1. Interpreting Key Features of Quadratic Functions

2. Warm-Up
Interpreting Key Features of Quadratic Functions

3. The object of a popular video game is to launch a boulder to knock over boxes, buildings, and other items. The graph below shows an obstacle on the left that the boulder must clear in order to knock over the stack of boxes on the right. The boulder will follow a parabolic path and will launch from (0, 0) and end at (8, 0).
Interpreting Key Features of Quadratic Functions

4. Will the boulder clear the obstacle? How do you know?
What are the x-intercepts for the parabola formed by the path of the boulder?
What is the axis of symmetry for the parabola formed by the path of the boulder? How do you know?
One possible path for the boulder is What is the vertex of the parabola created by this equation?
Will the boulder clear the obstacle? How do you know?
Will the boulder knock down the boxes? How do you know?
Interpreting Key Features of Quadratic Functions

5. The x-intercepts are 0 and 8.
What are the x-intercepts for the parabola formed by the path of the boulder?
The x-intercepts occur where the graph intersects the x-axis when y = 0.
The x-intercepts are 0 and 8.
Interpreting Key Features of Quadratic Functions

6. The midpoint between 0 and 8 is 4, so the axis of symmetry is x = 4.
What is the axis of symmetry for the parabola formed by the path of the boulder? How do you know?
The axis of symmetry is a vertical line through the middle of the parabola.
You can use the midpoint between the two x-intercepts to find the axis of symmetry.
The midpoint between 0 and 8 is 4, so the axis of symmetry is x = 4.
Interpreting Key Features of Quadratic Functions

7. The axis of symmetry goes through the vertex.
One possible path for the boulder is What is the vertex of the parabola created by this equation?
The axis of symmetry goes through the vertex.
The x-coordinate of the vertex is 4.
To find the y-coordinate, substitute 4 into the equation for the possible path of the boulder.
Interpreting Key Features of Quadratic Functions

8. Since y = 6, the vertex is (4, 6).
One possible path for the boulder is What is the vertex created by this equation?
Original equation
Substitute 4 for x.
Simplify.
Since y = 6, the vertex is (4, 6).
Interpreting Key Features of Quadratic Functions

9. Will the boulder clear the obstacle? How do you know?
Yes, it will. Sketch the boulder’s path using (0, 0),
(4, 6), and (8, 0), and you can see that the path is above the obstacle.
Interpreting Key Features of Quadratic Functions

10. Will the boulder knock down the boxes? How do you know?
No, it will not. The top right corner of the boxes is at the point (7, 2.5).
If you substitute 7 for x in the equation of the parabola, you find that the parabola contains the point (7, 2.625), which is slightly above the boxes.
The boulder will not come in contact with the boxes, though it is close.
Connection to the Lesson
Students will identify key features of quadratic functions such as the x-intercepts and vertex from the equation of the function.
Students will use key features to graph the quadratic function.
Interpreting Key Features of Quadratic Functions

11. Instruction
Interpreting Key Features of Quadratic Functions

12. Introduction
The tourism industry thrives on being able to provide travelers with an amazing travel experience. At the same time, tourism professionals need to know exactly how successful they are in their businesses.
Tour planners might use quadratic models and examine their key features to determine when profits are increasing or decreasing, when they maximized, and how they change month to month.
In this lesson, you will review the key features of a quadratic function and how to use graphs, tables, and verbal descriptions to identify and apply the key features.
Interpreting Key Features of Quadratic Functions

13. Key Concepts
The key features of a quadratic function are characteristics used to describe, draw, and compare quadratic functions.
These characteristics are: the x-intercept(s); the y-intercept; increasing and decreasing values of x; positive and negative graph intervals; minimum or maximum values; the axis of symmetry; and the end behavior of the graph of the function.
Recall each of the forms of quadratic functions, outlined on the following slides.
Interpreting Key Features of Quadratic Functions

14. Key Concepts, continued
Standard Form
The standard form, or general form, of a quadratic function is written as f(x) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term.
The y-intercept is the value of c.
The vertex of the function is found by first determining the x-coordinate, and then substituting this in the function to find the corresponding y-coordinate.
Interpreting Key Features of Quadratic Functions

15. Key Concepts, continued
Vertex Form
The vertex form of a quadratic function is written as f(x) = a(x – h)2 + k.
The vertex of the parabola is (h, k).
The axis of symmetry is identified from vertex form as x = h.
Factored Form
The factored form, or intercept form, of a quadratic function is written as f(x) = a(x – p) (x – q).
The x-intercepts of the function are p and q.
Interpreting Key Features of Quadratic Functions

16. Key Concepts, continued
The x-intercepts of a quadratic function occur when the parabola intersects the x-axis. In the graph at right, the x-intercepts are 2 and –2.
Interpreting Key Features of Quadratic Functions

17. Key Concepts, continued
All points on the x-axis have a y-coordinate of 0; therefore, the x-intercepts can also be found by identifying the value(s) of x when y = 0.
The table of values at right corresponds to the parabola shown on the previous slide. Notice that the x-intercepts, 2 and –2, can be found where the table shows that y is equal to 0. x y –4 12 –2 2 4
Interpreting Key Features of Quadratic Functions

18. Key Concepts, continued
The y-intercept is the y-coordinate of the point where the curve intersects the y-axis. The ordered pair that corresponds to a y-intercept is of the form (0, y).
Recall that the vertex is the maximum or minimum of the function.
The vertex is also the point where a parabola changes from increasing to decreasing or from decreasing to increasing.
Interpreting Key Features of Quadratic Functions

19. Key Concepts, continued
Increasing refers to the interval of a function for which the output values are becoming larger as the input values are becoming larger.
Decreasing refers to the interval of a function for which the output values are becoming smaller as the input values are becoming larger.
Recall that parabolas are symmetric to a vertical line that passes through the vertex, called the axis of symmetry.
Interpreting Key Features of Quadratic Functions

20. Key Concepts, continued
The reflection of every point on a parabola across the axis of symmetry is another point on the parabola. A point and its reflection are equidistant from the axis of symmetry.
Read the graph from left to right to determine when the function is increasing or decreasing.
If your pencil tip goes down as you move toward increasing values of x, then the function is decreasing.
If your pencil tip goes up as you move toward increasing values of x, then the function is increasing.
Interpreting Key Features of Quadratic Functions

21. Key Concepts, continued
If the graph of a quadratic function has a minimum value, it will start by decreasing toward the vertex; after the vertex, it will increase.
If the graph of a quadratic function has a maximum value, it will start by increasing toward the vertex; after the vertex, it will decrease.
The vertex is called an extremum. Extrema are the maxima or minima of a function.
Interpreting Key Features of Quadratic Functions

22. Key Concepts, continued
The concavity of a parabola refers to the arch of the curve, either upward or downward.
A quadratic function that has a minimum value has positive concavity and is concave up because the graph of the function is bent upward.
A quadratic function that has a maximum value has negative concavity and is concave down because the graph of the function is bent downward.
Interpreting Key Features of Quadratic Functions

23. Key Concepts, continued
The graphs that follow demonstrate examples of parabolas as they decrease and then increase, and vice versa.
Trace the path of each parabola from left to right with your pencil to see the difference.
Interpreting Key Features of Quadratic Functions

24. Key Concepts, continued
Decreasing then Increasing
Vertex: (0, –4); minimum
x < 0: decreasing
x > 0: increasing
Direction: concave up
Interpreting Key Features of Quadratic Functions

25. Key Concepts, continued
Increasing then Decreasing
Vertex: (0, 4); maximum
x < 0: increasing
x > 0: decreasing
Direction: concave down
Interpreting Key Features of Quadratic Functions

26. Key Concepts, continued
End behavior is the behavior of the graph as x approaches positive or negative infinity.
If the highest exponent of a function is even, and the coefficient of the same term is positive, then the function is approaching positive infinity as x approaches both positive and negative infinity.
If the highest exponent of a function is even, but the coefficient of the same term is negative, then the function is approaching negative infinity as x approaches both positive and negative infinity.
Interpreting Key Features of Quadratic Functions

27. Key Concepts, continued
Even and Positive
f(x) = x2 – 4
Highest exponent: 2
Coefficient of x2: positive
As x approaches positive infinity, f(x) approaches positive infinity.
As x approaches negative infinity, f(x) approaches positive infinity.
Interpreting Key Features of Quadratic Functions

28. Key Concepts, continued
Even and Negative
f(x) = –x2 + 4
Highest exponent: 2
Coefficient of x2: negative
As x approaches positive infinity, f(x) approaches negative infinity.
As x approaches negative infinity, f(x) approaches negative infinity.
Interpreting Key Features of Quadratic Functions

29. Common Errors/Misconceptions
incorrectly identifying when a function is increasing or decreasing
Interpreting Key Features of Quadratic Functions

30. Guided Practice Example 1
A local store’s monthly revenue from T-shirt sales, f(x), as a function of price, x, is modeled by the function f(x) = –5x x – 7. Use the equation and the graph on the next slide to answer the following questions:
At what prices is the revenue increasing? Decreasing? What is the maximum revenue?
What prices yield no revenue?
Interpreting Key Features of Quadratic Functions

31. Guided Practice: Example 1, continued
Interpreting Key Features of Quadratic Functions

32. Guided Practice: Example 1, continued
Determine when the function is increasing and decreasing.
Use your pencil to determine for which x-values the function is increasing and decreasing.
Moving from left to right, trace your pencil along the function.
The function increases until it reaches the vertex, then it decreases.
Interpreting Key Features of Quadratic Functions

33. Guided Practice: Example 1, continued
The vertex of this function has an x-value of 15.
The revenue is increasing when the price per shirt is less than $15, or when x < 15.
The revenue is decreasing when the price per shirt is more than $15, or when x > 15.
Interpreting Key Features of Quadratic Functions

34. Guided Practice: Example 1, continued Determine the maximum revenue.
Use the vertex of the function to determine the maximum revenue.
The T-shirt price that maximizes revenue is $15, or x = 15.
The maximum revenue is the corresponding y-value.
Since it is difficult to estimate accurately from this graph, substitute the value of x into the function and solve.
Interpreting Key Features of Quadratic Functions

35. Guided Practice: Example 1, continued The maximum revenue is $1,118.
f(x) = –5x x – 7
Original function
f(15) = –5(15) (15) – 7
Substitute 15 for x.
f(15) = 1118
Simplify.
Interpreting Key Features of Quadratic Functions

36. ✔ Guided Practice: Example 1, continued
Determine the prices that yield no revenue.
Identify the x-intercepts.
The x-intercepts are 0 and 30, so the store has no revenue when the shirts cost $0 and when the shirts cost $30 or more. ✔
Interpreting Key Features of Quadratic Functions

37. Guided Practice: Example 1, continued
Interpreting Key Features of Quadratic Functions

38. Guided Practice Example 2
A quadratic function has a minimum value of –5 and
x-intercepts of –8 and 4.
What is the value of x that minimizes the function?
For what values of x is the function increasing?
For what values of x is the function decreasing?
Interpreting Key Features of Quadratic Functions

39. Guided Practice: Example 2, continued
Determine the x-value that minimizes the function.
The graphs of quadratic functions are symmetric functions about the vertex and the axis of symmetry, the line that divides the parabola in half and passes through the vertex.
The x-value that minimizes the function is the midpoint of the two x-intercepts.
Interpreting Key Features of Quadratic Functions

40. Guided Practice: Example 2, continued
Find the midpoint of the x-intercepts by taking the average of the two values.
The value of x that minimizes the function is –2.
Interpreting Key Features of Quadratic Functions

41. ✔ Guided Practice: Example 2, continued
Determine when the function is increasing and decreasing.
Use the vertex to determine for which x-values the function is increasing and decreasing.
The minimum value is –5 and the vertex of the function is (–2, –5).
From left to right, the function decreases as it approaches the minimum and then increases.
The function is decreasing when x < –2 and increasing when x > –2. ✔
Interpreting Key Features of Quadratic Functions

42. Guided Practice: Example 2, continued
Interpreting Key Features of Quadratic Functions

43. Guided Practice Example 3 The table shows the predicted
temperatures for a summer
day in Woodland, California.
At what times is the temperature
increasing? Decreasing?
Time
Temperature (⁰F)
8 A.M. 52
10 A.M. 64
12 P.M. 72
2 P.M. 78
4 P.M. 81
6 P.M. 76
Interpreting Key Features of Quadratic Functions

44. Guided Practice: Example 3, continued
Use the table to determine approximate intervals of increasing and decreasing temperatures.
Examine what is happening to the temperatures as the day progresses from morning to evening.
The values are increasing when they get progressively larger in the table and decreasing when they get progressively smaller in the table.
At 4 P.M. the temperature is at its highest, 81°. Before this time, the temperatures are increasing, and after this, the temperatures are decreasing.
Interpreting Key Features of Quadratic Functions

45. Guided Practice: Example 3, continued
Use graphing technology to verify information that is assumed in the table.
On a TI-83/84:
Step 1: Press [STAT].
Step 2: Press [ENTER] to select Edit.
Step 3: Enter x-values into L1. Enter times based on a 24-hour clock for times after 12 P.M. For example, 1 P.M. should be entered as hour 13.
Step 4: Enter y-values into L2.
Step 5: Press [2ND][Y=].
Interpreting Key Features of Quadratic Functions

46. Guided Practice: Example 3, continued
On a TI-83/84, continued:
Step 6: Press [ENTER] twice to turn on Stat Plot.
Step 7: Press [ZOOM][9] to select ZoomStat.
Step 8: Press [STAT].
Step 9: Arrow to the right to select Calc.
Step 10: Press [5] to select QuadReg.
Step 11: Enter [L1][,][L2], Y1. To enter Y1, press [VARS] and arrow over to the right to “Y-VARS.” Select 1: Function. Select 1: Y1.
Step 12: Press [ENTER] to see the graph of the data and the quadratic equation.
Interpreting Key Features of Quadratic Functions

47. Guided Practice: Example 3, continued
On a TI-Nspire:
Step 1: Press the [home] key and select the Lists & Spreadsheet icon.
Step 2: Name Column A “time” and Column B “temperature.”
Step 3: Enter x-values under Column A. Enter times based on a 24-hour clock for times after 12 P.M. For example, 1 P.M. is entered as hour 13.
Step 4: Enter y-values under Column B.
Step 5: Select Menu, then 3: Data, and then 6: Quick Graph.
Interpreting Key Features of Quadratic Functions

48. Guided Practice: Example 3, continued
On a TI-Nspire, continued:
Step 6: Press [enter].
Step 7: Move the cursor to the x-axis and choose “time.”
Step 8: Move the mouse to the y-axis and choose “temperature.”
Step 9: Select Menu, then 4: Analyze, then 6: Regression, and then 4: Show Quadratic.
Step 10: Move the cursor over the equation and press the center key in the navigation pad to drag the equation for viewing, if necessary.
Interpreting Key Features of Quadratic Functions

49. Guided Practice: Example 3, continued
Interpreting Key Features of Quadratic Functions

50. Guided Practice: Example 3, continued State your conclusion.
The highest temperature in the table occurs at the maximum y-value on the curve, or in this case, the time at which the temperature goes from increasing to decreasing.
The highest temperature is 81°, and this occurs at 4 P.M. The maximum temperature appears to happen at hour 16, according to the quadratic model, or at around 4 P.M.
Interpreting Key Features of Quadratic Functions

51. ✔ Guided Practice: Example 3, continued State your conclusion.
The high temperature in the graphed quadratic model is approximately 79°, which is slightly less than 81°, the predicted temperature for that hour. ✔
Interpreting Key Features of Quadratic Functions