Exponential Functions
Write in Standard Form
y=(x-5)2-4
y=x2-10x-4
What is the shape of the graph for a quadratic function? (name and description)
Parabola, u-shaped
When you factor a quadratic, what does your equation need to be equal to?
zero (zero product property)
Factor the trinomial:
x^2 + 10x + 25
(x + 5)^2
What is the quadratic formula and when can you use it?
Works for any quadratic equation
What is the vertex and roots of this equation?
y=x2-8x+12
Vertex (4,-4)
Roots (6,0) and (2,0)
The minimum or maximum of a parabola is located at this point.
The vertex
TWO PARTS:
1.) Factor the equation y = x^2 + 10x + 21
2.) Solve using the zero product property
1.) y = (x + 7)(x + 3)
2.) 0 = x + 7 AND 0 = x + 3
x = -7 & -3
Find the possible values that could make this a perfect square trinomial, then factor.
x^2 + _______x + 256
x^2 + (-32)x + 256 -----> (x-16)^2
OR
x^2 + 32x + 256 -----> (x+16)^2
How can you use the quadratic formula to find out how many solutions your equation will have? Explain all options.
b^2 -4ac
positive = 2 solutions
negative = no solutions
zero = 1 solution
An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2+ 19.6t + 58.8, where s is in meters.
How long before the object hits the ground after launch?
-4.9(t2 – 4t – 12) = 0
(t – 6)(t + 2) = 0
t = 6 and t = -2
6 seconds
What the 3 forms of quadratic equations that we have learned? Give an example of each.
Standard, Factored, Vertex
TWO PARTS:
1.) Factor: 8x^2 = 30 + 43x
2.) Solve using the zero product property
1.) 0 = (8x + 5)(x - 6)
2.) x = -5/8 & 6
Fill in the blank with the value that would make this a perfect square trinomial, then factor it.
9x^2 + ______x + 25
9x^2 + 30x + 25
(3x + 5)^2
How many solutions does this equation have? Rational or irrational?
2x^2 − 7x − 13 = −10
2 irrational solutions
An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2+ 19.6t + 58.8, where s is in meters.
What is the maximum height of the object? When will it happen?
The maximum height of the object is 78.4. The object reaches its maximum height after 2 sec
TWO PARTS:
1.) The solutions for quadratics have many names, what are they and where are they found?
2.) How many solutions can a quadratic function have?
1.) Zeros, roots, solutions, x-intercepts. Found where the parabola crosses the x-axis
2.) 2, 1, 0
Name 5 values of c which make the expression factorable:
x^2 − 3x + c
Examples include: 0, 2, -4, -10, -18
Solve by completing the square- you MUST show your work.
x^2 − 12x + 23 = 0
Is the solution irrational or rational? Explain.
x = 9.61 & 2.39
Irrational
How many solutions will this equation have? Will they be rational or irrational? What are the solutions?
5x^2 = 80
2 rational solutions
x = 4,-4
Factor completely
y=2x3-16x2-24x
y=2x(x-6)(x-2)
TWO PARTS:
1.) How can you tell if a parabola will open upwards or downwards?
2.) How can you tell if the parabola will be wide or narrow?
1.) Negative a will result in a downward parabola. Positive a will be upward.
2.) The a value. The absolute value of the a value. A larger absolute will be narrower and smaller absolute will be wider
For what 6 values of b is the expression factorable?
x^2 + bx + 12
13, 8, 7, −13, −8, −7
Solve by completing the square- show ALL work
9x^2 − 21 = 13x
Is the solution rational or irrational? Explain.
x = 2.41 & -0.97
Irrational
Solve using the quadratic formula. Rational or irrational solutions?
x^2 − 31 − 2x = −6 − 3x^2 − 2x
x = 2.5 & -2.5