¨Wheres the graph!?¨
Quadratic Fundamentals I
Quadratic Fundamentals II
Parabolas & Vertexes
X Intercepts & Y Intercepts
100
f(x)=−(x−3)(x−5)
Find an equivalent form of the function f.
f(x)= -(x-3)(x-5)
f(x)=(-x+3)(x-5) <-- Negative is distributed
f(x)= -X² +5x +3x - 15 <-- Found using FOIL
f(x)= -X² +8x - 15 in Standard Form
or
f(x)= (x - 4)² + 1
h=-b/2a
h=-(-8) / (2*-1) = 4
Put x=4 into equation to find k:
k= -4² + (8 * 4) -15
k=1
100
Name three components needed to graph a quadratic function.
Axis of symmetry, vertex, T Chart, x-intercepts, y-intercept, the value of a
100
What is the quadratic formula?
100
What do p and q represent in intercept form?
y = a(x - p)(x - q)
p and q represent zeros of the quadratic function.
100
How do you find the x-intercept of a quadratic formula?
Entering zero in the place of y:
Ex. 0= ax² + bx + c
200
Find the factors of the following equation:
x²+3x+2 = 0
(x + 2)(x + 1)= x²+3x+2
This is because 2 and 1 are the only numbers that multiply into 2 and add up to 3.
200
Draw the graph of the following function:
-x² - 2x + 3 = f(x)
Line of symmetry falls on (-1,- 4) because of 2x value (as seen after experimenting on graph)
-x² implies negative parabola
200
What is Vertex Formula?
f(x) = a(x – h)2 + k
200
f(x) = (-x+2)²
g(x) = -4x² + 16x -16
Do the functions f(x) and g(x) share vertex coordinates?
Yes, we can see how g(x) is negative four times f(x), with g(x) being a reflection on the x-axis.
(-x+2)(-x+2)
x² - 2x -2x +4
f(x) = (-x+2)² = x² - 4x +4
-4(x² - 4x +4) = -4x² + 16x -16 = g(x)
Though f(x) is positive and g(x) is negative they share a vertex.
200
How do you find the y-intercept of a quadratic formula?
Entering zero in the place of all x-values:
Ex. f(0)= a(0)² + b(0) + c
300
Find the sum of all coefficients in the following function:
g(x)=7000x² + 34x - 400
7034
300
Draw the graph of the following equation:
f(x)= x² - 3
We can draw this function but first drawing f(x)=x², we can then translate the function f(x)=x² three units up the y-axis.
300
(v+1/5)²−9=0
Find the sum of all solutions to the function.
(v+1/5)²−9=0
+9 +9
(v+1/5)²=9
SquareRoot ((v+51)²)= SquareRoot (9)
v+1/5 = 3
v= +-3 - 1/5 or v= 3 +- 1/5
Sum:
(-1/5+ 3) + (-1/5-3)= (-2/5)
300
g(x) = -4x² + 16x -16
Find the vertex of g(x)
X-vertex = -b/2a
X-vertex = -16/(2*-4) --> X-vertex = 2
For Y vertex we can plug in X vertex
g(2)= -4(2²) + (16*2) - 16
g(2)= 0
Vertex: (2, 0)
300
Name the x-intercepts y=x² - 3x + 2
0= x² - 3x + 2
Using the quadratic formula we get:
3 +- SquareRoot (-3² - 4*1*2)/2
We get:
X= 1
X=2
400
What does the value of the coefficient a tell us about any given function?
A value smaller than 1 lead to a skinner parabola.
A value greater than 1 leads to a wider parabola.
400
Find g(f(5)) given:
g(x) = 2x²+3x+10
f(x)=x/10
2(5/10²) + 3(1/2)+ 10= 12
g(f(x))= 5
400
1−9b²=0
List all solutions to the function above.
1−9b²=0
-1 -1
-9b²=-1
9b²=1
SquareRoot (9b²)= SquareRoot (1)
3b = +-1
b= -1/3 and b = 1/3
400
p+1 and q-1 are roots to the function f(x)= x² + 4x - 12
Find p and q.
f(x)= x² + 4x - 12 factors to f(x) = (x-2)(x+6)
x=2
x=-6
Consider the first-factor p and the second-factor q:
(p+1) =2 --> p=1
(q-1) --> -6 --> q=-5
400
Name the y-intercept y=x² - 3x + 2
y=x² - 3x + 2
y=(0²) -3(0) +2
y= 2
500
Describe the relationship between the vertex and the axis of symmetry.
The axis of symmetry is the line of symmetry found in the middle of the parabola. The axis of symmetry is either one x value or one y value. The vertex is a coordinate that described the highest or lowest point in the function. It has an x and y value in the coordinate. The line of symmetry showcases one of the two values depending on how it is structured: In the case of quadratic functions, the line of symmetry is always on the x-axis.
500
What is the equation of this graph in standard form?
(X+1)(X-3)
Distribute using FOIL
X² -3x + 1x -3
X² -2x -3 = 0
500
Find the sum of the following answers:
a.) Mrs. Naik throws a ball into the air and times how long it takes for the ball to hit the ground, modeled by:
f(x)= -(x-3)² + 15
In how many seconds will the ball hit the ground?
b.) Hamza is studying ants in a part of an area (in hundreds) as a function of temperature in degrees Celsius modeled by:
g(x) = −4(x−6)²+400
What is the maximum number of ants in the area?
c.) Grace is flying a drone that that flys off of a platform. Its height (in feet) is measured by the function h(x), x seconds after take off, modeled by:
h(x)= −3(x−4)²+110
What is the height of the drone at the time of take-off?
a.) Using the Quadratic formula we get:
x=-6 +- SquareRoot(6^2 - (4*-1*6))/-2
Using GDC we get:
x=-0.87299
x=6.873 Seconds
Time cannot be negative: x = 6.873
b.) g(x) is in the Vertex form, therefore, we can identify the vertex as the maximum point of the parabola (g(x) also has a negative parabola).
x=400 Ants
c.) h(0) helps us find the height at the time of take off.
h(0)= -3(0-4)² +110
h(0)=-3(16) +110
h(0)= 62 feet
Sum: 6.873 + 400 + 62 = 468.873
500
Find the Vertex of the following function.
k(p)= 2k² - 3k + 1
h = -b/2*a
h = 3/2*2
h= 0.75
Plug into function:
2(0.75²) + -3(0.75) +1 = -0.125
(h,k)= (0.75, -0.125)
500
Find the sum of both x-intercepts and the y-intercept of the following function:
h(x)= x² + 7x -14
We can use the GDC to make the process more efficient:
x= -8.62
x= 1.623
y-intercept= -14
-8.62 + -14 + 1.623 = -20.997
Sum = -21