Chapter 1
Given that the two lines are parallel, find A: 10x+4y=20 and Ax+2y=12.
A=5
What is the relationship between the slopes of lines that are parallel/perpendicular?
Parallel lines have the same slope, perpendicular lines have slopes that are opposite reciprocals.
What defines a linear function?
Straight line, only one y-value for every x-value and vice versa, passes vertical line test, constant slope
What are the four methods that can be used for solving systems of equations?
Substitution, elimination, graphing, matrices
Write the equation of the function g(x)=-2x²+12x+4 in vertex form bu using the completing the square method, and, hence, find the vertex.
Vertex Form: g(x)=-2(x-3)²+22 Vertex: (3, 22)
Write the equation of the function f(x)=-3x²-12x-3 in vertex form by using the completing the square method, and, hence, find the vertex.
Vertex form: f(x)=-3(x+2)²+9 Vertex: (-2, 9)
Complex Numbers
a+bi (a and b are real numbers)
Find the equation of the parabola f(x) such that f(12)=152, f(22)=105, and f(30)=165 using a system of equations.
a=0.678, b=-27.4, c=387.3
General/Standard Form Equation
ax+by=c or ax+by+c=0
What is the discriminant?
b²-4ac
Perpendicular Bisector
cuts a line in half at a right angle
Find the discriminant and explain what this tells you about the graph of a parabola: f(x)=x²+2x-1
d=8, which means that the graph has a minimum
Distance Formula
d=√(x-x₁)²+(y-y₁)²
A baseball player tries to hit a ball over an outfield fence that is 6 ft. high and 358 ft. from home plate. The ball is hit 3 ft. above home plate and reaches it highest point 220 ft. above a point on the ground that is 180 ft. from home plate. Find an equation of the parabolic path of the baseball and write your answer in vertex form. Also, determine if the ball will go over the outfield fence.
f(x)=-0.007(x-180)²+220; yes, the ball will go over the outfield fence because 7.8 ft. is greater than 6 ft.
What are the 3 algebraic methods that can be used to solve quadratic equations?
factoring, completing the square, quadratic formula
Equation for dropped/falling objects
h(t)= -1/2gt²+h₀ *where g=gravity (9.8m/s² or 32 ft/s²), t=time, and h₀=initial height*
Equation for thrown objects
h(t)=-1/2gt²+v₀t+h₀ *where v₀=initial velocity*
A stone is thrown with an upward velocity of 14m/s from a cliff 30m high. Find its height above the ground t seconds later. When will the stone reach its highest elevation? When will the stone hit the ground?
h(t)=-4.9t²+14t+30; 1.43 seconds; 4.29 seconds
i¹=
i
i¹³=
i
i¹⁰⁵=
i
i⁵=
i
i⁻³=
i
Find the length and midpoint of a line that has endpoints of (-1, 9) and (4, -3).
length=13, midpoint=(3/2, 3)
If b²-4ac=0, then how many solutions does ax²+bx+c=0 have?
one solution (zero imaginary solutions)
Find p and q: (3+pi)+(2q+4i)=11+12i
p=8, q=4
What does this say about the graph of a parabola: a<0
the graph has a maximum at its vertex
What does this say about the graph of a parabola: a>0
the graph has a minimum at its vertex
If b²-4ac>0, then how many solutions does ax²+bx+c=0 have?
two solutions (zero imaginary solutions)
Intercept Form Equation
x/a + y/b=1 (b is y intercept and a is x intercept)
Solve by factoring: 3x²+10x-8=-11
x=-1/3 and x=-3
Find the equation for the axis of symmetry: f(x)=x²+2x-1
x=-2/2(1)
Solve: (x+2)²=3
x=-2±√3
Solve: x²-9x=0
x=0 and x=9
Solve using the quadratic formula: 2x²-3x=-1
x=1 and x=1/2
Solve by completing the square: 3x²-12x+6=0
x=2±√2
Solve by completing the square: 2x²-12x-8=0
x=3±√13
What is the equation and slope for vertical lines?
x=constant, slope is undefined
Solve: √2x+5=x+1
x=±2
Point-Slope Form Equation
y-y1=m(x-x1)
Quadratic Equation Vertex Form
y=(x-h)²+k *(h, k) is the vertex*
Write an equation for the line that is perpendicular to y=1/2x+4 and has a y-intercept of 3.
y=-2x+3
Write an equation of the perpendicular bisector of the segment joining points (-1, 4) and (2, -2).
y=1/2x + 3/4
Find the equation of the line that goes through the points (2,1) and (3,6) using any form.
y=5x-9
Quadratic Equation Standard Form
y=ax²+b+c
What is the equation and form for horizontal lines?
y=constant, slope=0
Slope-Intercept Form Equation
y=mx+b
If b²-4ac<0, then how many solutions does ax²+bx+c=0 have?
zero solutions (2 imaginary solutions)
Find the coordinates of all points where the lines f(x)=2x+5 and g(x)=8-x² intersect.
(-3, -1) and (1, 7)
What is the vertex: h(x)=2(x+4)²-5
(-4, -5)
Solve this system: 3x+2y=4 and 2x+y=8
(12, -16)
What is the vertex: g(x)=2(x-4)²+5
(4, 5)
Midpoint Formula
(x₁+x₂)/2, (y₁+y₂)/2
i²=
-1
i⁶=
-1
i⁻²=
-1
(√-25)(√-9)=
-15
Simplify: 5/3i
-5/3i
Simplify: √-16 -2√25
-6i
i²³=
-i
i³=
-i
i⁷=
-i
i⁻¹=
-i
i⁰=
1
i⁴=
1
i⁻⁴=
1
(4+6i)+(8+10i)=
12+16i
What are the steps to solving a quadratic equation using the completing the square method? (four steps total)
1: divide both sides by the coefficient of x² 2: subtract the constant term from both sides 3: add the square of one half of the coefficient of x to both sides 4: factor and solve
How do you find the equation of the parabola f(x) by using the regression features on your calculator?
1: go to the Stat feature 2: enter the x-coordinates in L₁ 3: enter the y-coordinates in L₂ 4: go to Calc in Stat 5: go to QuadReg and press enter
Florence purchased a number of hand-held electrical games for $180. She decided to keep one for herself and sell the rest. She sold the remaining games for $1.00 more than she bought them for. After keeping one game for herself, Florence made a profit of $10.00. How many games did she purchase?
20
Simplify: 1/(4-3i)
4/25 + 3/25i
y=ax²+8x+c has two equal roots at x=4. Show that a=-1 and then find c.
4=-8/2a; c=-16
f(x) is a quadratic function which has a vertex at (4, 8) and a root at x=7. What is the distance between the x-intercepts on the graph?
6
Simplify: √-36
6i
Francesco walks 24 km every day. He always maintains a constant speed. if he had walked 2km/h faster than he usually does, he would have completed his walk one hour earlier. At what speed does Francesco usually walk?
6km/h
Use an algebraic process to find two positive numbers that differ by 2 and have a product of 63.
7 and 9
(3+2i)(5+4i)=
7+22i