Math 2 Fall Semester Final Review

Selected Response problems can have more than 1 correct answer. Select ALL the factors of the quadratic function: 𝑓 𝑥 = 𝑥 2 −6𝑥+5 a. x+3 b. x-5 c. x+5 d. x+1 e. x-2 f. x-1 Solve the following quadratic equation: (𝑥+3) 2 −24=0 . 𝑆𝑒𝑙𝑒𝑐𝑡 𝒂𝒍𝒍 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠. a. 𝑥=−3 24 b. 𝑥=3 ± 24 c. 𝑥=−3 ± 24 d. 𝑥=−3 ±4 6 e. 𝑥=−3 ±2 6 f. 𝑥=3 24

Selected Response problems can have more than 1 correct answer. Select ALL the factors of the quadratic function: 𝑓 𝑥 = 𝑥 2 −6𝑥+5 a. x+3 b. x-5 c. x+5 d. x+1 e. x-2 f. x-1 Solve the following quadratic equation: (𝑥+3) 2 −24=0 . 𝑆𝑒𝑙𝑒𝑐𝑡 𝑎𝑙𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠. a. 𝑥=−3 24 b. 𝑥=3 ± 24 c. 𝑥=−3 ± 24 d. 𝑥=−3 ±4 6 e. 𝑥=−3 ±2 6 f. 𝑥=3 24

Selected Response problems can have more than 1 correct answer. Select ALL the factors of the quadratic function: 𝑓 𝑥 = 𝑥 2 −6𝑥+5 a. x+3 b. x-5 c. x+5 d. x+1 e. x-2 f. x-1 Solve the following quadratic equation: (𝑥+3) 2 −24=0 . 𝑆𝑒𝑙𝑒𝑐𝑡 𝑎𝑙𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠. a. 𝑥=−3 24 b. 𝑥=3 ± 24 c. 𝑥=−3 ± 24 d. 𝑥=−3 ±4 6 e. 𝑥=−3 ±2 6 f. 𝑥=3 24

Selected Response problems can have more than 1 correct answer. Select ALL the factors of the quadratic function: 𝑓 𝑥 = 𝑥 2 −6𝑥+5 a. x+3 b. x-5 c. x+5 d. x+1 e. x-2 f. x-1 Solve the following quadratic equation: (𝑥+3) 2 −24=0 . 𝑆𝑒𝑙𝑒𝑐𝑡 𝑎𝑙𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠. a. 𝑥=−3 24 b. 𝑥=3 ± 24 c. 𝑥=−3 ± 24 d. 𝑥=−3 ±4 6 e. 𝑥=−3 ±2 6 f. 𝑥=3 24

Selected Response problems can have more than 1 correct answer. Select ALL the factors of the quadratic function: 𝑓 𝑥 = 𝑥 2 −6𝑥+5 a. x+3 b. x-5 c. x+5 d. x+1 e. x-2 f. x-1 Solve the following quadratic equation: (𝑥+3) 2 −24=0 . 𝑆𝑒𝑙𝑒𝑐𝑡 𝑎𝑙𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠. a. 𝑥=−3 24 b. 𝑥=3 ± 24 c. 𝑥=−3 ± 24 d. 𝑥=−3 ±4 6 e. 𝑥=−3 ±2 6 f. 𝑥=3 24

Some Problems are Matching. For each quadratic , match the correct minimum value. 1. 𝑓 𝑥 = (𝑥−3) 2 −5 a. -2 b. -5 c. -8 d. 2 e. 8 f. 5 2. 𝑓 𝑥 = (𝑥+5) 2 −8 3. 𝑓 𝑥 = (𝑥+1) 2 +2

Some Problems are Matching. For each quadratic , match the correct minimum value. 1. 𝑓 𝑥 = (𝑥−3) 2 −5 a. -2 b. -5 c. -8 d. 2 e. 8 f. 5 2. 𝑓 𝑥 = (𝑥+5) 2 −8 3. 𝑓 𝑥 = (𝑥+1) 2 +2

Some Problems are Matching. For each quadratic , match the correct minimum value. 1. 𝑓 𝑥 = (𝑥−3) 2 −5 a. -2 b. -5 c. -8 d. 2 e. 8 f. 5 2. 𝑓 𝑥 = (𝑥+5) 2 −8 3. 𝑓 𝑥 = (𝑥+1) 2 +2

Some Problems are Matching. For each quadratic , match the correct minimum value. 1. 𝑓 𝑥 = (𝑥−3) 2 −5 a. -2 b. -5 c. -8 d. 2 e. 8 f. 5 2. 𝑓 𝑥 = (𝑥+5) 2 −8 3. 𝑓 𝑥 = (𝑥+1) 2 +2

Some Problems are Matching. For each quadratic , match the correct minimum value. 1. 𝑓 𝑥 = (𝑥−3) 2 −5 a. -2 b. -5 c. -8 d. 2 e. 8 f. 5 2. 𝑓 𝑥 = (𝑥+5) 2 −8 3. 𝑓 𝑥 = (𝑥+1) 2 +2

Which function has the narrowest graph? b. 𝑦= 𝑥 2 +2𝑥−6 c. 𝑦= 1 5 𝑥 2 +2𝑥−6

Which function has the narrowest graph? b. 𝑦= 𝑥 2 +2𝑥−6 c. 𝑦= 1 5 𝑥 2 +2𝑥−6

Which function has the narrowest graph? How do you Know? b. 𝑦= 𝑥 2 +2𝑥−6 c. 𝑦= 1 5 𝑥 2 +2𝑥−6

Which function has the narrowest graph? The larger the coefficient of the 𝑥 2 term the more stretched out the graph is which makes it appear more narrow. a. 𝑦= 3𝑥 2 +2𝑥−6 b. 𝑦= 𝑥 2 +2𝑥−6 c. 𝑦= 1 5 𝑥 2 +2𝑥−6 How do you Know?

𝑥 2 −5𝑥+6=0 Solve the following quadratic equations by factoring. 𝑥 2 +2𝑥−8=0 𝑥 2 +9𝑥+20=0

𝑥 2 −5𝑥+6=0 Solve the following quadratic equations by factoring. 𝑥−3 𝑥−2 =0 𝑥 2 +2𝑥−8=0 𝑥 2 +9𝑥+20=0

𝑥 2 −5𝑥+6=0 Solve the following quadratic equations by factoring. 𝑥=3 𝑥=2 𝑥−3 𝑥−2 =0 𝑥 2 +2𝑥−8=0 𝑥 2 +9𝑥+20=0

𝑥 2 −5𝑥+6=0 Solve the following quadratic equations by factoring. 𝑥=3 𝑥=2 𝑥−3 𝑥−2 =0 𝑥 2 +2𝑥−8=0 𝑥+4 𝑥−2 =0 𝑥 2 +9𝑥+20=0

𝑥 2 −5𝑥+6=0 Solve the following quadratic equations by factoring. 𝑥=3 𝑥=2 𝑥−3 𝑥−2 =0 𝑥 2 +2𝑥−8=0 𝑥=−4 𝑥=2 𝑥+4 𝑥−2 =0 𝑥 2 +9𝑥+20=0

𝑥 2 −5𝑥+6=0 Solve the following quadratic equations by factoring. 𝑥=3 𝑥=2 𝑥−3 𝑥−2 =0 𝑥 2 +2𝑥−8=0 𝑥=−4 𝑥=2 𝑥+4 𝑥−2 =0 𝑥 2 +9𝑥+20=0 𝑥+4 𝑥+5 =0

𝑥 2 −5𝑥+6=0 Solve the following quadratic equations by factoring. 𝑥=3 𝑥=2 𝑥−3 𝑥−2 =0 𝑥 2 +2𝑥−8=0 𝑥=−4 𝑥=2 𝑥+4 𝑥−2 =0 𝑥 2 +9𝑥+20=0 𝑥+4 𝑥+5 =0 𝑥=−4 𝑥=−5

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥= −(−5)± 5 2 −4∙1∙6 2∙1 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥= 5± 25−24 2 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥= 5± 25−24 2 𝑥= 5± 1 2 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥= 5± 1 2 𝑥= 5±1 2 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥= 5+1 2 𝑥= 5−1 2 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 𝑥= 4 2 =2 𝑥= 6 2 =3 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 {3 , 2} 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 {3 , 2} 𝑥= −2± 2 2 −4∙1∙−8 2∙1 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 {3 , 2} 𝑥= −2± 4+32 2 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 {3 , 2} 𝑥= −2± 36 2 𝑥 2 +2𝑥−8=0

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 {3 , 2} 𝑥= −2±6 2 𝑥 2 +2𝑥−8=0 𝑥= −2+6 2 𝑥= −2−6 2

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 {3 , 2} 𝑥= −2±6 2 𝑥 2 +2𝑥−8=0 𝑥= 4 2 =2 𝑥= −8 2 =−4

Solve the following quadratic equations by using the quadratic formula. 𝑥 2 −5𝑥+6=0 {3 , 2} 𝑥 2 +2𝑥−8=0 {-4 , 2}

Can you put a function into vertex form? 𝑓 𝑥 = 𝑥 2 +12𝑥−8

Can you put a function into vertex form? 𝑓 𝑥 = 𝑥 2 +12𝑥−8 Take half of middle term and square it. 12 2 = (6) 2 =36 Add it, then subtract it 𝑥 2 +12𝑥+36 −36−8 Now factor and simplify 𝑓 𝑥 =(𝑥+6) 2 −44

Can you put a function into vertex form? Now you try . . . 𝑓 𝑥 = 𝑥 2 +8𝑥+5

Can you put a function into vertex form? Now you try . . . 𝑓 𝑥 = 𝑥 2 +8𝑥+5 (𝑥 2 +8𝑥+16)−16+5 𝑓 𝑥 =(𝑥+4) 2 −11

Write the equation for this graph.

Write the equation for this graph. Identify the x intercepts if possible.

Write the equation for this graph. Identify the x intercepts if possible x = 1 and x = 3

Write the equation for this graph. Identify the x intercepts x = 1 and x = 3 Use the intercepts to write factors

Write the equation for this graph. Identify the x intercepts if possible x = 1 and x = 3 Use the intercepts to write factors. (x – 1)(x – 3)

Write the equation for this graph. Identify the x intercepts if possible x = 1 and x = 3 Use the intercepts to write factors. (x – 1)(x – 3) Multiply out to get standard form.

Write the equation for this graph. Identify the x intercepts if possible x = 1 and x = 3 Use the intercepts to write factors. (x – 1)(x – 3) Multiply out to get standard form. 𝑥 2 −4𝑥+3=0 Don’t forget, there is a stretch of 2.

Write the equation for this graph. Identify the x intercepts if possible x = 1 and x = 3 Use the intercepts to write factors. (x – 1)(x – 3) Multiply out to get standard form. 𝑥 2 −4𝑥+3=0 Don’t forget, there is a stretch of 2. 2(𝑥 2 −4𝑥+3)= Distribute the 2 . . .

2 𝑥 2 −8𝑥+6=0 Write the equation for this graph. Identify the x intercepts if possible x = 1 and x = 3 Use the intercepts to write factors. (x – 1)(x – 3) Multiply out to get standard form. 𝑥 2 −4𝑥+3=0 Don’t forget, there is a stretch of 2. 2(𝑥 2 −4𝑥+3)= 2 𝑥 2 −8𝑥+6=0

Write the equation for this graph. You can also use the vertex and stretch to write the equation. Do you remember vertex form?

Write the equation for this graph. You can also use the vertex and stretch to write the equation.

Write the equation for this graph. You can also use the vertex and stretch to write the equation. The vertex is (2 , -2)

Write the equation for this graph. You can also use the vertex and stretch to write the equation. The vertex is (2 , -2) The stretch is 2 (over 1 up 2)

Write the equation for this graph. You can also use the vertex and stretch to write the equation. The vertex is (2 , -2) The stretch is 2 (over 1 up 2) 𝑓 𝑥 = 2(𝑥−2) 2 −2

𝑓 𝑥 =2 𝑥 2 −8𝑥+6 Write the equation for this graph. You can also use the vertex and stretch to write the equation. The vertex is (2 , -2) The stretch is 2 (over 1 up 2) 𝑓 𝑥 = 2(𝑥−2) 2 −2 Now multiply into Standard Form 𝑓 𝑥 =2 𝑥 2 −4𝑥+4 −2 𝑓 𝑥 =2 𝑥 2 −8𝑥+8−2 𝑓 𝑥 =2 𝑥 2 −8𝑥+6

Find the x and y intercepts

Find the x and y intercepts Y=6 or (0 , 6) is the y intercept

Find the x and y intercepts y=6 or (0 , 6) is the y intercept x=1 and x=3 or (1 , 0) and (3 , 0) are the x-intercepts.

Is this a maximum or a minimum?

Is this a maximum or a minimum? This has a minimum.

Is this a maximum or a minimum? This has a minimum. It opens up.

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function.

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function. Find first difference,

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function. Find first difference,

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function. Find first difference. How about second difference?

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function. Find first difference. How about second difference?

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function. Find first difference. How about second difference? Use recursive pattern to write function.

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function. Use recursive pattern to write function. 𝑓 1 =3 𝑓 𝑛 =𝑓 𝑛−1 +(2𝑥−1)

Write the recursive equation for this function. x y 1 3 2 6 11 4 18 5 27 38 Write the recursive equation for this function. Use recursive pattern to write function. 𝑓 1 =3 𝑓 𝑛 =𝑓 𝑛−1 +(2𝑥−1)

Find the function for this piece-wise graph.

Find the function for this piece-wise graph. You need slope and y-intercept for both pieces.

Find the function for this piece-wise graph. You need slope and y-intercept for both pieces. Let’s find slopes first.

Find the function for this piece-wise graph. You need slope and y-intercept for both pieces. One piece has a slope of − 4 2 =−2 The other piece has a slope of 1 4

Find the function for this piece-wise graph. You need slope and y-intercept for both pieces. One piece has a slope of − 4 2 =−2 The other piece has a slope of 1 4

Find the function for this piece-wise graph. You need slope and y-intercept for both pieces. One piece has a slope of − 4 2 =−2 The other piece has a slope of 1 4 How about y intercepts?

One piece has a slope of − 4 2 =−2 The other piece has a slope of 1 4 Find the function for this piece-wise graph. You need slope and y-intercept for both pieces. One piece has a slope of − 4 2 =−2 The other piece has a slope of 1 4 How about y intercepts? One piece has y-intercept of 1

One piece has a slope of − 4 2 =−2 The other piece has a slope of 1 4 Find the function for this piece-wise graph. You need slope and y-intercept for both pieces. One piece has a slope of − 4 2 =−2 The other piece has a slope of 1 4 How about y intercepts? One piece has y-intercept of 1 Using the slope, the other piece has a y-intercept of -2.

Find the function for this piece-wise graph. One piece has a slope of − 4 2 =−2 Its y-intercept is 1 The other piece has a slope of 1 4 Its y-intercept is -2 Now use the information to write the equations.

{ −2𝑥+1 𝑓 𝑥 = 1 4 𝑥−2 Find the function for this piece-wise graph. One piece has a slope of − 4 2 =−2 Its y-intercept is 1 The other piece has a slope of 1 4 Its y-intercept is -2 Now use the information to write the equations. { −2𝑥+1 𝑓 𝑥 = 1 4 𝑥−2

{ −2𝑥+1 𝑓 𝑥 = 1 4 𝑥−2 Find the function for this piece-wise graph. One piece has a slope of − 4 2 =−2 Its y-intercept is 1 The other piece has a slope of 1 4 Its y-intercept is -2 Now use the information to write the equations. { −2𝑥+1 We still need to state the domain for each piece. 𝑓 𝑥 = 1 4 𝑥−2

Find the function for this piece-wise graph. One piece has a slope of − 4 2 =−2 Its y-intercept is 1 The other piece has a slope of 1 4 Its y-intercept is -2 Now use the information to write the equations. { −2𝑥+1: 0≤𝑥≤3 𝑓 𝑥 = 1 4 𝑥−2: 4≤𝑥≤8

{ Find the function for this piece-wise graph. −2𝑥+1: 0≤𝑥≤3 𝑓 𝑥 = −2𝑥+1: 0≤𝑥≤3 𝑓 𝑥 = 1 4 𝑥−2: 4≤𝑥≤8

Absolute value Functions are similar to Quadratic Functions when you graph them.

You can Identify the vertex from the graph.. Absolute value Functions are similar to Quadratic Functions when you graph them. You can Identify the vertex from the graph..

You can Identify the vertex from the graph. Absolute value Functions are similar to Quadratic Functions when you graph them. You can Identify the vertex from the graph. You can Identify the vertex from the equation.

You can Identify the vertex from the graph. Absolute value Functions are similar to Quadratic Functions when you graph them. You can Identify the vertex from the graph. You can Identify the vertex from the equation.

You can Identify the vertex from the graph. Absolute value Functions are similar to Quadratic Functions when you graph them. You can Identify the vertex from the graph. You can Identify the vertex from the equation. We took the outside number just like a quadratic! We took the opposite just like a quadratic!

The Shape is always a V from the vertex. Absolute value Functions are similar to Quadratic Functions when you graph them. You can Identify the vertex from the graph. You can Identify the vertex from the equation. The Shape is always a V from the vertex. We took the outside number just like a quadratic! We took the opposite just like a quadratic!

The Shape is always a V from the vertex. Absolute value Functions are similar to Quadratic Functions when you graph them. You can Identify the vertex from the graph. You can Identify the vertex from the equation. The Shape is always a V from the vertex. The slope is the coefficient of x. We took the outside number just like a quadratic! We took the opposite just like a quadratic!

You Try matching a few … a. 𝑓 𝑥 = 𝑥−5 −4 b. 𝑓 𝑥 = 𝑥+5 +4 𝑑. 𝑓 𝑥 = 𝑥−5 +4

You Try matching a few … a. 𝑓 𝑥 = 𝑥−5 −4 b. 𝑓 𝑥 = 𝑥+5 +4 𝑑. 𝑓 𝑥 = 𝑥−5 +4 Identify the vertex.

a. 𝑓 𝑥 = 𝑥−5 −4 b. 𝑓 𝑥 = 𝑥+5 +4 c. 𝑓 𝑥 = 𝑥+5 −4 𝑑. 𝑓 𝑥 = 𝑥−5 +4 Identify the vertex. (-5 , -4)

a. 𝑓 𝑥 = 𝑥−5 −4 b. 𝑓 𝑥 = 𝑥+5 +4 c. 𝑓 𝑥 = 𝑥+5 −4 Identify the vertex. (-5 , -4) Now write the equation. 𝑑. 𝑓 𝑥 = 𝑥−5 +4

You Try: match the graph with the correct function. b. 𝑓 𝑥 = 𝑥+5 +4 c. 𝑓 𝑥 = 𝑥+5 −4 𝑑. 𝑓 𝑥 = 𝑥−5 +4

a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 Match the graph with the correct function. a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3

a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 Match the graph with the correct function. a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 The Vertex is . . .

a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 Match the graph with the correct function. a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 The Vertex is . . . (1 , 3)

a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 Match the graph with the correct function. a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 The Vertex is . . . (1 , 3) The equation is . . .

a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 Match the graph with the correct function. a. 𝑓 𝑥 = 𝑥−1 −3 b. 𝑓 𝑥 = 𝑥+1 −3 c. 𝑓 𝑥 = 𝑥+1 +3 d. 𝑓 𝑥 = 𝑥−1 +3 The Vertex is . . . (1 , 3) The equation is . . .

Write the explicit and recursive equation for the pattern above.

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 What is the pattern?

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 Add 5, then add 7, then add 9 . . .

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 Add 5, then add 7, then add 9 . . . Finish the table . . .

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 Add 5, then add 7, then add 9 . . . Finish the table . . .

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Add 5, then add 7, then add 9 . . . Finish the table . . .

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Lets get the recursive equation.

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65

Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Second difference is 2 so we can write the recursive now.

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+?) Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Second difference is 2 so we can write the recursive now. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+?)

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+?) Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Second difference is 2 so we can write the recursive now. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+?)

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Second difference is 2 so we can write the recursive now. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1)

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Let’s start with a table. Write the explicit and recursive equation for the pattern above. Let’s start with a table. t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Second difference is 2 so we can write the recursive now. Now let’s work on explicit. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1)

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit.

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit.

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit.

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit.

𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit.

𝑦=𝑥 𝑥+2 +2 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 𝑦=𝑥 𝑥+2 +2

𝑦=𝑥 𝑥+2 +2 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 𝑦=𝑥 𝑥+2 +2 Convert to standard form . . .

𝑦=𝑥 𝑥+2 +2 𝑦= 𝑥 2 +2𝑥+2 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) t 1 2 3 4 5 6 7 d 10 17 26 37 50 65 Write the explicit and recursive equation for the pattern above. 𝑓 1 =5;𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1) Now let’s work on explicit. 𝑦=𝑥 𝑥+2 +2 Convert to standard form . . . 𝑦= 𝑥 2 +2𝑥+2

Write the explicit and recursive equation for the pattern above. 1 2 3 4 5 6 7 d 10 17 26 37 50 65 𝑦= 𝑥 2 +2𝑥+2 𝑓 1 =5; 𝑓 𝑥 =𝑓 𝑥−1 +(2𝑥+1)