THE GRAPH OF A QUADRATIC FUNCTION (DAY 1) – 2016-17
· The Quadratic Equation is written as: ______; this equation has a degree of ______.
§ Where a, b and c are integer coefficients (where a 0)
· The graph of this equation is called a ______; it is ______.
· Parabolas are functions because they ______.
Draw in the line of symmetry of the parabola on the grid to the left.
This line of symmetry is called the ______
· It is always a vertical line that goes through the turning point of the curve.
Formula: Axis of Symmetry:
1. What is an equation of the axis of symmetry of the parabola represented by
y = ?
2. What is the equation for the axis of symmetry for :
(1) x = 1 (2) x = – 1 (3) y = 1 (4) y = – 1
Turning Point: Is another term for the vertex of the parabola.
The “vertex” has the coordinates of .
To Find Turning Point (T.P.) - Algebraically
·
3. Find the axis of symmetry AND turning point for y = -x2 + 4x – 5.
4. What are the coordinates of the turning point of the parabola whose equation is
y = x2 – 4x + 4?
(1) (2, 0) (2) (–2, 16) (3) (2, –4) (4) (–2, 0)
Zeros (roots) of the equation are the points where the parabola ______
the x – axis, so y = ______.
5. What are the zeros of the parabola on the grid?
6. If the roots of a quadratic equation are –2 and 3, the equation can be written as:
(1) (x – 2)(x + 3) = 0 (3) (x + 2)(x + 3) = 0
(2) (x + 2)(x – 3) = 0 (4) (x – 2)(x – 3) = 0
How to use the zeros to write a
QUADRATIC equation:
GRAPHING QUADRATIC FUNCTIONS (DAY 2)
1. Graph:
State the following: Domain: ______Range: ______
Is the above graph a function?, Explain. ______
2. Graph: () ¬ This is called
an interval, which means your table should cover the x values of -4 to 2.
State the following: Domain: ______Range: ______
Is the above graph a function? Explain. ______
3. Graph:
State the following: Domain: ______Range: ______
Is the above graph a function?, Explain. ______
4. What is the turning point of the parabola represented by y = ?
5. Write a quadratic equation for a function that has zeros of 5 and -2.
EXPLORING THE GRAPHED QUADRATIC EQUATION (DAY 3)
Quadratic functions are written in the form: ______
The x – intercepts (when y = 0) of the parabola are called the ______or ______of the equation ()
Determine the number of roots illustrated in each graph below
1. Given the following graph .
What is the axis of symmetry? ______
What are the coordinates of the turning point?______
Is the T.P. a max or minimum point? ______
How many zeros are there?______
What are the solutions of this equation? ______
What would have been the factors of the equation for this graph?
What if we told you that the equation of the above graph was y = x2 – 7x + 10? Now, find the zeros algebraically. What did you notice?
2. Graph:
What are the solutions of this equation? ______
Given the two graphs below, write an equation for each.
3. 4.
EXPLORING FACTORS AND ZEROS OF QUADRATIC EQUATIONS (DAY 4)
1. The zeros of the function are
(1) -1 and -2 (2) -1 and 2 (3) 1 and -2 (4) 1 and 2
2. A polynomial function has zeros of -5, 2, and 9. What would be the factors of this polynomial function?
3. A polynomial function contains the factors x, x – 3, and x + 6. Which graph(s) below could represent the graph of this function? Explain your answer.
(1) (2) (3)
4. Which equation has the same solutions as
(1) (3)
(2) (4)
5. Given the following three functions, which one has the least minimum? Justify
y = x2 + 4x + 4 f(x) = x2 + 2x – 24
6. A polynomial function is graphed to the right.
a) State the Domain ______
b) State the Range ______
c) Find f(0) ______
What is the name for this point ______
d) For what interval(s) is this function decreasing?
e) For what interval(s) is this function increasing?
f) What are the zeros of this function?
g) State the factors that would represent this polynomial function.
h) Which equation(s) represent the graph? Explain your reasoning.
7. Which functions below have the largest maximum?
(1) (2) g(x)= (5 – x)(3 + x) (3) f(x) = –2x– 8x +3 (4)
SOLVING QUADRATIC – LINEAR SYSTEMS (DAY 5)
Recall: Systems of Equations:
Solutions to Systems of Equations:
Determine the solution(s) to the systems graphed below.
1. Solve the following system of equations graphically
To check on your graphing calculator (find intersection):
1) Go to (Calculate) and pick (intersection)
2) Move cursor to wanted intersection point and hit
2. Solve the following system of equations graphically and check.
3. Which is a solution of the following system of equations?
(1) (3, –9) (2) (0, 0) (3) (5, 5) (4) (6, 0)
4. If and , for which values of x is f(x) = g(x)?
(1) -1.75 and -1.438 (3) -1.75 and 4
(2) -1.438 and 0 (4) 4 and 0
5. If and, which pair below makes h(x) = w(x)?
(1) (5, 5) (4) (6, 0)
(2) (0, 0) (3) (3, –9)
SOLVING QUADRATIC – LINEAR SYSTEMS
ALGEBRAICALLY (DAY 6)
Examples:
1. Solve the following system:
2. Find the solutions of:
3. Solve for the solutions:
APPLICATIONS WITH PARABOLIC FUNCTIONS (DAY 7)
1. A football player attempts to kick a football over a goal post. The path of the football can be modeled by the function , where x is the horizontal distance from the kick, and h(x) is the height of the football above the ground, when both are measured in feet.
On the set of axes below, graph the function y = h(x) over the interval
Determine the vertex of y = h(x). Interpret the meaning of this vertex in context with the problem.
The goal post is 10 feet high and 45 yards away from the kick. Will the ball be high enough to pass over the goal? Justify your answer.
2. A company is considering building a manufacturing plant. The determine the weekly production cost at site A to be while the production cost at site B is , where x represents the number of products, in hundreds, and A(x) and B(x) are the production costs, in hundreds of dollars.
Graph the production cost functions on the set of axes below and label them site A and site B.
State the positive value(s) of x for which the production costs at the two sites are equal. Explain how you determined your answer.
If the company plans on manufacturing 200 products per week, which site should they use? Justify your answer.
Quadratic Application Word Problems (DAY 8)
1. Let represent the height of an object above the ground after t seconds. Determine the number of seconds it takes to achieve its maximum height. Justify your answer.
State the time interval, in seconds, during which the height of the object decreases. Explain your reasoning.
2. After t seconds, a ball tossed in the air from the ground level reaches a height of h feet given by the function .
a. What is the height of the ball after 3 seconds?
b. What is the maximum height the ball will reach?
c. After how many seconds will the ball hit the ground before rebound?
3. A toy rocket is launched from the ground straight upward. The height of the rocket above the ground, in feet, is given by the equation , where t is time in seconds. Determine the domain for this function in the given context. Explain your reasoning.
4. A rocket carrying fireworks is launched from a hill 80 feet above a lake. The rocket will fall into the lake after exploding at its maximum height. The rocket’s height above the surface of the lake is given by the function .
a. What is the height of the rocket after 1.5 seconds?
b. What is the maximum height reached by the rocket?
c. After how many seconds after it is launched will the rocket hit the lake?
5. A rock is thrown from the top of a tall building. The distance, in feet, between the rock and the ground t seconds after it is thrown is given by . How long after the rock is thrown is it 370 feet from the ground?
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