Part 1
Ask students the following questions: “What do you know about a quadratic function? What does one look like? Where have you seen a quadratic function? What does ‘parabola’ mean? How does it relate to a quadratic function? When would you need to use a parabola? Who uses parabolas on a daily basis when solving problems? In what forms can we represent a quadratic function?” [IS.4 - All Students]
Students should respond that quadratic functions can be represented in equation form, as tables and graphs. Quadratics can also be described using words.
“Before we explore the various representations of quadratic functions, let’s make sure we understand the terminology of quadratic function and parabola. A quadratic function is the actual function being described in various forms. A parabola is the shape of the graph. The parabola is the shape of the graph of a quadratic function.”
“Can someone provide a rough definition for a quadratic function?”
Students might respond that a quadratic function is a function that is not linear and one that increases or decreases more rapidly from the vertex of the graph.
“What would the equation look like?”
Students should respond that the leading term will have a variable of degree two. Examples include: y = 3x², y = –x², y = 12x² + 2, y = x² – 3, and so on. [IS.5 - All Students]
“What would the table look like? In other words, how will the change in y-values compare to the change in x-values? How will the change in y-values from a quadratic function compare to the change in y-values from a linear function?”
Students should respond that the change in y-values is greater than the change in
x-values. Also, the change in y-values from a quadratic function decreases/increases more rapidly than those from a linear function. The rate of change is not constant with a quadratic function.
“What would the graph look like? What would the shape be?”
Students should predict the parabola u-shape. Encourage students to draw it as a parabola that opens up and a parabola that opens down. If the parabola opens up, it has a minimum point; if it opens down, it has a maximum point.
“Now, that we’ve made some predictions, let’s explore and determine various representations of quadratic functions and the parabolas thereof.”
“First, let’s examine the forms of a quadratic function. A quadratic function can take the form y = ax² + bx + c, y = ax² + c ”
Students making a table of values to investigate the relationship will add to their understanding of the behavior of the function.
x
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y= x²
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−3
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9
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−2
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4
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−1
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1
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0
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0
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1
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1
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2
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4
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3
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9
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Students should notice that the y-values are all positive; thus the graph will be in the shape of a U, opening up. In other words, the orientation will be positive. Students should also note the origin is at (0, 0). Ask students to plot the ordered pairs, using the table above. (Make comparisons between the table of values for the absolute value parent function and this table, as well as comparisons between the graph of a quadratic and graph of an absolute value function.)
Students should help create the graph below.
Review with students that the points should be connected to demonstrate the function. Doing so will create the graph below.
Ask students to describe the quadratic parent function in words.
Students should be able to explain that the quadratic parent function y = x2 takes each input (x), squares it, and gives the output (y). The rate of change decreases and then increases more rapidly than that of a linear function. The vertex is at the origin, or (0, 0), and the domain is all real numbers, or (−∞, ∞), with a range of all real numbers greater than or equal to 0, or [0, ∞). Thus, this graph has a minimum, and it is the vertex, or
(0, 0).
Examples might include tossing a ball, shooting an arrow, other projectile-type problems, modeling profits, losses, or costs, etc.
Sample problem: Kevin tosses a coin off of his hotel balcony onto the sidewalk below. The balcony is 45 feet above the ground. The height of the coin after t seconds is given by the function h = 36t2 + 45.
Create a table of values, graph the function, and describe the overall graph, including vertex, domain, and range.
t
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h = 36t2 + 45
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0
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45
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1
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9
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2
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−99
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Remind students that the graph represents the behavior of the measurements of the coin (distance and time) as it falls, not the path of the coin.
There are several questions we need to answer from this graph:
· What happens at 0 feet?
· What does the negative h-value represent?
· When exactly does the coin hit the sidewalk? How do we determine the answer?
· What is the domain and range? What is the reasonable domain and range, related to the context of the problem?
· What is the vertex and what does it mean? What is the orientation?
Let’s answer each question.
· What happens at 0 feet?
At 0 feet, the coin hits the sidewalk.
· What does the negative point mean?
The negative point would mean that after 2 seconds, the coin is 99 feet below the ground. Such a point is not reasonable within the context of the problem.
· When exactly does the coin hit the sidewalk? How do we determine the answer?
In order to find out when the coin hits the sidewalk, we must set the function equal to 0. In other words, we will replace h with 0. So,
After 1.12 seconds, the coin hits the sidewalk.
· What is the domain and range? What is the reasonable domain and range, related to the context of the problem?
The domain is [0, 2). The range is (−∞, 45]. However, the reasonable domain and range are quite different. The reasonable domain is [0, 1.12]. The reasonable range is [0, 45]. is an exact solution.
In other words, reasonably, you only have seconds of 0 through 1.12 until the coin hits the ground. The distance goes from 45 feet above the ground to 0 feet above the ground, or ground level.
· What is the vertex and what does it mean? What is the orientation?
The vertex is the highest or lowest point of the parabola. In this case, the vertex is at (0, 45). It is a maximum, meaning the parabola opens down. It is the point of the maximum height of the coin before it is dropped. The orientation is negative, since the graph opens down. The coin is dropped and thus the height decreases as time passes.
“What else can be discerned from the graph?”
We can determine that there is definitely not a constant rate of change, meaning the coin does not travel from his hand to the sidewalk at a constant rate of speed.
“How would we know that this sort of scenario would best be modeled by a quadratic function, or parabola? Why not a linear function, or absolute value function? Why not an exponential function?” Note that students may not have experience with these types of functions.
Place each student with a partner and have the group create a real-world quadratic function, table of values, and graph. Ask students to be as specific about all of the details of the graph and table as possible. How do they relate to the problem? How do you describe them in everyday language?
“Let’s now look at some transformations of quadratic functions. When I say ‘transformations,’ what comes to mind? What examples can you give? When would we possibly need to know about transformations or use transformations to solve a problem?”
Review the quadratic parent function and expose students to a variety of transformed quadratic functions. A parent function is a function that has not been transformed.
“Let’s create a list of possible transformed quadratic functions. Imagine that you are transforming the quadratic parent function. How could you do that? How many ways? Let’s create some of them.”
Here are some examples:
y = 4x²
y = 2x²
y = –4x²
y = –2x²
y = –x²
y = x² + 4
y = (x + 4)²
y = x² – 4
y = (x – 4)²
y = (x + 4)² – 4
y = –4(x – 4)²
“Let’s look at each type of transformation, starting with simple transformations and ending with combinations of shifts.”
Have students predict what each transformation will look like. Divide students into groups of three to four. Ask students to fill in the chart below. Note: Students will draw the predicted graph before they create the table of values and construct the actual graph. Students will be comparing each transformed function to the quadratic parent function. Use any or all of the examples provided above. Ask students to investigate other functions by adding more rows to the table.
Function
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Predicted Graph
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Table of Values
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Actual Graph
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After students graph each function by hand, have them check their graphs using either the graphing calculator or GeoGebra. With the graphing calculator, students can graph using either the table feature, or by entering the equation for the function into the y = screen. GeoGebra will allow students to plot points and graph the function.
Students can have a nice discussion, concerning the reasons for their predictions. Students will discover reasons for their thinking regarding shrinking and stretching a graph. Were their intuitions correct? “How can we understand the connection of the values in the equation of a stretched/compressed graph with the actual appearance of the graph? In other words, how can we understand the relationship without memorizing a rule?”
“What did we learn about shifts within and outside the parentheses? Were the results what you thought they would be? Why or why not? What did you learn? How did you make the connections, thus developing your conceptual understanding?”
“What happened with combinations of transformations with the quadratic functions? Which transformation did you do first? Did the order matter? Why or why not?”
Hold the discussion first, and then offer a more guided examination of each piece. Doing so allows students to voice their ideas/concerns first, and thus provides a place from which to catapult.
Distribute copies of the Lesson 2 Graphic Organizer (M-A2-7-2_Lesson 2 Graphic Organizer.doc) for practice on recognizing the characteristics of quadratic functions.
“Let’s go through and compare the equations, tables, and graphs for each of our examples. We will also provide a description of what happened using words. Underneath each graph, we will note the domain, range, and vertex.”
Transformations of the Quadratic Parent Function y = x2
Function
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Table
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Graph
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What Happened?
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y = −x2
A
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x
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y
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−2
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−4
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−1
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−1
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0
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0
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1
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−1
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2
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−4
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|
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The graph was reflected across the x-axis. In other words, each positive y was replaced by a negative y.
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y = 4x2
B
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x
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y
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−2
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16
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−1
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4
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0
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0
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1
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4
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2
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16
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The graph was made narrower because each squared x-value was multiplied by 4, thus forcing a higher y-value for each x-value.
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y = 2x2
C
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x
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y
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−2
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8
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−1
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2
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0
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0
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1
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2
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2
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8
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The graph was made narrower because each squared x-value is multiplied by 2, thus forcing a higher y-value for each x-value. However, the graph is not as narrow as the preceding one because the coefficient is smaller.
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y = −4x2
D
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x
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y
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−2
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−16
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−1
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−4
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0
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0
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1
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−4
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2
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−16
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The graph is narrower and reflected across the x-axis.
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y = −2x2
E
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x
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y
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−2
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−8
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−1
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−2
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0
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0
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1
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−2
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2
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−8
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|
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The graph is narrower and reflected across the x-axis. Note: The graph is not as narrow as the previous one.
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y = x2 +4
F
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x
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y
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−2
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8
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−1
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5
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0
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4
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1
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5
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2
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8
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The graph is simply shifted up 4 units. In other words, each y-value increased by 4 units.
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y = (x + 4)2
G
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x
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y
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−8
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16
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−6
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4
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−4
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0
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−2
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4
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−1
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9
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0
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16
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1
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25
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2
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36
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The graph is shifted to the left 4 units. The x-intercept is (−4, 0), showing a shift of 4 units to the left from the origin.
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y = x2 − 4
H
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x
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y
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−2
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0
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−1
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−3
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0
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−4
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1
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−3
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2
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0
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|
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The graph is shifted down 4 units. Each y-value is 4 units less than the corresponding y-value from the parent function.
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y = (x – 4)2
I
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x
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y
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−1
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25
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0
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16
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1
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9
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2
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4
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4
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0
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6
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8
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8
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16
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The graph is shifted 4 units to the right. The x-intercept is (4, 0), showing a shift of 4 units to the right of the origin.
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y = (x + 4)2 − 4
J
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x
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y
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−8
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12
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−6
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0
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−2
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0
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−1
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5
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0
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12
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1
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21
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The graph is shifted 4 units to the left and shifted 4 units down.
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y = −4(x – 4)2
K
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The graph is reflected across the x-axis, made narrower, and shifted to the right 4 units.
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“Now, let’s look at all functions graphed on the same graph, including the parent quadratic function. We will label each function with a letter.” See the chart above.
Note: Graphs of A, B, C, D, E, F, G, H, I, J, K are dark blue, green, red, light blue, purple, yellow, light green, blue-green, pink, purple, and green respectively.
Divide students into partners. Have each group determine the domain, range, and vertex for the transformed functions above. The domain and range should be written in both interval notation and word form.
Part 2
Have students explore quadratic functions and transformations of functions using NLVM’s virtual Grapher applet available at http://nlvm.usu.edu/en/nav/frames_asid_109_g_4_t_2.html?open=activities&from=category_g_4_t_2.html.
Students should use the applet to explore the parent function y = x² and transformations, including reflection across the x-axis, y = –x². Students should also become adept at other transformations, such as slides to the right and left and shifts up and down. By exploring and discovering with the applet, students should record notes and observances related to connections between the function equation and resulting graph. Have students explore min/max (vertex), domain, range, and rate of change. Students should also use the applet to explore the changes in the width of the function when varying the value of the coefficient of a.
Distribute copies of the Quadratic Modeling Worksheet (M-A2-7-2_Quadratic_Modeling_Worksheet.docx and M-A2-7-2_Quadratic_Modeling_Worksheet_Key.docx).
Review
1. Provide a matching activity in which students match the graphs and tables of various quadratic functions with their equations. Include 5–10 graphs and tables.
2. Have students write rules for transformations of quadratic functions. Algorithms should be detailed and specific.
3. Ask students to write another word problem that involves a quadratic function. Students should solve the problem using a variety of representations.
Extension:
· Have students develop and model quadratic functions of the form
y = ax² + bx + c. Ask students to create a real-world problem with such a function form. Students should explain the meaning of a, b, and c within the context of the problem.