1. Determining Intervals of a Quadratic Functions
Quadratics Unit
Determining Intervals of a Quadratic Functions
Medina

2. Key Points in Determining Intervals of a Quadratic Function
Vertex:
The vertex helps determine which part of the quadratic function is increasing from left to right.
and which part of the quadratic function is decreasing, from left to right.

3. Key Points in Determining Intervals of a Quadratic Function
𝑥-intercept(s):
The 𝑥-intercept(s) helps determine which part of the quadratic function is positive.
★Any value(s) above the 𝑥-axis. + + +
𝑥-axis − − −
★Any value(s) below the 𝑥-axis.
and which part of the quadratic function is negative.

4. Determining Intervals
𝑦 = 𝒙² + 2𝒙 −𝟑
Step 1: Graph the function
Step 2: Identify key points of intervals + +
Vertex:
( −𝟏 , −𝟒 )
𝑥-intercept(s): −
(−𝟑 , 𝟎 )
(𝟏 , 𝟎 )
State the Domain
Increasing
Decreasing
State the Range
Positive
Negative
{ −∞ ≤ 𝑥 ≤ +∞ } or
(−∞, ∞ )
{𝑥 >−1} or
(−1, ∞ )
{ 𝑥 <−1 } or
(−∞, −1 )
{ 𝑦 ≥−4} or
−4 , ∞ )
{𝑥<−3} and 𝑥>1 or
(−∞, −3 ) and (1, ∞ )
{−3<𝑥<1} or
(−3, 1)

5. Determining Intervals
𝑦 =−𝒙² +𝟒𝒙 −𝟑
Step 1: Graph the function +
Step 2: Identify key points of intervals
Vertex:
( 𝟐, 𝟏)
𝑥-intercept(s):
(𝟏 , 𝟎 )
(𝟑 , 𝟎 ) − −
State the Domain
Increasing
Decreasing
State the Range
Positive
Negative
{ −∞ ≤ 𝑥 ≤ +∞ } or
(−∞, ∞ )
{𝑥 <2} or
(−∞, 2 )
{ 𝑥 >2 } or
(2, ∞ )
{ 𝑦 ≥−4} or
−4 , ∞ )
{1<𝑥<3} or
{𝑥<1} and 𝑥>3 or

6. Determining Intervals

7. Determining Intervals
𝑦 = 𝒙 2 +𝟒𝒙+𝟒
Step 1: Graph the function
Step 2: Identify key points of intervals + +
Vertex:
( 𝟐, 𝟎 )
𝑥-intercept(s):
(𝟐, 𝟎 )
State the Domain
Increasing
Decreasing
State the Range
Positive
Negative
{ −∞ ≤ 𝑥 ≤ +∞ } or
(−∞, ∞ )
{𝑥 >2} or
(2, ∞ )
{ 𝑥 <2} or
(−∞, 2)
{ 𝑦 ≥2} or
2 , ∞ )
{𝑥<2} and 𝑥>2 or
(−∞, 2) and (2, ∞ )
There is no negative interval because there are no values below the 𝑥-axis.

8. Determining Intervals of Real World Quadratics
Billy hits a baseball into the air with the height of ℎ 𝑡 =− 𝑡− ,
where ℎ 𝑡 is measure in feet and 𝑡 is measure in seconds:
Step 1: Identify key points of intervals
Vertex:
( 𝟑, 𝟏𝟑)
𝑥-intercept(s): +
Starting point of the situation
Ending point of the situation
(𝟎 , 𝟒 )
(𝟔.𝟔 , 𝟎 )
State the Domain
Increasing
Decreasing
State the Range
Positive
Negative
{ 𝟎≤𝒕≤𝟔.𝟔 } or
𝟎 , 𝟔.𝟔
{ 𝟎<𝒕<𝟑 } or
𝟎 , 𝟑
{ 𝟑<𝒕<𝟔.𝟔 } or
𝟑 , 𝟔.𝟔
{ 𝟎≤𝒉(𝒕)≤𝟏𝟑 } or
𝟎 , 𝟏𝟑
{ 𝟎<𝒕<𝟔.𝟔 } or
𝟎 , 𝟔.𝟔
There is no negative interval because time and height in a real world problem can never be negative.