By: Harisha Yasothananthan
What is Quadratics?
Learning Goals for vertex form
- I'm able to label and understand the parts of a parabola.
- I can graph a parabola and an equation.
- I can identify the vertex, y-intercept and x-intercept on a graph or on a equation.
- I understand how to use mapping notation and step pattern.
- I can make connections and think outside of the box for solutions.
Vertex form y=a(x-h)²+k
- Y-intercept- when the graph intercepts the y axis. In each parabola graphed, there is always a y intercept crossing the graph.
- X-intercept- when you let y=0. when a curve goes through the x intercept of the graph. For example (0,2) or (4,0)
- Vertex- where the point of the parabola crosses the axis of symmetry. If the coefficient of x² is positive, the vertex of the parabola will be pointing down on the graph.
- Axis of Symmetry- The axis of symmetry has a vertical line that passes through the vertex of the parabola, which cutting the parabola in half. The x- intercept will always have the same distance from the axis of symmetry.
What do we need to know about vertex form?
- Vertex- the point of the parabola that passes through the axis of symmetry. To find the vertex of the parabola, you just need to look at the highest point on the graph. Also, you can find the vertex by the equation y=a(x-h)²+k. To find the vertex in the equation, you need to know the "h" and "k". The "h" can be changed easily by it becoming positive or negative. For example (h,k) [(3,7) (-3,7)]. The positive 3 will automatically change into negative 3.
- Y-intercept- substituting x=0 in the equation given
- X-intercept- substituting y=0 in the equation given
Graphing an Equation
Example using vertex form
Vertex- (h,k) (5,-7)
X-intercept- (when y=0) x=-39 (-39,0)
Y-intercept- (when x=0) y=-57 (0,-57)
Find the equation for the parabola with vertex (2,6) that passes through the point (5,3).
y=a(x-2)²+6 (plugging in vertex into the equation)
3=a(5-2)²+6 (plugging in coordinates that passes through the point)
3=a(3)²+6 (solving what's in the bracket first and squaring it)
3=a(9)+6 (multiplying 9 and "a" )
3=9a+6 (bring 9a to the other side of the equal sign)
-9a=6-3 (solving for what 6-3 is)
-9a=3 (dividing -9 by 3)
a=3/-9 (solving for "a" and simplifying 3/-9 is)
a=-1/3 ( put the value that "a" represents into the equation we made)
The "h" means that it moves the parabola left or right of the graph. Also known as horizontal shift. But when the "h" is negative, it moves to the right and when the "h" is positive, it moves to the left.
The "k" represents vertical shifts of the parabola on the graph (up and down).
The "a" shows that it controls the graph by it indicating if it opens up or down. Also, if its narrow or compressed.
example using mapping notation
It starts of at (0,0) and continues in a pattern. For example you can start at -3 squared to get 9 and etc...
Also known as you going over 1 and up 1, over 2 and up 4, over 3 and up 9 and etc.
learning goals for factored form
- I'm able to factor different expression in 6 different ways.
- I know all 6 types of factors.
- I can make up a question and factor accordingly.
- I understand how to make and solve equations.
Factored form y=ax^2+bx+c
Types of factores
There are 6 different ways to factor a quadratic equation. For example Common Factoring, Factor by Grouping, Simple Trinomials, Complex Trinomials, Difference of Squares and Perfect Squares.
expand and simplify trinomials
=2(4x+3) - (if you divide the 8x by 2, you get 4x and if u divide 6 by 2, you get 3)
TO CHECK : YOU DO DISTRIBUTIVE PROPERTIES
FINAL ANSWER= 8X+6
FACTOR BY GROUPING
=x(x-4)-4(x-4) - divide the first term by (x)
=x(x-4)-4(x-4) - divide the second term by (4)
=(x-4) (x-4) - both brackets have the same terms, (x-4) and outside the brackets, if you divide it by (x-4), you get (x-4)
=(x-4)² - if you simplify it, you get (x-4)²
TO CHECK: DISTRIBUTIVE PROPERTIES:
double the brackets (x-4) (x-4) and multiply the brackets.
Example 2: gh+fd+fh+gd
=gh+gd+fh+fd - collect like terms
=g(h+d) +f(h+d) - divide the 2 terms by g and f and you get (h+d) in both brackets
=(h+d) (g+f) -divide the terms you get (h+d) and (g+f)
TO CHECK: DISTRIBUTIVE PROPERTIES:
multiply (h+d) (g+f) and you will get gh+gd+fh+fd
difference of squares
special cases: Perfect square trinomials
A rectangle has an area of x²-2x-35. What are the possible dimensions of the rectangle? Use factoring.
x²-2x-35 p= (1) (-35) =-35
-7+5=-2 and -7*5=-35
= x²-7x+5x-35 ( factor by grouping)
=(x-7) (x+5) (Final Answer)
axis of symmetry
learning goals for standard form
2) I understand how to solve by using quadratic formula and discriminant
3) I can identify the y-intercept
4) I can change standard form into vertex form
standard form y=ax²+bx+c
"a"- gives you the shape and direction of opening of the graph
"c"- states the y-intercept
Also, standard form indicates the maximum and minimum of the graph by completing the square. By using completing the square, you can change standard form into vertex form and ind the vertex of the graph.
completing the square
steps on how to solve by completing the square
steps + examples on completing the square
example of quadratic formula
EXAMPLEs of quadratic formula
word problem- quadratic formula
word problems- completing by square
Learning quadratics was interesting and different because I learned something that will help me in grades 11 and 12. I found that Unit 2 of quadratics was more intriguing. I learned how to common factor, factor by grouping, about simple trinomials, complex trinomials, perfect squares and the difference of squares. I understood it and I gained knowledge and understanding throughout Unit 1, 2 and 3 of quadratics. The unit was very fascinating and educational. In my opinion, the easiest aspect of this unit was finding the x intercept, finding the vertex, and graphing. The hardest or more complex aspect was completing the square. Although, the unit had its ups and downs, the test on factored form seemed straight forward because I have practiced and showed my dedication to my work. An example, is that I did poorly on finding the expression of a shaded area. I feel like my strengths on the test was in solving equations and expressions and maximum and minimum word problems. While I had my strengths on the test, my weakness was expanding and explaining my thoughts on questions that were difficult. Overall, quadratics was an interesting and exciting unit that I have learned and I think that this would be helpful in the future with my pathway and career.