolute Value Functions

This week we started our new unit on absolute value and reciprocal functions. We extended our understanding of absolute value functions, and how they would look on a graph as a linear function and a quadratic function.

An absolute value is a distance between a number to zero, therefore the absolute value of a negative number will always be positive.

linear function: y=mx+b

quadratic function: y= a(x-p)^2 + q

When the absolute value symbols are added to the functions, it makes the negative points of y become positive, which changes the parabola.

The x-intercept of a graph is a critical point, also known as the point of inflection. It is where the graph of the function changes direction dramatically.

Image result for linear absolute value function

Piecewise notation: the original part of the graph without the absolute value symbols that stay the same after the symbols are put in are first written, and then the part that changes and reflected through the absolute value symbols are written after.

Linear function

y= |2x – 1|

  • point of inflection: (1.5, 0)
  • y-intercept: 1
  • Domain: XER
  • Range: y ≥ 0
  • piecewise notation: f(x)= {2x-1, x ≥ 1.5, -(2x-1), x< 1.5

Quadratic function (parabola)

y= |2x² – 1|

  • piecewise notation: f(x)= { 2x² – 1, -0.8 ≥ x, x ≥ 0.8, -(2x² – 1), -0.8 < x < 0.8

 

 

Week 11- Solving Quadratic Inequalities With One Variable

This week in pre-calculus 11, we learned how to solve quadratic inequalities and graph them. This was a review from last year’s math.

y=mx+b is the equation used for linear graphs, which are graphs with straight lines that increase or decrease by the same number each time. b is the y-intercept, and m represents the slope of the graph.

A quadratic inequality is just like a quadratic equation, but instead of an equal sign, there’s an equality.

*always flip signs when dividing by a b=negative number

Solving the Inequality:

  1. Factor the inequality
  2. Find the Values of x
  3.  select numbers in between the x values and test it in the inequality

Graphing

*the value of a in y= ax^2 +bx + c, can tell us if the graph opens up or down, which tells us the positive and negative values of the quadratic inequality when the it intercepts with x.

signs:

  • ≥ greater than or equal to has a line, shaded dot
  • ≤ less than or equal to has a line, shaded dot
  • > greater than has a dotted line, unshaded dot
  • <less than has a dotted line, unshaded dot
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