Linear Systems
What is the point called when two linear equations meet on a graph?
What is: the point of intersection (P.O.I)
What is the arc-looking shape found on a graph when doing quadratic expressions?
What is: A parabola.
What are the three ways to solve a quadratic equation?
What is: factoring, completing the square, and the quadratic formula.
What are the three patterns called to help find similarities between angles in triangles?
What is: the x pattern, f pattern, and z pattern.
Chapter 1: Elimination
Solve the system of linear equations. 3x + y = 19 & 4x - y = 2.
What is: (3,10)
3x + y = 19 / 4x - y = 2
7x = 21 x = 21/7 x = 3
Substitute x = 3 into equation 1 to find the corresponding y-value. 3 x + y = 19
3(3) + y = 19
9 + y = 19
y = 10
What are the three most common ways to solve linear equation problems?
What is: Solving by graphing, solving by substitution, and solving by elimination.
What are the three formulas a quadratic relation can be written in?
What is: standard form, vertex form, and factored form.
What is the quadratic formula?
What is: x = -b +- √(b2 - 4ac) /(2a)
What is the acronym used to describe ways to solve for sides on a right triangle, and what do the letters O, H, and A mean on the triangle?
What is: SOH CAH TOA. and O = Opposite, H = Hypotenuse, and A = Adjacent.
Chapter 4: Quadratic relations
Describe the transitions from y=x2 for each parabola
a) y = x2+2
b) y = -1/3x2
c) y = x2
d) y = (x - 2)2
What is:
a) Vertical shift 2 units up
b) Reflection on x-axis & vertical compression
c) No change
d) Horizontal shift 2 units right
What strategy would you use and explain why?
-5x - 5y = 10
y = -4x -17
substitution because y is already solved for.
Which form does the quadratic formula use, and what does that form look like?
What is: Standard form & (y = ax^2 + bx + c)
True or False? Using the quadratic formula and factoring will give you different results.
What is: False, the quadratic formula is just an easier option for hard trinomials in standard form.
When using the _____ _____, what is the formula to calculate the length of the hypotenuse?
What is: The Pythagorean Theorem & The formula a2+ b2 = c2
Chapter 5: Quadratic expressions
Find each binomial product using F.O.I.L.
a) (x + 2)( x + 5)
b) (x - 2)( x + 4)
c) (3x + 7)( x - 5)
What is:
a) x2 + 5x + 2x + 10 or x2 +7x + 10
b) x2 + 4x - 2x - 8 or x2 + 2x - 8
c) 3x2 - 15x + 7x -35 or 3x2 - 8x - 35
a graph with two linear equations can have _____ solution, _____ solutions, or _____ solutions.
what fills in the blanks and when the graph has that many solutions?
What is: a graph with two linear equations can have One solution, Infinite solutions, or No solutions.
this depends if the graph has the linear equations intersect, parallel, or overlap directly.
If a parabola is represented by the equation y=-2(x+6)2+3, what are the coordinates of the vertex?
What is: (-6,3)
What are the 5 potential features of a parabola?
What is: y-intercepts, x-intercepts, vertex, max or min, and axis of symmetry.
what does the symbol θ stand for, and what are the equations for SOH CAH TOA
What is: Theta,
Sin(θ) = O/H
Cos(θ) = A/H
Tan(θ) = O/A
Chapter 6 quadratic equations
Rewrite y = x2 + 8x + 5 in the form y = a(x - h)2 + k. Then write the coordinates of the vertex of the parabola.
What is:
y = x2 + 8x + 5
= (x2 + 8x) + 5
= (x2 + 8x + 42 + 42 ) + 5
= (x + 4)2 - 16 + 5
= (x + 4)2 - 11
The vertex is (h, k), or ( -4, -11).
A total of 76 teachers attended a strategy meeting. The number of females exceeded the number of males by 2. How many of each gender were at the meeting?
What is:37 male teachers & 39 female teachers.
M+(M+2)=76
Combine like terms:
2M+2=76
Subtract 2 from both sides:
2M=74
Divide by 2:
M=37
Now that we have the number of male teachers (M=37), we can find the number of female teachers (F) using the second equation:
F=M+2 F=37+2 F=39
So, there were 37 male teachers and 39 female teachers at the meeting.
A plot of land for sale has a width of x ft, and a length that is 8ft less than its width. A farmer will only purchase the land if it measures 240 square feet. What value of x will cause the farmer to purchase the land?
What is: 20ft
Let's say the width of the land as x feet. According to the given information, the length of the land is 8 feet less than its width, so the length is x−8 feet.
The area (A) of the rectangular land is given by the product of its length and width:
A=length×width
In this case, A = (x)(x−8).
The farmer will only purchase the land if its area is 240 square feet, so we can set up the following equation:
(x)(x−8)=240
Now, let's solve for x: x2−8x−240=0
To factor the quadratic equation, we look for two numbers whose product is (−240) and whose sum is (−8). These numbers are −20 and +12. Therefore, we can factor the quadratic as:
(x−20)(x+12)=0
Setting each factor equal to zero gives us two possible solutions:
Since the width (x) cannot be negative in this context, we discard the x=−12 solution.
So, the width of the land that will cause the farmer to purchase it is x=20 feet.
Solve the following quadratic equation by using the quadratic formula: 4x2 - 17x - 15 = 0
What is: x = 5, x = -3/4 (-0.75)
Solution: x = -(-17) +- √(-17)2 - 4(4)(-15) / 2(4)
x = 17 +- √289 - (-240) / 8
x = 17 +- √529 / 8
x = 5, x = -3/4 (-0.75)
Samantha is standing 20 feet away from a large tree. The angle of elevation from her eyes, which are 5 feet above the ground, to the top of the tree is 57 degrees. How tall is the tree? (to the nearest tenth of a foot)
What is: 35.8ft
The given angle is 57 degrees, the Adjacent side is 20ft, and the height above ground is 5ft, The opposite side is needed so, tan(57)/1 = x/20
cross multiplication gives you 20tan(57) = x
The height of the tree (opposite side) is 20tan(57) + 5 = 35.8ft
Chapter 7: trigonometry
Captain Jack is navigating his ship to Port Harbour, which is directly north of the ship’s location. To compensate for an easterly current, he aims for a point on shore that is 5 km west of Port Harbour. Assuming that the point on shore is 20 km from his position now, at what bearing must Jack head his ship?
What is: N15°W.
Captain Jack’s ship, Port Harbour, and the western target form a right triangle. For the unknown bearing angle, the opposite and hypotenuse sides are known. Apply the sine ratio to find Jack’s bearing.
sin theta = opp/hyp = 5/20 = 0.25
To find theta, calculate the inverse sine of 0.25. theta sin-1 (0.25)
= 14.477° Jack must head his ship on a bearing of approximately N15°W.