List the following words and give a mathematical definition in your own words on your wikispace. Remember, you may edit your definitions after we begin the unit.
quadratic- something involving the squaring of a number
vertex- the highest or lowest point
x-intercept- where the line intersects the x axis
y-intercept- where the line intersects the y axis
increasing- gaining value
decreasing- losing value
maximum-highest point
minimum- lowest point
parabola- u shaped line
5.2:
Summarize the similarities and differences between linear functions and quadratic functions. Discuss the graphs, the equations and the properties of each function.
One difference is that linear functions are in the from y=mx+b while quadratic functions are in the form y=ax^2+bx+c. Also, the graphs are different, because a quadratic forms a parabola, which is a u shaped line. A linear function forms a straight line.
5.3:
Joe is standing at the end zone of a football field and throws the football across the field. The function below models the path that football is thrown, in feet. f(x)=-2(x-75)squared +22
Answer the following questions in your wikispace.
What graphical shape did the football create as it flew through the air? - it forms a u shape
Identify the vertex. (
Describe, in context, what the x-coordinate of the vertex represents. - the amount of time it takes the football to reach a certain height
Describe, in context, what the y-coordinate of the vertex represents. - the height after a certain amount of time
Find f(2). Describe what your answer means in the context of the problem.
5.4:
Complete the three graphs and tables in the document below.
In your wikispace journal, describe the similarities and differences between the three graphs and equations. Be sure to compare the following features; y-intercept, x-intercepts, direction of the graph. Find the connection between these features and their equations - what causes them?
Function 1 : No x intercepts, upward facing parabola, y intercept: (0,4)
Function 2: X intercepts = (2,0) (4,0), Downward facing parabola, y intercept: (0, -8)
Function 3: One x intercept= (-3,0), upward facing parabola, y intercept: (0,9)
The discriminant of each function determines their x intercepts. Function one has a negative discriminant. Function 2 has a positive discriminant. Function 3 has a discriminant of 0.
5.5:
You and your friend are playing a game of tennis. Your friend throws the ball in the air, hitting the ball when it is 3 ft above the court with an initial velocity of 40 ft/sec. The height h(t) of the ball can be modeled by the function h(t) = -16t^2+40t+3, where t is the elapsed time in seconds after the dive.
Answer the following questions in your wikispace.
What shape does the path of the tennis ball make while traveling in the air. - a downward facing U
Find h(1). Describe what h(1) means in the context of the problem- in context, h(1) means that after 1 second, it reaches a height of 27ft.
What is the y-intercept of h(t). In context, what does the y-intercept represent? - (0,3) In context, the ball is at a height of 3ft before its hit.
Identify the vertex. Describe in context what the x-coordinate of the vertex represents. Describe in context what the y-coordinate of the vertex represents.- vertex(1.25,28) in context, the x coordinate represents that amount of time it takes for the tennis ball to reach its maximum height. the y coordinate 28 represents the maximum height that the tennis ball reaches.
What is the x-intercept(s) of h(t). - In context what does the x-intercept(s) represent? (-0.1,0) and (2.6,0) in context, the only logical x intercept would be (2.6,0) and in context, this means that after 2.6 seconds, the ball reaches the ground.
5.6:
In your own words, explain what the Zero Property Rule is and how and when it is used. Given the equation 0 = (2x-3)(x+5), verbally explain the step-by-step process to solve for x as you would to a brand new student entering our class. - the zero property rule is when you set a function equal to 0 to find the x intercepts. Given the equation above, first, you should either take 2x-3 or x+5 and set it equal to 0. Example: 2x-3=0 or x+5=0 . Then, you would solve it. 2x-3=0, add 3 to zero, then divide both sides by 2. x+5=0, subtract 5 from both sides, and then you're finished. You end up with x= 1.5 or x= -5 .
You looked over at Joey's paper and noticed he had written 3 and -5 as his two solutions. Explain where Joey may have made his mistake? How would you prove to Joey that his solutions are not true? - Joey made his mistake with the equation 2x-3=0 . He added 3 to zero, but then forgot to divide by 2.
5.7:
Listed below are 4 graphs and 12 equations. Some equations are written in intercept form, some in standard form and some in vertex form. A single graph will match one of each type of equation (so 3 equations per graph).
In your wikispace, explain your thought process and order of matching the equations and graphs together.
What properties did you look at first? What types of equation did you match first?
What type of equation was the hardest to match?
How did you narrow down your choices?
The properties i looked at first were the y-intercepts, x-intercepts, or the vertex. I matched the equations in standard form first. The hardest equation to match were the equations in factored form. I narrowed down my choices by finding the matches for the equations in vertex form and standard form because they were easiest, then whatever was left, was for the equations in factored form.
1-A
2-D
3-A
4-C
5-C
6-B
7-D
8-B
9-A
10-C
11-B
12-D
Unit 5: Quadratic Functions5.1:
List the following words and give a mathematical definition in your own words on your wikispace. Remember, you may edit your definitions after we begin the unit.5.2:
Summarize the similarities and differences between linear functions and quadratic functions. Discuss the graphs, the equations and the properties of each function.One difference is that linear functions are in the from y=mx+b while quadratic functions are in the form y=ax^2+bx+c. Also, the graphs are different, because a quadratic forms a parabola, which is a u shaped line. A linear function forms a straight line.
5.3:
Joe is standing at the end zone of a football field and throws the football across the field. The function below models the path that football is thrown, in feet. f(x)=-2(x-75)squared +22Answer the following questions in your wikispace.
5.4:
Complete the three graphs and tables in the document below.In your wikispace journal, describe the similarities and differences between the three graphs and equations. Be sure to compare the following features; y-intercept, x-intercepts, direction of the graph. Find the connection between these features and their equations - what causes them?
Function 1 : No x intercepts, upward facing parabola, y intercept: (0,4)
Function 2: X intercepts = (2,0) (4,0), Downward facing parabola, y intercept: (0, -8)
Function 3: One x intercept= (-3,0), upward facing parabola, y intercept: (0,9)
The discriminant of each function determines their x intercepts. Function one has a negative discriminant. Function 2 has a positive discriminant. Function 3 has a discriminant of 0.
5.5:
You and your friend are playing a game of tennis. Your friend throws the ball in the air, hitting the ball when it is 3 ft above the court with an initial velocity of 40 ft/sec. The height h(t) of the ball can be modeled by the function h(t) = -16t^2+40t+3, where t is the elapsed time in seconds after the dive.
Answer the following questions in your wikispace.
5.6:
In your own words, explain what the Zero Property Rule is and how and when it is used. Given the equation 0 = (2x-3)(x+5), verbally explain the step-by-step process to solve for x as you would to a brand new student entering our class.- the zero property rule is when you set a function equal to 0 to find the x intercepts. Given the equation above, first, you should either take 2x-3 or x+5 and set it equal to 0. Example: 2x-3=0 or x+5=0 . Then, you would solve it. 2x-3=0, add 3 to zero, then divide both sides by 2. x+5=0, subtract 5 from both sides, and then you're finished. You end up with x= 1.5 or x= -5 .
You looked over at Joey's paper and noticed he had written 3 and -5 as his two solutions. Explain where Joey may have made his mistake? How would you prove to Joey that his solutions are not true?
- Joey made his mistake with the equation 2x-3=0 . He added 3 to zero, but then forgot to divide by 2.
5.7:
Listed below are 4 graphs and 12 equations. Some equations are written in intercept form, some in standard form and some in vertex form. A single graph will match one of each type of equation (so 3 equations per graph).
In your wikispace, explain your thought process and order of matching the equations and graphs together.
The properties i looked at first were the y-intercepts, x-intercepts, or the vertex. I matched the equations in standard form first. The hardest equation to match were the equations in factored form. I narrowed down my choices by finding the matches for the equations in vertex form and standard form because they were easiest, then whatever was left, was for the equations in factored form.
1-A
2-D
3-A
4-C
5-C
6-B
7-D
8-B
9-A
10-C
11-B
12-D