Transcript Characteristics of Quadratic Functions

Characteristics of Quadratic
Functions
Section 2.2 beginning on page 56
The Big Ideas
In this section we will learn about….
• The properties of parabolas
o Axis of symmetry
o Vertex
• Finding the maximum and minimum values of a quadratic function
o The vertex is the maximum or minimum
o The x-value of the vertex is the location of the max/min and the yvalue is the max/min. (This is a concept often used in solving real-world
problems. )
o The function will be increasing on one side of the vertex and decreasing
on the other side of the vertex.
• Graphing quadratic functions using x-intercepts
o The x-intercepts are the values of x that make y=0
o In real-world problems the x-intercepts are often starting and/or ending
points.
Core Vocabulary
Previously Learned:
New Vocabulary:
• x-intercept
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Axis of symmetry
Standard form
Minimum value
Maximum value
Intercept form
Properties of Parabolas
** This is good info for your notebook
The axis of symmetry is a line that divides a parabola into mirror images.
The axis of symmetry passes through the vertex.
Vertex form : 𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 +𝑘
The vertex is at the point (ℎ, 𝑘)
The axis of symmetry is the line 𝑥 = ℎ.
Using Symmetry to Graph a Parabola
𝑥 = −3
(−3,4)
** Just list the basic steps in your notebook to
refer to when doing similar problems.
Standard Form
Quadratic equations can also be written in standard form, 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
When given an equation in standard form you identify the key characteristics of the
parabola in a different way.
𝒂 has the same meaning in vertex form and standard form.
if 𝒂 > 𝟎 the parabola is concave up and the vertex is a minimum
if 𝒂 < 𝟎 the parabola is concave down and the vertex is a maximum
The x-value of the vertex and the axis of symmetry can be found using the formula:
𝒙=
−𝒃
𝟐𝒂
The y-value of the axis of symmetry is found by plugging
this x-value into the original equation.
The value of 𝒄 in standard form is the y-intercept. (when 𝑥 = 0, 𝑦 = 𝑐)
Standard Form
Sound
familiar??
Graphing a Quadratic Function in
Standard Form
Example 2: Graph 𝑓 𝑥 = 3𝑥 2 − 6𝑥 + 1
Step 1: Identify a, b, and c
Step 2: Find the vertex
−𝑏
𝑥=
2𝑎
6
𝑥=
2(3)
𝑎 = 3, 𝑏 = −6, 𝑐 = 1
𝑥=1
𝑦 = 𝑓 1 = 3(1)2 −6 1 + 1
𝑦 = −2
Step 3: Plot the vertex and the axis of symmetry
(1, −2)
𝑥=1
Step 4: Plot the y-intercept and its reflection in
the axis of symmetry
(0,1)
𝑐=1
Step 5: Find another point to plot along with its
reflection 𝑥 = 3
(3,10)
𝑓 3 = 10
Step 6: Draw a parabola through the points
Graphing Quadratic Functions
2) 𝑔 𝑥 = 2(𝑥 − 2)2 +5
Maximum and Minimum Values
Because the vertex is the highest or lowest point on a parabola, its y-coordinate is the maximum
value (when 𝒂 < 𝟎) or the minimum value (when 𝒂 > 𝟎) of the function.
The vertex lies on the axis of symmetry so the function is increasing on one side of the axis of
symmetry and decreasing on the other side.
Finding a Minimum or Maximum Value
1
Example 3: Find the minimum or maximum value of 𝑓 𝑥 = 2 𝑥 2 − 2𝑥 − 1.
Describe the domain and range of the function and where the function is increasing
and decreasing.
-Is there a maximum or minimum? 𝑎 > 0 , there is a minimum
-Find the vertex (the y-value is the max/min)
1
𝑦 = 𝑓 2 = (2)2 −2 2 − 1
2
−𝑏
𝑥=
2𝑎
2
𝑥=
2(1/2)
𝒙=𝟐
𝒚 = −𝟑
-The Domain: All Real Numbers
-The Range:
𝒚 ≥ −𝟑
Since we have a minimum value, all of the y values
will be at or above that minimum value.
-Increasing/Decreasing? Because this function has a minimum, it is decreasing to the
left of 𝑥 = 2 (the axis of symmetry) and increasing to the
right of 𝑥 = 2.
Finding a Minimum or Maximum
Graphing Quadratic Functions Using
x-intercepts
When the graph of a quadratic function has at least one x-intercept, the function can
be written in intercept form, 𝑓 𝑥 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞) where 𝑎 ≠ 0.
Graphing a Quadratic Function in
Intercept Form
Step 1: Identify the x-intercepts.
𝑝 = −3
(−3,0)
𝑞=1
(1,0)
Step 2: Find the coordinates of the vertex.
𝑥=
𝑝+𝑞
2
=
−3 + 1
2
=
−2
2
= −1
𝑦 = 𝑓 −1 = −2(−1 + 3)(−1 − 1)
= −2 2 (−2)
𝒙 = −𝟏
𝒚=𝟖
(−1,8)
Step 3: Draw a parabola through the vertex and the points where the x-intercepts occur.
Graphing a Quadratic Function in
Intercept Form
Modeling With Mathematics
Example 5: The parabola shows the path of your fist golf shot, where x is the horizontal
distance (in yards) and y is the corresponding height (in yards). The path of your second
shot can be modeled by the function 𝒇 𝒙 = −𝟎. 𝟎𝟐𝒙 𝒙 − 𝟖𝟎 . Which shot travels
farther before hitting the ground? Which travels higher?
We are comparing the maximum heights and the distance the
ball traveled. One shot is represented as a graph, and the other
as an equation.
The graph shows us that the maximum height is ….
25 yards
The y value of the vertex is the maximum (50,25).
The graph shows us that the distance travelled is ….
100 yards
The difference in the x-values is the distance the ball traveled. (0,0) and (100,0)
100 − 0 = 0
Modeling With Mathematics
Example 5: The parabola shows the path of your fist golf shot, where x is the horizontal
distance (in yards) and y is the corresponding height (in yards). The path of your second
shot can be modeled by the function 𝒇 𝒙 = −𝟎. 𝟎𝟐𝒙 𝒙 − 𝟖𝟎 . Which shot travels
farther before hitting the ground? Which travels higher?
To find the max height and distance traveled with the equation
we can look at the equation in intercept form.
𝒇 𝒙 = −𝟎. 𝟎𝟐(𝒙 − 𝟎) 𝒙 − 𝟖𝟎
Find the x-intercepts….
Height : 25 yards
Distance : 100 yards
𝟎, 0 and (𝟖𝟎, 0)
Identify the distance travelled… 80 − 0 = 80
Distance traveled = 80 yards
Use the x-intercepts to calculate the maximum height …
𝑝+𝑞
𝑥=
2
=
0 + 80
2
=
80
2
= 45
𝑦 = 𝑓 45 = −0.02(45)(45 − 80)
𝒙 = 𝟒𝟓
𝒚 = 𝟑𝟐
Maximum height = 32 yards
The first shot travels
further but the second
shot travels higher.