Characteristics of Quadratic Functions

Section 2.2 beginning on page 56

The Big IdeasIn this section we will learn about….

• The properties of parabolaso Axis of symmetry o Vertex

• Finding the maximum and minimum values of a quadratic functiono The vertex is the maximum or minimumo The x-value of the vertex is the location of the max/min and the y-value

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-interceptso The x-intercepts are the values of x that make y=0o In real-world problems the x-intercepts are often starting and/or ending

points.

Core Vocabulary

Previously Learned:

• x-intercept

New Vocabulary:

• Axis of symmetry• Standard form• Minimum value• Maximum value• Intercept form

Properties of Parabolas

The axis of symmetry is a line that divides a parabola into mirror images.

The axis of symmetry passes through the vertex.

Vertex form :

The vertex is at the point

The axis of symmetry is the line .

** This is good info for your notebook

Using Symmetry to Graph a Parabola

(−3,4)

𝑥=−3

** Just list the basic steps in your notebook to refer to when doing similar problems.

Standard FormQuadratic equations can also be written in standard form, .

When given an equation in standard form you identify the key characteristics of the parabola in a different way.

has the same meaning in vertex form and standard form.

if the parabola is concave up and the vertex is a minimumif 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:

The value of in standard form is the y-intercept. (when )

The y-value of the axis of symmetry is found by plugging this x-value into the original equation.

Standard Form

Sound familiar??

Graphing a Quadratic Function in Standard Form

Example 2: Graph

Step 1: Identify a, b, and cStep 2: Find the vertex

Step 3: Plot the vertex and the axis of symmetry

Step 4: Plot the y-intercept and its reflection in the axis of symmetry

Step 5: Find another point to plot along with its reflection

Step 6: Draw a parabola through the points

𝑎=3 ,𝑏=−6 ,𝑐=1

𝑥=−𝑏2𝑎 𝑥=

62(3) 𝑥=1

𝑦= 𝑓 (1 )=3(1)2−6 (1 )+1 𝑦=−2

(1 ,−2) 𝑥=1

𝑐=1 (0,1)

𝑥=3 (3,10)𝑓 (3 )=10

Graphing Quadratic Functions2)

Maximum and Minimum ValuesBecause the vertex is the highest or lowest point on a parabola, its y-coordinate is the maximum value (when ) or the minimum value (when ) 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 ValueExample 3: Find the minimum or maximum value of . Describe the domain and range of the function and where the function is increasing and decreasing.

-Is there a maximum or minimum?

-Find the vertex (the y-value is the max/min)

-The Domain:

-The Range:

-Increasing/Decreasing?Since we have a minimum value, all of the y values will be at or above that minimum value.

, there is a minimum

𝑥=−𝑏2𝑎

𝑥=2

2(1/2) 𝒙=𝟐𝑦= 𝑓 (2 )=1

2(2)2−2 (2 )−1 𝒚=−𝟑

All Real Numbers

𝒚 ≥−𝟑

Because this function has a minimum, it is decreasing to the left of(the axis of symmetry) and increasing to the right of .

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, where .

Graphing a Quadratic Function in Intercept Form

Step 1: Identify the x-intercepts.

Step 2: Find the coordinates of the vertex.

Step 3: Draw a parabola through the vertex and the points where the x-intercepts occur.

𝑝=−3 𝑞=1(−3,0) (1,0)

𝑥=𝑝+𝑞2

¿−3+12

¿−22 ¿−1 𝒙=−𝟏

𝑦= 𝑓 (−1 )=−2(−1+3)(−1−1) 𝒚=𝟖¿−2 (2 )(−2) (−1,8)

Graphing a Quadratic Function in Intercept Form

Modeling With MathematicsExample 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 . 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 ….

The graph shows us that the distance travelled is ….

The y value of the vertex is the maximum (50,25).

25 yards

The difference in the x-values is the distance the ball traveled. (0,0) and (100,0) 100−0=0

100 yards

Modeling With MathematicsExample 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 . Which shot travels farther before hitting the ground? Which travels higher?

Height : 25 yardsDistance : 100 yards

To find the max height and distance traveled with the equation we can look at the equation in intercept form.

Find the x-intercepts….

Identify the distance travelled…

Use the x-intercepts to calculate the maximum height …

𝒇 (𝒙 )=−𝟎 .𝟎𝟐(𝒙−𝟎) (𝒙−𝟖𝟎 )(𝟎 ,0 )∧(𝟖𝟎 ,0)

80−0=80 Distance traveled = 80 yards

𝑥=𝑝+𝑞2

¿0+802

¿802 ¿ 45 𝒙=𝟒𝟓

𝑦= 𝑓 (45 )=−0.02(45)(45−80) 𝒚=𝟑𝟐

Maximum height = 32 yards

The first shot travels further but the second shot travels higher.