Name____________________________ Period______ Date_________
Introduction to Quadratic Functions – Activity 2
Imagine yourself standing on the roof of the 1450-foot-high Willis Tower (formerly called the Sears
Tower) in Chicago. When you release and drop a baseball from the roof of the tower, the ball’s height
above the ground, H (in feet), can be described as a function of the time, t (in seconds), since it was
dropped. This height function is defined by:
𝐻(𝑡) = −16𝑡 2 + 1450
1. Sketch a diagram illustrating the Willis Tower and the path of the baseball as it falls to the ground.
2. a. Complete the following table.
TIME, t(sec)
0
1
2
3
4
5
6
7
8
9
10
𝐻(𝑡) = −16𝑡 2 + 1450
b. How far does the baseball fall during the first second?
c. How far does it fall during the interval from 1 to 3 seconds?
3. Using the height function 𝐻(𝑡) = −16𝑡 2 + 1450 , determine the average rate of change of 𝐻 with
respect to 𝑡 over the given interval. Remember:
average rate of change =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑜𝑢𝑡𝑝𝑢𝑡
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑝𝑢𝑡
a. 0 ≤ 𝑡 ≤ 1
b. 1 ≤ 𝑡 ≤ 3
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c. Based on the results of parts a and b, do you believe that 𝐻(𝑡) = −16𝑡 2 + 1450 is a linear
function? Explain.
4. a. What is the value of 𝐻 when the baseball strikes the ground? Use the table in Problem 2a to
estimate the time when the ball is at ground level.
b. What is the practical domain of the height function?
c. Determine the practical range of the height function.
d. On the following grid, plot the points in Problem 2a that satisfy part b (practical domain) and
sketch a curve representing the height function.
e. Is the graph of the height function in part d the actual path of the object (see Problem 1)?
Explain.
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Some interesting properties of the function defined by 𝐻(𝑡) = −16𝑡 2 + 1450 arise when you ignore
the falling object context. Replace 𝐻 with 𝑦 and 𝑡 with 𝑥 and consider the general function defined by
𝑦 = −16𝑥 2 + 1450
5.
a. The graph of the function looks like:
b. Describe the important features of the graph of 𝑦 = −16𝑥 2 + 1450. Discuss the shape,
symmetry, and intercepts.
Quadratic Functions
The graph of the function defined by 𝑦 = −16𝑥 2 + 1450 is a parabola. The graph of a parabola is a ⋃
shaped figure that opens upward, ⋃ , or downward, ∩, . Parabolas are graphs of a special category of
functions called quadratic functions.
Definition
Any function defined by an equation of the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 or 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
where 𝑎, 𝑏, and 𝑐 represent real numbers and 𝑎 ≠ 0, is called quadratic function. The output
variable 𝑦 is defined by an expression having three terms: the quadratic term, 𝑎𝑥 2 , the linear
term, 𝑏𝑥, and the constant term, 𝑐. The numerical factors of the quadratic and linear terms, 𝑎 and
𝑏, are called the coefficients of the terms.
𝑯(𝒕) = −𝟏𝟔𝒕𝟐 + 𝟏𝟒𝟓𝟎 defines a quadratic function. The
quadratic term is – 𝟏𝟔𝒕𝟐 . The linear term is 𝟎𝒕, although it is not written as part of the
expression defining 𝑯(𝒕). The constant term is 1450. The numbers – 𝟏𝟔 and 𝟎 are the
coefficients of the quadratic and linear terms, respectively. Therefore, 𝒂 = −𝟏𝟔,
𝒃 = 𝟎, 𝒂𝒏𝒅 𝒄 = 𝟏𝟒𝟓𝟎.
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6. For each of the following quadratic functions, identify the value of a, b, and c.
Quadratic Function
a
b
c
2
𝑦 = 3𝑥
𝑦 = −2𝑥 2 + 3
𝑦 = 𝑥 2 + 2𝑥 − 1
𝑦 = −𝑥 2 + 4𝑥
The Constant Term 𝒄: A Closer Look
Consider once again the height function 𝐻(𝑡) = −16𝑡 2 + 1450 from the beginning of the activity.
7.
a.
What is the vertical intercept of the graph? Explain how you obtained the results.
b. What is the practical meaning of the vertical intercept in this situation?
c. Predict what the graph of 𝐻(𝑡) = −16𝑡 2 + 1450 would look like if the constant term 1450
were changed to 800. That is, the baseball is dropped from a height of 800 feet rather than
1450 feet. Verify your prediction by graphing 𝐻(𝑡) = −16𝑡 2 + 800. What does the
constant term tell you about the graph of the parabola?
The constant term c of a quadratic function 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 always indicates the vertical
intercept of the parabola. The vertical intercept of any quadratic function is (0, 𝑐) since 𝑓(0) =
𝑎(0)2 + 𝑏(0) + 𝑐 = 𝑐.
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8. Graph the parabolas defined by the following quadratic equations. Note the similarities and
differences among the graphs, especially the vertical intercepts.
Use desmos.com on your mobile device or download the app.
a. 𝑓1(𝑥) = 1.5𝑥 2
b. 𝑓2(𝑥) = 1.5𝑥 2 + 2
c. 𝑓3(𝑥) = 1.5𝑥 2 + 4
d. 𝑓4(𝑥) = 1.5𝑥 2 − 4
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The Effects of the Coefficient 𝒂 on the Graph of
𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
9. a. Graph the quadratic function defined by 𝑓1(𝑥) = 16𝑥 2 + 1450 on the same screen as
𝑓2(𝑥) = −16𝑥 2 + 1450. Click on the wrench and change the y-axis to −2000 ≤ 𝑦 ≤ 2000 and
use your touchscreen to make adjustments.
b. What effect does the sign of the coefficient of 𝑥 2 appear to have on the graph of the parabola?
c. Graph the functions in the 𝑓3(𝑥) = −16𝑥 2 + 100,
𝑓4(𝑥) = −6𝑥 2 + 100 ,
𝑓5(𝑥) = −40𝑥 2 + 100 in the same window. What effect does the magnitude of the
coefficients of 𝑥 2 (namely, |−16| = 16, |−6| = 6, |−40| = 40 appear to have on the graph
of that particular parabola?
The results from Problems 9 regarding the effects of the coefficient 𝑎 can be summarized as follows.
The graph of a quadratic function defined by 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 is called a parabola.
If 𝑎 > 0, the parabola opens upward.
If 𝑎 < 0, the parabola opens downward.
The magnitude of 𝑎 affects the width of the parabola. The larger the absolute value of 𝑎, the
narrower the parabola.
•
•
•
10. a. Is the graph of ℎ(𝑥) = 0.3𝑥 2 wider or narrower than the graph of 𝑓(𝑥) = 𝑥 2 ?
b.
How do the output values of ℎ and the output values of 𝑓 compare for the same input value?
c. Is the graph of 𝑔(𝑥) = 3𝑥 2 wider or narrower than the graph of 𝑓(𝑥) = 𝑥 2 ?
d.
How do the output values of 𝑔 and 𝑓 compare for the same input value?
e. Describe the effect of the magnitude of the coefficient 𝑎 on the width of the graph of the
parabola.
f.
Describe the effect of the magnitude of the coefficient of 𝑎 on the output value.
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The Effects of the Coefficient 𝒃 on the Turning Point
Assume for the time being that you are back on the roof of the 1450-foot Willis Tower. Instead of merely
releasing the ball, suppose you throw it down with an initial velocity of 40 feet per second. Then the
function describing its height above ground as a function of time is modeled by
𝐻𝑑𝑜𝑤𝑛 (𝑡) = −16𝑡 2 − 40𝑡 + 1450
If you tossed the ball straight up with an initial velocity of 40 feet per second, then the function
describing its height above ground as a function of time is modeled by
𝐻𝑢𝑝 (𝑡) = −16𝑡 2 + 40𝑡 + 1450
11. Predict what features of the graphs of 𝐻𝑑𝑜𝑤𝑛 and 𝐻𝑢𝑝 have in common with 𝐻(𝑡) = −16𝑡 2 + 1450.
12. a. Graph the three functions 𝐻(𝑡). 𝐻𝑑𝑜𝑤𝑛 , and 𝐻𝑢𝑝 using 𝑥 for 𝑡 and the window settings
Xmin= −10, Xmax= 10, Ymin= −50 and Ymax= 3000
b. What effect do the −40𝑡 and 40𝑡 terms seem to have upon the turning point of the
graphs?
If 𝑏 = 0, the turning point of the parabola is located on the vertical axis. If 𝑏 ≠ 0, the turning
point will not be on the vertical axis.
13. For each of the following quadratic functions, identify the value of b and then, without graphing,
determine whether or not the turning point is on the y-axis. Verify your conclusion by graphing the
given function using desmos.
a. 𝑦 = 𝑥 2
b. 𝑦 = 𝑥 2 − 4𝑥
c. 𝑦 = 𝑥 2 + 4
d. 𝑦 = 𝑥 2 + 𝑥
e. 𝑦 = 𝑥 2 − 3
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14. Match each function with its corresponding graph below, and then verify using desmos. Explain how
you chose your answer.
a. 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 4 b. 𝑓(𝑥) = 0.2𝑥 2 + 4 c.ℎ(𝑥) = −𝑥 2 + 3𝑥
Explanations:
Summary
1. The equation of a quadratic function with 𝑥 as the input variable and 𝑦 as the output variable
has the standard form
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
where 𝑎, 𝑏, and 𝑐 represent real numbers and 𝑎 ≠ 0.
2. The graph of a quadratic function is called a parabola.
3. For the quadratic function defined by 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐:
a. If 𝑎 > 0, the parabola opens upward.
b. If 𝑎 < 0, the parabola opens downward.
The magnitude of 𝑎 affects the width of the parabola. The larger the absolute value of 𝑎, the
narrower the parabola.
4. If 𝑏 = 0. the turning point of the parabola is located on the vertical axis. If 𝑏 ≠ 0, the turning
point will not be on the vertical axis.
5. The constant term 𝑐 of a quadratic function 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 always indicates the vertical
intercept of the parabola. The vertical intercept of any quadratic function is (0, 𝑐).
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Practice
1. Complete the following table for 𝑓(𝑥) = 𝑥 2
𝑥
𝑓(𝑥) = 𝑥 2
−3
−2
−1
0
1
2
3
b. Use the results of part a to sketch a graph 𝑦 = 𝑥 2 . Verify using desmos.
c. What is the coefficient of the term 𝑥 2 ?
d. From the graph, determine the domain and range of the function.
e. Create a table similar to the one in Exercise 1a to show the output for 𝑔(𝑥) = −𝑥 2 .
𝑥
𝑔(𝑥) = −𝑥 2
f.
−3
−2
−1
0
1
2
3
Sketch the graph of 𝑔(𝑥) = −𝑥 2 on the same coordinate axis in part b. Verify using desmos
g. What is the coefficient of the term −𝑥 2 ?
h. How can the graph of 𝑦 = −𝑥 2 be obtained from the graph of 𝑦 = 𝑥 2 ?
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2. In each of the following functions defined by an equation of the form = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 , identify the
value of 𝑎, 𝑏, and 𝑐.
2
a. 𝑦 = −2𝑥 2
b. 𝑦 = 5 𝑥 2 + 3
𝑐. 𝑦 = −𝑥 2 + 5𝑥
d. 𝑦 = 5𝑥 2 + 2𝑥 − 1
3. Predict what the graph of each of the following quadratic functions will look like. Use desmos to
verify your prediction.
a. 𝑓(𝑥) = 3𝑥 2 + 5
b. 𝑔(𝑥) = −2𝑥 2 + 1
𝑐. ℎ(𝑥) = 0.5𝑥 2 − 3
4. Graph the following pairs of functions, and describe any similarities as well as any differences that
you observe in the graphs.
a. 𝑓(𝑥) = 3𝑥 2 , 𝑔(𝑥) = −3𝑥 2
1
b. ℎ(𝑥) = 2 𝑥 2 , 𝑓(𝑥) = 2𝑥 2
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c. 𝑓(𝑥) = 5𝑥 2 , 𝑔(𝑥) = 5𝑥 2 + 2
d. 𝑓(𝑥) = 4𝑥 2 − 3, 𝑔(𝑥) = 4𝑥 2 + 3
e. 𝑓(𝑥) = 6𝑥 2 + 1, ℎ(𝑥) = −6𝑥 2 − 1
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5. Use desmos to graph the two functions 𝑓1(𝑥) = 3𝑥 2 and 𝑓2(𝑥) = 3𝑥 2 + 2𝑥 − 2.
a. What is the vertical intercept of the graph of each function?
b. Compare the two graphs to determine the effect of the linear term 2𝑥 and the constant
term −2 on the graph of 𝑓1(𝑥) = 3𝑥 2 . Discuss this below.
For Exercises 6–10, determine
a. whether the parabola opens upward or downward and
b. the vertical intercept.
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6. 𝑓(𝑥) = −5𝑥 2 + 2𝑥 − 4
7. 𝑔(𝑡) = 2 𝑡 2 + 𝑡
a.
b.
8. ℎ(𝑣) = 2𝑣 2 + 𝑣 + 3
a.
b.
9. 𝑟(𝑡) = 3𝑡 2 + 10
a.
b.
a.
b.
10. 𝑓(𝑥) = −𝑥 2 + 6𝑥 − 7
a.
b.
11. Does the graph of 𝑦 = −2𝑥 2 + 3𝑥 − 4 have any horizontal intercepts? Explain.
12.
a.
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Is the graph of 𝑦 = 5 𝑥 2 wider or narrower than the graph of 𝑦 = 𝑥 2 ? Why?
b. For the same input value, which graph would have a larger output value? Prove this using an
example.
13. Put the following in order from narrowest to widest.
a. 𝑦 = 0.5𝑥 2
b, 𝑦 = 8𝑥 2
c. 𝑦 = −2.3𝑥 2
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