UNIT 1 - Linear
Systems
Systems
If the sum of John's age and twice Peter's age is fifty, what is that as an equation?
J + 2P = 50
Is the following linear, quadratic or neither?
y=4x2 - 2x
The graph of y=x2 is translated 3 units to the left. The equation for this new graph is?
y=(x+3)2
What are the three ways to prove similarity?
1. If the ratios of corresponding sides are equal the triangles are similar.
2. If two corresponding angles are equal
3. If the ratios of two corresponding sides are equal and 1 pair of corresponding angles are equal.
What is the median of a triangle?
Line segement that joins a vertex to the midpoint of the opposite side
What is the number of solutions found in the linear system: y = 2x + 4 and 2y = 8 + 4x
Infinite Solutions
The graph of y=x2 is translated 1 unit to the right. What is the equation of the new graph?
y=(x-1)2
When the equation of a quadratic function is in factored form, which features in most easily determined?
X-intercept
When do you use Sine Law?
Use Sine Law for non-right triangles when you have a side-angle pair
What is the altitude of a triangle?
It is a line segment that passes through a vertex and is perpendicular to the opposite side
Solve the following linear system by substitution. Show your work.
2x + y = 4
4x - y = 2
y=-2x+4
4x-y=2
4x-1(-2x+4)=2
4x+2x-4=2
6x/6=6/6
x=1
2x+y=4
2(1)+y=4
2+y=4
y=2
POI is (1,2)
The graph of y=x2 is vertically stretched by a factor of 2. The equation for this new graph is?
y=2x2
Describe the transformations that must be applied to the graph of y=x2 to obtain the graph of y=-2(x+3)2+6
1. Reflect in the x-axis
2. VS by a factor of 2
3. HT 3 units left
4. VT 6 units up
What are the primary TRIG Ratios?
SOH CAH TOA
SIN COS TAN
What is the length of a line segmnet formula?
d = √(x2-x1)2 + (y2-y1)2
Solve the following linear equation by elimination.
1. 5x+2y=-11
2. 3x+2y=-9
5x+2y=-11
- 3x+2y=-9
2x/2 = -2/2
x=-1
3(-1) + 2y = -9
2y/2 = -6/2
y = -3 POI is (-1,-3)
Factor Fully:
x(a-3) + 4(a-3)
Ans:
(x+4) (a-3)
Complete the square:
y = x2-6x-1
y=(x2-6x)-1
=(x2-6x+9)-9-1
=(x-3)2-10
What is Sine Law?
a/SIN A = b/SIN B = c/SIN C
Using the formula determine the length of the line segemnent formed:
A(4,-3), B(-8,-5)
d = (-8-4)2 + (-5+3)2
d = (-12)2 + (-2)2
d = √144 + 4
d = √148
d = 12.17
Shyam wisely decides to save some money after working over the summer. He has a total of $2000 to invest and decides to put some money at RBC which offers 2% interest and the rest of his money at TD which offers 3% interest. If the total amount of interest Darshan earns from both banks is a combined $55, how much does Darshan put into each bank?
Let x be the amount of money in RBC. Let y be the amount of money in TD.
2x+2y=4000
- 2x+3y=5500
- 1y/-1=-1500/-1
y = 1500
x+y=2000
x+1500=2000
x = 500
Ans: He invested $1500 in TD and $500 in Royal Bank.
Solve this equation by using the Quadratic formula:
0=2x2+8x-5
x1=-8+√124/6
x2=-8-√124/6
Write down the key features of this parabola:
y=0.5(x-2)2+3
Vertex: (2,3)
Axis of Symmetry: x=2
Direction of Opening: up
Max/Min Value: min=3
Domain: {x∈R}
Range: {y∈R|y≥3}
Sarah and Rachel stood at opposite ends of Mitchell's main street about 950m apart. A hot air balloon hovered over the main street between them, Rachel measured an angle of elevation of 72 to the balloon while Sarah measured a 59 angle of elevation. How high about Main Street was the balloon?
2232.2 meters
Find the euqation of the altitude from vertex C in the triangle ABC, defined by the following points: A(0,3), B(6,0) and C(4,6)
MAB = 0-3/6-0
MAB = -1/2
MCD = 2
Finding B:
Point: (4,6)
y = mx + b
6 = 2(4) + b
-2 = b
Therfore y = 2x -2