Question

1.) The height h in feet of a baseball on Earth after t seconds can be modeled by the function h(t) = -16(t – 1.5)2 + 36, where -16 is a constant in ft/s2 due to Earth's gravity. The gravity on Mars is only 0.38 times that on Earth. If the same baseball were thrown on Mars, it would reach a maximum height 59 feet higher and 2.5 seconds later than on Earth. Write a height function for the baseball thrown on Mars.

Answer 1

2.) A short stop makes an error by dropping the ball. As the ball drops, its height h in feet as a function of time, t, is modeled by h(t) = -16t2 + 3. A slow motion replay of the error shows the play at half speed. What function describes the height of the ball in the replay? (Hint: The function squares the time, so half the time is also squared in the new function.)

Answer 2

hellur

Answer 3

hi

Answer 4

need some help @CamPayne

Answer 5

very much so yes

Answer 6

OKAY

Answer 7

yay

Answer 8

Graph the parent quadratic function .
x −3 −2 −1 0 1 2 3
y 9 4 1 0 1 4 9
Step One: Make a table of values (t-chart).

Step Two: Plot the points on a coordinate grid and connect to draw the parabola.






Note: The vertex is a minimum at , and the axis of symmetry (the vertical line that passes through the vertex) is .
What is the domain (the input values) of ? {all real numbers} What is the range (the output values)?

For what value of is increasing? For what values of x is it decreasing?
is increasing for and decreasing for .

Why is the vertex of the graph of ? It is the lowest point on the parabola.

Answer 9

Graphing Quadratic Functions Using a Table
Ex 2
Graph by using a table.
Make a table. Plot enough ordered pairs to see both sides of the curve.




1


2


3


4


5



Ex 3
Graph the parabola
x −3 −2 −1 0 1 2 3
y −9 −4 −1 0 −1 −4 −9
Step One: Make a table of values (t-chart).

Step Two: Plot the points on a coordinate grid and connect with a smooth curve to draw the parabola.
Note: The vertex is a maximum at , and the axis of symmetry is .
Describe the transformation of the parent graph .
It is a reflection across the x axis.

What is the domain of ? {all real numbers} What is the range?

For what value of is increasing? For what values of x is it decreasing?
is increasing for and decreasing for .

Why is the vertex of the graph of ? It is the highest point on the parabola.

Answer 10

Reflection, Shrinking and Stretching, Vertical and Horizontal Translations
Comparing and : The vertex is , and the axis of symmetry is for both graphs. When a is positive, the parabola opens up and its vertex is a minimum; when a is negative, the parabola opens down and its vertex is a maximum.
DISCOVERY:
Calculator Exploration: Transformations with and .
Use the graphing calculator to investigate the graphs of quadratic functions. Describe the effect on the graphs of and . (Note: In the calculator graphs shown, or is graphed as a dotted line.)

1. Compare to
Vertex: Same Opens narrower than
2. Compare to
Vertex: Same Opens wider than
3. Compare to
Vertex: Up 3 Opens the same as
4. Compare to
Vertex: Down 4 Opens the same as
5. Compare to
Vertex: left 3 Opens the same as
6. Compare to
Vertex: right 3 Opens the same as

Answer 11

okay then

Answer 12

still need help

Answer 13

cause here is alot of info

Answer 14

yes

Answer 15

wait

Answer 16

Transformations of Quadratic Functions




Vertical Stretch: Vertical Shrink:
Reflection over x-axis: Reflection over y-axis:
Horizontal Translation: moves left moves right
Vertical Translation: moves down moves up
Other: Axis of Symmetry Vertex is
If a is positive the parabola opens up. If a is negative the parabola opens down.

Ex 4 Describe the shift, reflect and stretch of the parent function.
a.
• Vertical Shrink by a factor of (wider)
• Horizontal Shift 5 units left.

b.
• Horizontal Shift 2 units right
• Vertical Shift 1 units down

c.
• Reflect across the x axis
• Vertical Stretch by a factor of 2
• Horizontal Shift 3 units right
• Vertical Shift 4 units up

Answer 17

Ex 5 Graphing a Quadratic Function in Vertex Form
Graph the quadratic function . State the vertex and axis of symmetry.
Step One: Determine if the graph opens up or opens down. Because is the graph opens down.
Step Two: Identify the vertex and axis of symmetry. Note: Another way of writing the function is . So the vertex is and the axis of symmetry is .
Step Three: Plot the vertex and sketch the graph. Notice the parent graph is reflected over the x axis, is wider (vertical shrink), shifted to the left 3 (opposite of what you think), and is translated up four units.
Optional: Make a table of values. When choosing x-values for the T-table, use the vertex, a few values to the left of the vertex, and a few values to the right of the vertex. (Note: Because of the fraction, you may want to choose values that will guarantee whole numbers for the y-coordinates.) Five to seven points will give a nice graph of the parabola.
x −9 −7 −5 −3 −1 1 3
y −14 −4 2 4 2 −4 −14







Ex 6 Writing the Equation of a Quadratic Function in Vertex Form


Write an equation for the parabola in vertex form.

The vertex is at . So the vertex form of the equation is .
To solve for a, we will choose a point on the parabola and substitute it into the equation for .
Choose .
So the vertex form of the equation is .

Ex 7 Automotive Application
The minimum braking distance d in feet for a vehicle on dry concrete is approximated by the function , where v is the vehicle’s speed in miles per hour. If the vehicle’s tires are in poor condition, the braking-distance function is . What kind of transformation describes this change and what does the transformation mean?

Answer 18

fast type writer

Answer 19

hold on

Answer 20

kay

Answer 21

okay back

Answer 22

yay

Answer 23

Examine both functions in vertex form.

The value has increased from .035 to .065. The increase indicates a vertical stretch.
Find the stretch factor by comparing the new a-value to the old a value:

The function represents a vertical stretch of d by a factor of approximately 1.9. Because the value of each function approximates braking distance, a vehicle with tires in poor condition takes about 1.9 times as many feet to stop as a vehicle with good tires does.
Graph: Graph both functions on a graphing calculator. The graph of appears to be vertically stretched compared with the graph of .



Explore: The height h in feet of a baseball on Earth after t seconds can be modeled by the function where is a constant in due to Earth’s gravity.

a. What if…? The gravity on Mars is only 0.38 times that on Earth. If the same baseball were thrown on Mars, it would reach a maximum height 59 feet higher and 2.5 seconds later than on Earth. Describe the transformations that must be applied to make the function model the height of the baseball on Mars.
b. Write a height function for the baseball thrown on Mars.


QOD: Describe the shift, reflect, and stretch of a quadratic equation in vertex form.

Write About It: Describe the graph of without graphing it.
The graph would by a very narrow parabola opening upward with its vertex at (-5, 5)

Closure: What type of graph would a function of the form have if ? What type of function would it be?

Answer 24

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Answer 25

yeah and still more to go to

Answer 26

do have to much info

Answer 27

far too much to take in at once

Answer 28

still need more

Answer 29

i think ill try to figure it out from here

Answer 30

thank you so much tho

Answer 31

your welcome

Answer 32

Graphing a Quadratic Function in Standard Form
Standard Form of a Quadratic Function , when ; a, b, and c are real numbers
Vertex: the vertex is the point Axis of Symmetry: y-intercept: c
Minimum Value: When a parabola opens upward, the y-value of the vertex is the minimum value.
Maximum Value: When a parabola opens downward the y-value of the vertex is the maximum value.
Axis of Symmetry: the vertical line that passes through the vertex of a quadratic function.











Ex 8
Graph the quadratic function . State the vertex and axis of symmetry.
Step One: Determine whether the graph opens upward or downward. If is positive the graph opens upward. If is negative the graph opens downward. Since =1 the parabola opens upward.
Step Two: Find the axis of symmetry. (the x-coordinate of the vertex)
The axis of symmetry is the line

Answer 33

Step Three: Find the vertex.
The vertex lies on the axis of symmetry, so the x-coordinate is 2. The y-coordinate is the value of the function at this x-value, or .

The vertex is
Step Four: Find the y-intercept.
Because , the y-intercept is .
Step Five: Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point . Use the axis of symmetry to find another point on the parabola. Notice that is 3 units left of the axis of symmetry. The point on the parabola symmetrical to is 3 units to the right of the axis at . Connect points with a smooth curve to draw the parabola.
Vertex:
Axis of Symmetry:
































































Optional: To check, make a table of values. When choosing x-values for your T-table, use the vertex, a few values to the left of the vertex, and a few values to the right of the vertex.
x 0 1 2 3 4 5 6
y −1 −6 −9 −10 −9 −6 −1


Note: When calculating the y-coordinate of points to the right and left of the vertex, notice the symmetry.

Ex 9
Find the minimum or maximum value of State the domain and range.
Step One: Determine whether the function has a minimum or maximum value.
Because is positive, the graph opens upward and has a minimum value.

Answer 34

Step Two: Find the x-value of the vertex. Step Three: Then find the y-value of the vertex, .

Minimum Value: or 4.5.
Domain: all real numbers, . Range: all real numbers greater than or equal to or
Check with Graphing Calculator. Graph The graph and table support the answer.

Using a Quadratic Model
Ex 10
A basketball’s path can be modeled by , where x represents time (in seconds) and y represents the height of the basketball (in feet). What is the maximum height that the basketball reaches?
Press the “Y=” key and graph the function in Y1 and find the maximum (in the CALC menu). The maximum is the vertex. The maximum height of the basketball is the y-coordinate of the vertex, which is approximately .




Ex 11
A baseball is thrown with a vertical velocity of 50 ft/sec from an initial height of 6 ft. The height h in feet of the baseball can be modeled by , where t is the time in seconds since the ball was thrown.
Approximately how many seconds does it take the ball to reach it maximum height?
About 1.6 seconds
What is the maximum height that the ball reaches?
About 45 ft.

Answer 35

hey @zepdrix trying help while he is gone

Answer 36

Closure: Show equivalent quadratic functions in both standard form and vertex form. Review how to use each form to determine the y-intercept, axis of symmetry, vertex, and maximum/minimum value. Use a graph to check.



SAMPLE EXAM QUESTIONS
1. What is the equation of the parabola shown?







(A) (B)
(C) (D)
Ans: B
2. Find the vertex of and state if it is a maximum or a minimum.
(A) (-1, -4); maximum
(B) (-1, -4); minimum
(C) (-4, -1); maximum
(D) (-4, -1); minimum

Ans: B



3. Create a table for the quadratic function f(x) = 5x2 + 9x + 1, and use it to graph the function.
a. c.
b. d.



Ans: C

4. Graph the function . Label the vertex and axis of symmetry.



Ans:

Answer 37

6. How would the graph of the function be affected if the function were changed to ?
a. The graph would shift 2 units up.
b. The graph would shift 5 units up.
c. The graph would shift 2 units to the right.
d. The graph would shift 5 unit down.

Ans: B

7. How would you translate the graph of to produce the graph of
a. translate the graph of down 4 units
b. translate the graph of up 4 units
c. translate the graph of left 4 units
d. translate the graph of right 4 units
Ans: A

8. Which transformation from the graph of a function f(x) describes the graph of ?
a. horizontal shift left unit c. vertical compression by a factor of
b. vertical shift up unit d. vertical shift down unit

Ans: C

9. Identify the vertex of .
a. ( , ) c. (14, )
b. ( , 8) d. (14, 8)

Ans: A

10. Use this description to write the quadratic function in vertex form:
The parent function is vertically stretched by a factor of 3 and translated 8 units right and 1 unit down.
a. c.
b. d.

Ans: C

11. What is the maximum of the quadratic function ?
A. C.
B. D.
Ans: D

Answer 38

3. Create a table for the quadratic function f(x) = 5x2 + 9x + 1, and use it to graph the function.
a. c.
b. d.



Ans: C

4. Graph the function . Label the vertex and axis of symmetry.



Ans:









5. Use a table of values to graph





Ans:



6. How would the graph of the function be affected if the function were changed to ?
a. The graph would shift 2 units up.
b. The graph would shift 5 units up.
c. The graph would shift 2 units to the right.
d. The graph would shift 5 unit down.

Ans: B

7. How would you translate the graph of to produce the graph of
a. translate the graph of down 4 units
b. translate the graph of up 4 units
c. translate the graph of left 4 units
d. translate the graph of right 4 units
Ans: A

8. Which transformation from the graph of a function f(x) describes the graph of ?
a. horizontal shift left unit c. vertical compression by a factor of
b. vertical shift up unit d. vertical shift down unit

Ans: C

9. Identify the vertex of .
a. ( , ) c. (14, )
b. ( , 8) d. (14, 8)

Ans: A

10. Use this description to write the quadratic function in vertex form:
The parent function is vertically stretched by a factor of 3 and translated 8 units right and 1 unit down.
a. c.
b. d.

Ans: C

11. What is the maximum of the quadratic function ?
A. C.
B. D.
Ans: D


12. Use this description to write the quadratic function in vertex form:
The parent function is vertically compressed by a factor of and translated 11 units left and 5 units down.
a. c.
b. d.

Ans: B
13. Which graph represents ?

Ans: C

14. Consider . What are its vertex and y-intercept?

a. vertex: ( , ), y-intercept: (0, 2) c. vertex: (1, 1), y-intercept: (0, 2)
b. vertex: ( , 2), y-intercept: (0, ) d. vertex: ( , 1), y-intercept: (0, )

Ans: A


Sample Essay Question:

15. Rick uses 1800 feet of fencing to build a rectangular pen. He divides the pen into two sections that have the
same area. Let x be the width (in feet) of the pen, as shown in the drawing.

Part A: Write an expression to represent the length of the pen in terms of x. Justify your work.
Part B: Write an equation for the area y of the pen in terms of x. Graph the equation using a graphing calculator.
Part C: Does the function in Part B have a maximum or a minimum value? Explain.
Part D: Rick wants the pen to have the largest possible area. What width x should he use? What is the area of the pen with the largest area?






Ans:
Part A: 900 – 1.5x; Rick has 1800 feet of fencing. He uses 3x feet to make two sides and a divider for the pen. So, the amount of fencing he has to make the other two sides of the fence is 1800 – 3x. Divide 1800 – 3x by 2 to get the length of one of the two sides: 900 – 1.5x.
Part B: ; see graph below. ; use x-scale with intervals of 50, and y-scale with intervals of 20,000.]
Part C: The function has a maximum value because the value of the coefficient of x2 is negative.
Part D: 300 feet; 135,000 square feet

Answer 39

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Answer 40

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Answer 41

okay @CamPayne

Answer 42

okay

Answer 43

Medal earned @Loser66

Answer 44

shut up @loser66

Answer 45

i weak af