Week 10 – Precalc 11

This week in Precalculus 11, we started the Graphing Inequalities & Systems of Equations unit. We first learned about solving quadratic inequalities in one variable.

quadratic inequality: an expression with degree 2 that is not equal to 0

It can be written in general form as ax^2 + bx + c >/</≥/≤ 0

To solve a quadratic inequality, we first rearrange the inequality to general form. We then write and solve its corresponding quadratic equation. We label a number line with the roots. We then test a number from each interval by substituting it into the original inequality.

Example:

This was new to me, as I had learned how to solve linear inequalities, but not quadratic inequalities.

Week 9 – Precalc 11

This week in Precalculus 11, we learned about equivalent forms of a quadratic function.

quadratic function:

x-intercept: point on a graph crossing the x-axis

root: x-value to an equation when solved

y-intercept: point on a graph crossing the y-axis

vertex: maximum (opening down) or minimum (opening up) point on a parabola

A quadratic function has 3 forms: factored, general, and standard.

factored form: y = a(x – x_1)(x – x_2)

  • x_1 & x_2 are the graph’s x-intercepts, or the equation’s roots

general form: y = ax^2 + bx + c

  • c is the graph’s y-intercept

standard form: y = a(x - p)^2 + q

  • p is the vertex’s x-value
  • q is the vertex’s y-value

For all 3 forms, a determines whether the parabola opens up (positive) or down (negative) and its stretch (a>1) or compression (a<1) value.

By converting from one form to another, we can determine all elements of the equation.

Week 8 – Precalc 11

This week in Precalculus 11, we started the Analyzing Quadratic Functions unit. We first learned about the properties of a quadratic function.

quadratic function: function that can be written in general form

general form: y = ax^2 + bx + c, where a ≠ 0

parabola: the curve of a quadratic function’s graph

vertex: a parabola’s highest (minimum) or lowest (maximum) point (if the coefficient of x^2 is positive, the vertex opens up. If it is negative, the vertex opens down)

axis of symmetry: intersects parabola at its vertex

domain: all possible x values

range: all possible y values

x-intercepts: where the parabola touches the x axis, the roots of the quadratic equation

y-intercepts: where the parabola touches the y axis.

 

Example: y = 2x^2 + 8x + 6

x: -5, -4, -3, -2, -1, 0

y: 16, 6, 0, -2, 0, 6

vertex: (-2, -2) minimum/opens up

x-intercept: (-1, 0) & (-3, 0)

y-intercept: (0, 6)

axis of symmetry: x = -2

D: {x∈R}

R: {y ≥ -2}

 

I had not learned about many of these terms before and found it interesting that you could find so much information from the vertex.

Week 7 – Precalc 11

This week in Precalculus 11, we learned about interpreting the discriminant.

Discriminant/Radicand: the number underneath a radical sign (√)

Real root: square root of a positive number

In the quadratic formula, the discriminant is b^2 – 4ac. By solving the discriminant, we can indicate how many real roots the equation has.

If b^2 – 4ac > 0, the equation has 2 real roots.

Example: 5x^2 – 9x + 4 = 0

b^2 – 4ac

= (-9)^2 – 4(5)(4)

= 81 – 20 · 4

= 81 – 80

= 1 -> 2 real roots

If b^2 – 4ac = 0, the equation has 1 real root.

Example: 2x^2 + 16x + 32 = 0

b^2 – 4ac

= (16)^2 – 4(2)(32)

= 256 – 8 · 32

= 256 – 256

= 0 -> 1 real root

If b^2 – 4ac < 0, the equation has 0 real roots.

Example: 6x^2 + 7 = 0

b^2 – 4ac

= (0)^2 – 4(6)(7)

= 0 – 24 · 7

= 0 – 168

= -168 -> 0 real roots

By interpreting the discriminant beforehand, we can determine whether or not to bother solving a quadratic equation.

 

 

Week 6 – Precalc 11

This week in Precalculus 11, we learned about solving quadratic equations by factoring.

Solving: finding a variable’s value

Quadratic equation: an equation with a degree of 2

Degree: sum of the variables’ exponents in a term

Factoring: separating an expression into its components

Variable: an unknown number represented by a letter

The zero-product property is one strategy to solve quadratic equations. If the product of two numbers is 0, than one or both of the numbers have a value of 0. To solve a quadratic equation using this strategy, we first move all terms to one side of the equation, so the other has a value of 0. 

Example:

x^2 – 6x = 27

x^2 – 6x – 27 = 0

We then factor the expression.

(x – 9)(x + 3) = 0

Then we make one of the components have a value of zero and solve for the variable.

x – 9 = 0

x = 0 + 9

x = 9

OR

x + 3 = 0

x = 0 – 3

x = -3

We then verify the solutions by replacing x with the solutions in the original equation.

x^2 – 6x = 27

(9)^2 – 6(9) = 27

81 – 54 = 27

27 = 27

OR

x^2 – 6x = 27

(-3)^2 – 6(-3) = 27

9 + 18 = 27

27 = 27

This is an interesting new way to solve equations, especially since I did not know a variable could have more than one solution. 

 

 

What Darwin Never Knew

How did the discovery of DNA prove that Darwin’s theory of evolution was correct and how does it change the way we view evolution today and into the future?

 

Michael Nackman of the University of Arizona studied how some of the Pincate Desert’s pocket rock mice evolved to blend in with volcanic rock. A mutation in the mouse’s DNA changed its fur colour from light to dark. Now, rock pocket mice exist with light or dark fur.

 https://uanews.arizona.edu/story/coats-different-color-desert-mice-offer-new-lessons-survival-fittest

Sean Caroll studied why one fruit fly species has spots on its wings, but another does not when the two species both have the spot-coding gene. One DNA portion differed in the spotted fly. Injecting this DNA portion into the non-spotted fly resulted in spotted wings. These DNA portions are called switches, and they activate and deactivate genes.

 https://hort.uwex.edu/articles/spotted-wing-drosophila/

David Kingsley and Dolph Schluter studied why the ocean stickleback has spikes on its belly, but the lake stickleback does not. A mutation in the lake stickleback’s DNA broke the switch that activates the spike-coding gene. The gene that codes for the stickleback’s spikes also codes for hind legs in other animals.

 

https://learn.genetics.utah.edu/content/selection/stickleback/

Arkhat Abzhanov and Cliff Tabin studied how the Galapagos finches have different beaks by looking at the birds’ embryos. The birds had the same beak-coding genes, but they differed by how much and when the switch activated the gene.

https://en.m.wikipedia.org/wiki/Darwin%27s_finches

Neil Shubin of the University of Chicago studied how fish evolved to walk on land. He found a fossil of a flat-headed fish with leg-like fins. They had the same bone structure as the limbs of all other four-legged animals. In the paddlefish, a relative of the fossil Tiktaalik, the same genes code for fins as they code for limbs in humans. A few mutations changed the ancient fish’s fins to limbs.

https://phys.org/news/2014-01-discovery-tiktaalik-roseae-fossils-reveals.html

Through the study of DNA, scientists have discovered that mutations in the genome cause evolutionary change. Switches in an organism’s DNA activate and deactivate hox genes that code for physical features. Mutations can affect these switches, therefore affecting the organism’s physical features. This proves that Charles Darwin’s theory of evolution was correct. We can now discover how almost any organism differs from another, and how it evolved and came to be.

 

Sources

• What Darwin Never Knew notes & transcript

 

Week 5 – Precalc 11

This week in Precalculus 11, we started the Solving Quadratic Equations Unit. We first reviewed factoring polynomial expressions.

Factoring: separating an expression into its components

Polynomial expression: an expression of numbers and variables being added, subtracted or multiplied (example: 2x)

Greatest common factor: the greatest number that can divide into all the terms in the expression (example: 2, 4, 6, 8, 10      GCF = 2)

Term: a number or a group of numbers being multiplied (example: 2x^2)

Binomial: a polynomial expression with 2 terms (example: x + 1)

Difference of squares: a polynomial expression in which subtraction takes place between 2 perfect square terms (example: 4x – 1)

Trinomial: a polynomial expression with 3 terms (example: x^2 + x + 1)

Conjugates: 2 terms with opposite addition/subtraction signs (example: 1 + 1 & 1 – 1)

To factor a polynomial expression, we first look for the greatest common factor between the terms and divide each term by that number.

Example:

2x + 2

= 2(x + 1)

If the expression is a binomial, we check if it is a difference of squares. When factored, a difference of squares results in conjugates.

Example:

4x – 1

= (2x + 1)(2x – 1)

If the expression is a trinomial, we check if it is in the form ax^2 + bx + c. If a = 1, we separate bx into 2 terms that multiply to c and add to bx. 

Example:

x^2 + 2x + 1

= (x + 1)(x + 1)

= (x + 1)^2

If a ≠ 1, we separate bx into 2 terms that multiply to ac and add to bx, then find the greatest common factor of each side, and divide each term by that number.

Example:

2x^2 + 4x + 2

= 2x^2 + 2x + 2x + 2

= 2x(x + 1) + 2(x + 1)

= (2x + 2)(x + 1)

If the expression is in a different form than ax^2 + bx + c, the expression can sometimes be changed to this form.

Example:

(x + 1)^2 + 2(x + 1) + 1

= a^2 2a + 1

= (a + 1)(a + 1)

= (a + 1)^2

= [(x + 1) + 1]^2

= (x + 2)^2

Week 4 – Precalc 11

This week in Precalculus 11, we learned about adding and subtracting radical expressions.

Radical expression: the term a\sqrt[b]{c}

Radical: a root sign (represented by √)

Radicand: the number underneath the radical

Coefficient: the number outside the radical, it is being multiplied by the radical

When adding or subtracting radical expressions, the radicals (the radicand and the root) must be like terms. Some radicals can be simplified to achieve like terms. The coefficients are then added or subtracted, and the radical remains the same.

Example:

\sqrt{2} + \sqrt{8}

= \sqrt{2} + \sqrt{4}\sqrt{2}

= \sqrt{2} + 2\sqrt{2}

= 3\sqrt{2}

 

Week 3 – Precalc 11

This week in Precalculus 11, we started the Absolute Value and Radicals unit. We first learned about the absolute value of a real number.

Absolute value of a real number: its distance from zero on a number line/the principal square root of its square

Principal square root: the positive (+) square root 

Example: \sqrt{1} = ±1     +1 = principal square root     -1 = negative (-) square root

The symbols “||” represent absolute value. 

Example: <-0-1-2-3-4-5->     |0| = 0

|0|

= \sqrt{0^2}

= 0 

This is a new concept I had not yet learned about and can add to my knowledge of Pre-Calculus.

 

Week 2 – Precalc 11

This week in Precalculus 11, we learned about geometric sequences.

Geometric sequence: a list of terms in which each term is multiplied by a common ratio to equal the next term 

Term: a number (represented by “t_n“, “n” being the term’s place in the geometric sequence)

Common ratio: a common number multiplied by each term to equal the next term (represented by “r”)

Example: 1, 2, 4, 8, 16     r = 2

To find the value of a term in a geometric sequence, we use the formula t_n = t_1 · r^{n - 1}.

Example: Find t_{10}: 1, 2, 4, 8, 16

t_{10} = ?     t_1 = 1     r = 2     n = 10

t_n = t_1 · r^{n - 1}

(t_{10}) = (1)(2)^{(10) - 1}

t_{10} = 1 · 2^9

t_{10} = 1 · 512

t_{10} = 512

This method of finding the value of terms in an geometric sequence is faster than extending the list.