This week in math, we learned more about quadratic functions and how to find the x intercepts through factored form. We learned through general form we can find the y intercept and through standard form we can find the vertex and the axis of symmetry. Taking this information, we can take the general form of the equation and convert it into either standard form or factored form to find the information that we need.
Standard form(completing the square): +
Factored form: = () ()
Ex) General form to factored and standard form
= 2 -14 + 16
- y-intercept: 16
- opens up, minimum value
- congruent: =2
= (2-16) (+1)
- x-intercepts: (8, 0) (-1, 0)
- axis of symmetry(x) = 3.5
= 2 -14 + 16
= 2( -7+49/4 -49/4)+ 16
= 2( -7+49/4)-98/4+ 64/2
= 2( -7/2)2 -34/4
= 2( -7/2)2 -17/4
- vertex (3.5, 8.5)
This week, we learned how to solve quadratic equations by factoring, completing the square, and using the quadratic formula.
Factoring and using the zero product property is one of the easiest ways to solve a quadratic formula, and it works with rational roots.
- A quadratic equation is any equation that can be written in the form a + bx + c= 0.
- Zero product property: if ab=0, then a=0 and b=0, or both a, b=0
3 – 21x- 54= 0
- Factor, remembering CDPEU
3( – 7x- 18)= 0
- Use the zero product property
(x+2) (x-9)= 0
Either x+2= 0 or x-9= 0
x= -2, x= 9
Completing the Square
When the equation doesn’t factor, completing the square is a great way to solve the quadratic equation, and it works well with irrational roots.
- Rearrange the equation if the coefficient of is negative
- Make sure to get rid of the coefficient of by dividing each side by the coefficient (when the equation equals to 0)
- Move the constant term to the right of the equation
- complete the perfect square by dividing the 2nd term and multiplying and squaring it, and adding it to both sides of the equation
- factor the perfect square, and simplify the right side
- rake the square root of each side, and isolate the x
12x= -2 + 54
2 + 12x= 54
+ 6x= 27
+ 6x + 9= 27+ 9
√ = √36
x= 3 6
The Quadratic formula is easy to use if it is arranged in the right form.
- Quadratic formula:
- Identify the values of a,b, and c( the coefficients) a + bx+ c= 0
- Plug a,b, and c in the formula and solve the equation
This week in pre-calculus 11, we reviewed and extended our understanding of factoring polynomials.
To remember the steps of factoring polynomials is simple. Just remember can divers pee easily underwater:)
Common GCF (remainder)
Difference of squares
Easy (when the variable doesn’t have a coefficient)
Ugly (if the variable has a coefficient)
-17, -11, -5, 1, 7…
= + (n- 1)d
= -17+ (50- 1)6
= -17+ (49)6
= -17+ 294
Sn= (2 + (n-1)d)
In the poem, “I Know Why the Caged Bird Sings,” by Maya Angelou, contains both a connotative and denotative meaning. The literal poem is about a caged bird that sings, because its voice is the only thing left that it has due to all the restrictions. The caged bird cries to get someone’s attention and recognition, while the free bird gets to fly and do what it pleases to, but the caged bird never gives up and keeps singing for the hope of being free. The connotative behind the poem is actually about the black slaves in the United States, during the civil rights movements. The caged bird represents the African Americans that were lacking human rights at the time, while the free bird represents the Caucasians. It is saying that the Black Americans are struggling to get their point across the White people who aren’t recognizing their problems, but the Black Americans don’t stop using their voice to make a change to their unfair situation. The title is significant, because Maya Angelou was a African American women, who experienced the racism and understood the feeling of being restricted. The whole poem is basically a metaphor, because it contains a different connotative meaning behind it. End rhyme is seen in the poem in the parts “still” and “hill”. Near rhyme is also used im “breeze” and “trees”. Personification is present when the author describes the caged bird’s shadow shouts on a nightmare. It is personification because a shadow doesn’t shout. This poem contains a deeper meaning that can help us reflect on the historical mistakes that we’ve made in the past, and understand how the Black Americans felt on a more personal level.
This week in math we started to learn about relations and functions. We reviewed linear relations from grade 9. And we learned how to find the intercepts of x and y on a graph.
To find the y intercept you would multiply x by 0, and devide both sides by 7 to single out the y.
With the two answers you are now ready to draw the graph.
Example 2) Find the possible 2 integers that equal to 24 when multiplied together. Pick the pair of integers that equal to 11 when added together. Make sure to see if the integers have to be possitive or negative depending on the middle and last number (11, 24). If there is no pair that equals to the middle number when added together, it can’t be factored. The integers can be written in any