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Determining Intervals of a Quadratic Functions

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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.


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